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8th Conference
GEOMETRY
AND
TOPOLOGY
OF
MANIFOLDS
Lie Algebroids, Dynamical Systems and ApplicationsLuxembourg � Polish � Ukrainian Conference
Under the auspices of Prof. Jan KrysinskiRector of the Technical University of Lodz
Przemysl (Poland)� L�viv (Ukraine), 30.04.2007� 6.05.2007
bmw
Contents
Forewordp. 10
Organizers and Scienti�c Committeepp. 11 - 12
Sponsorsp. 13
List of participantspp. 14 - 23
Titles of lecturespp. 24 - 26
A b s t r a c t s
On equivariant Lipschitz homeomorphisms of G-manifolds withcodimension one orbitp. 28Authors: Kojun Abe; Kazuhiko Fukui
Electron motion and ruled surfacespp. 29 - 30Author: Yuriy Aminov
Poisson cohomology in arbitrary dimensionp. 31Authors: Mourad Ammar; Norbert Poncin
Homeomorphism groups of non-compact manifoldsp. 31Authors: T. Banakh; K. Sakai; K. Mine
Topological classi�cation of the hyperspaces of closed convex subsets ofa Banach spacep. 32 - 33Authors: T. Banakh; K. Sakai; M. Yaguchi; I. Zarichnyi
Hyperspaces of max-plus convex sets which are in�nite-dimensional manifoldsp. 34Author: Lidia Bazylevych
Vertex Operator construction of coupled soliton hierarchiesp. 35 - 36Author: Paolo Casati
Hilbert di¤eomorphism groups and their geometry for open manifoldsp. 37Author: Jürgen Eichhorn
The Poincaré conjecture and a geometric model of "black hole"p. 38Author: Alexander A. Ermolitski
S-cohomologie vs Diagram cohomologyp. 38Author: Yael Fregier
Higher vector bundles and multi-graded manifolds(joint work with Miko÷aj Rotkiewicz)p. 39Author: Janusz Grabowski
Singular symplectic forms and round functionsp. 40Author: Bogus÷aw Hajduk
The Equivalence Principle and Projective Structure in 4-dimensionalLorentz manifoldspp. 41 - 42Author: Graham Hall
The topology and geometry of polygon spacesp. 43Author: Jean-Claude Hausmann
The degree of maps of manifolds with actions of compact Lie groupsp. 43Author: Jan Jaworowski
Examples of multiple solutions for the Yamabe problem on scalar curvaturep. 44Author: Guy Kass
On some generalizations of the Poisson sigma model(joint with T. Strobl and, partially, with P. Schaller)p. 44Author: Alexei Kotov
4
On the reconstruction problem for some nontransitive homeomorphism groupsp. 45Authors: Agnieszka Kowalik; Ilona Michalik; Tomasz Rybicki
Remarks on Extended Operator Calculuspp. 46 - 47Author: Ewa Krot-Sieniawska
Holonomy representations admitting two pairs of supplementary invariantsubspacesp. 48Author: Thomas Krantz
To Invariant Normales of Vector Field in Four-Dimmentional A¢ ne Spacsubspacesp. 48Author: Olga Kravchuk
Flat connections with singularities in some Lie algebroidsp. 48Author: Jan Kubarski
Ivan Bernoulli Series Universalissimap. 49Author: A. Krzysztof Kwasniewski
Deformation quantization in in�nite dimensional analysisp. 49Author: Rémi Léandre
Fragmentations of the second kind in some di¤eomorphism groupsp. 50Authors: Jacek Lech; Tomasz Rybicki
Jetbundles on �agvarietiesp. 50Author: Helge Maakestad
Presentation for the fundamental groups of orbits of Morse functinoson surfacesp. 51Author: Sergiy Maksymenko
A note on the Lagrange di¤erential in bigraded module of vertical tangentbundle valued formsp. 52Author: Roman Matsyuk
Hyperspaces of Riemannian manifolds related to the Hausdor¤ dimensionp. 53Author: Nataliia Mazurenko
5
Underdetermined systems of ODEs �the geometric approach of E.Cartanp. 54Author: Piotr Mormul
Surgery on strati�ed manifoldsp. 55Author: Yuri V. Muranov
Orbits structure of co-adjoint action and bystages hypothesispp. 56Author: Ihor V. Mykytyuk
Di¤erential of superorder following Koszul and KV cohomologyfollowing Nijenhuisp. 57Author: Michel Ngui¤o Boyom
A¢ ne special Lagrangian submanifolds of Cnp. 57Author: Barbara Opozda
Projective structures and invariant di¤erential operatorsp. 58Author: Valentin Ovsienko
An analogue of the Kazhdan�s property (T) for operator algebras(joint with Evgenij Troitsky)p. 58Author: Alexander Pavlov
Node points of Sen-Witten equations and positive energy theoremp. 59Author: Wolodymyr Pelykh
Around Birkho¤Theoremp. 60Authors: O. Petrenko; I. V. Protasov
Totally singular Lagrangians and a¢ ne hamiltonianspp. 61 - 62Authors: Paul Popescu; Marcela Popescu
Topological groups through the looking-glassp. 63Author: I. V. Protasov
Generalized retracts concerned to free topoplogical groupsp. 64Author: Nazar M. Pyrch
6
Natural and projectively equivariant quantizationp. 65Author: Fabian Radoux
On in�nite Lie pseudo-groups and �ltered Lie algebraspp. 66 - 67Author: Alexandre A. Martins Rodrigues
A note on the reconstruction problem for factorizable homeomorphismgroups and foliated manifoldsp. 68Author: Matatyahu Rubin
Geometric calculus associated to second-order dynamics and applicationsp. 69Author: Willy Sarlet
Manifolds modeled on the direct limits of Tychonov cubesp. 70Author: Oryslava Shabat
Vector �elds on 4-manifoldsp. 70Author: Vladimir Sharko
The order of the di¤erentablity of horizonsp. 71Author: David Szeghy
On stable orbit types of isometric actions on Lorentz manifoldsp. 71Author: János Szenthe
The Bases of Di¤erential Geometry of Vector Field in n-dimensional spaceof A¢ ne Connectionp. 72Author: Petro Tadeev
Modi�ed Hochschild and Periodic Cyclic Homologyp. 72Author: Nicolae Teleman
On the Sectional Curvatures of the Time-like Generalized Ruled Surfacein IRn1pp. 73 - 75Authors: Murat Tosun; Soley Ersoy
ANR-property of hyperspaces with the Attouch-Wets topologypp. 76 - 77Author: Rostislav Voytsitskyy
7
About topological equivalence of some functionsp. 78 - 79Author: Irina Yurchuk
Universal maps of in�nite-dimensional manifoldsp. 80Author: Mykhailo Zarichnyi
b m w
8
FOREWORD
This is the conference of the cycle initiated in 1998 with a meeting in Konopnica(http://im0.p.lodz.pl/konferencje/) and is organized in two towns, the �rst part is orga-nized in Przemysl, Poland, at the State High School of East Europe, the second part isorganized in L�viv, Ukraine, at the Ivan Franko National University of L�viv.The main aim of the conference series is to present and discuss new results on geometry
and topology of manifolds with the particular attention being paid to applications of algebraicmethods. The topics usually discussed include:
y dynamical systems on manifolds and applications,
y Lie groups (including in�nite dimensional), Lie algebroids and their generalizations,Lie groupoids,
y Characteristic classes, index theory, K-theory, Fredholm operators,
y Singular foliations, cohomology theories for foliated manifolds and their quotients,
y Symplectic, Poisson, Jacobi and special Riemannian manifolds,
y topology of in�nite-dimensional manifolds,
y applications to mathematical physics.
ORGANIZERS AND SCIENTIFIC COMMITTEE
8th Conference on
GEOMETRY AND TOPOLOGY OF MANIFOLDSis organized by
z Jan Kubarski, chairman, ×ódz, Poland, email: [email protected]� Institute of Mathematics of the Technical University of Lodz, ×ódz.
z Robert Wolak, Kraków, Poland, email: [email protected]�Institute of Mathematics of the Jagiellonian University, Kraków
z Tomasz Rybicki, Kraków, Poland, email: [email protected]�Faculty of Applied Mathematics at the AGHUniversity of Science and Technology, Kraków
z Michael Zarichny, email: [email protected]�Institute of Mathematics, Rzeszow University, Rzeszów
�Faculty of Mechanics and Mathematics of L�viv Ivan Franko National University, L�viv
z Norbert Poncin, email: [email protected]�Institute of Mathematics, University of Luxembourg, Grand�Duchy of Luxembourg
z Andriy Panasyuk, email: [email protected]�Institute for Applied Problems of Mechanics and Mathematics of National Academy of
Sciences of Ukraine, L�viv
�Department of Mathematical Methods in Physics, University of Warsaw, Warsaw
z Vladimir Sharko, email: [email protected]�Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev
11
Scienti�c CommitteeYu. Aminow (Ukraine)B. Bojarski (Poland)A. Borisenko (Ukraine)R. Brown (UK)S. Brzychczy (Poland)J. Grabowski (Poland)J. Kubarski (Poland)P. Lecomte (Belgium)A. S. Mishchenko (Russia)V. Yu. Ovsienko (France)J. Pradines (France)A. O. Prishlyak (Ukraine)A. M. Samoilenko (Ukraine)V. Sharko (Ukraine)N. Teleman (Italy)M. Zarichny (Poland, Ukraine)N. T. Zhung (France)
Przemysl Local Organizing Committee:R. Wolak, the chairman of the Przemysl part, Jagiellonian University, CracowR. Choma, the Mayor of the PrzemyslJ. Draus, the Rector of the State High School of East Europe, PrzemyslJ. Musia÷, the Chancellor of the State High School of East Europe, Przemysl
L�viv Local Organizing Committee:A. Panasyuk, Warsaw�L�viv the chairmenT. Banakh, L�viv of the L�viv partR.M.Kushnir, The Director of the Pidstryhach Institute for the Applied Problems
of Mechanics and Mathematics, L�vivV.M.Kyrylych, The Vice�Rector of L�viv Ivan Franko National University, L�vivR. Matsyuk, Institute for Applied Problems of Mechanics and Mathematics of
National Academy of Sciences of Ukraine, L�vivI. Mykytyuk, Rzeszow University, Rzeszów; Pidstryhach Institute for the Applied
Problems of Mechanics and Mathematics, L�vivV.O.Pelykh, The Vice�Director of the Pidstryhach Institute for the Applied
Problems of Mechanics and Mathematics, L�vivM. Zarichny, Institute of Mathematics, Rzeszow University, Rzeszów; The Dean of
the Faculty of Mathematics and Mechanics of L�viv Ivan FrankoNational University, L�viv
12
SPONSORS
The organizers of the conference are grateful to the following sponsors
� The Committee on Mathematics of the Polish Academy of Sciences
� Rector of the Technical University of Lodz
� Rector of the Jagiellonian University
� Rector of the AGH University of Science and Technology
� Rector of the Rzeszów University
� Vice-Rector of Lviv Ivan Franko National University, L�viv, Ukraine
� Dean of the Faculty of Technical Physics, Computer Science andApplied Mathematics OF the Technical University of Lodz
� Dean of the Faculty of Mathematics and Natural Sciences of the RzeszówUniversity
� Head of the Institute of Mathematics, University of Luxembourg, Lux-embourg, Grand-Duchy of Luxembourg
� The Director of the Pidstryhach Institute for the Applied Problemsof Mechanics and Mathematics of the National Academy of Sciencesof Ukraine, L�viv, Ukraine
� The Director of the Institute of Mathematics of the National Acad-emy of Sciences of Ukraine, Kyiv, Ukraine
� Head of the Department of Mathematical Methods in Physics, Facultyof Physics, University of Warsaw, Warsaw
� Bank "Dnister", L�viv
� "Kredo Bank", L�viv
13
LIST OF PARTICIPANTS
1. Abe, Kojun, Matsumoto, Nagano Prefecture, JapanShinshu University, Matsumoto, Nagano Prefecture, Japane-mail: [email protected]
2. Abib, Renée, Rouen, FranceUMR 6085 CNRS-Université de RouenAvenue de l�Université, BP.1276801 Saint Etienne de Rouvray, Francee-mail: [email protected]
3. Aminov, Yuriy, Kharkiv, UkraineB.I.Verkin Istitute for Low Temperature Ph. and Eng. of NAS of UkraineKharkiv, Lenin av. 47, 61103, Ukrainee-mail: [email protected]
4. Ammar, Mourad, Luxembourg, Grand-Duchy of LuxembourgInstitute of Mathematics, University of Luxembourg,Campus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
5. Balcerzak, Bogdan, ×ódz, PolandInstitute of Mathematics, Technical University of Lodzul. Wólczanska 215, 90�924 ×ódz, Polande-mail: [email protected]
6. Banakh, Taras, L�viv, Ukraine;L�viv National University, Universytetska 1, L�viv 79000, Ukraine;Akademia Swi ¾etokrzyska, Kielce, Polande-mail: [email protected]
7. Bazylevych, Lidia, L�viv, UkraineNational University ("L�viv Polytechnica")Vyhovskoho 89/104, 79021 L�viv, Ukrainee-mail: [email protected]
8. Bojarski, Bogdan, Warszawa, PolandMathematical Institute, Polish Academy of Sciences, Warszawa, Polande-mail: [email protected]
14
9. Bokalo, Bogdan, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
10. Casati, Paolo, Milano, Italy
Dipartimento di Matematica e applicazioni
II Università di Milano Bicocca�Ed. U5
Via R. Cozzi 53, I-20126 Milano, Italy
e-mail: [email protected]
11. Domitrz, Wojciech, Warszawa, Poland
Warsaw Univesity of Technology, Faculty of Mathematics and Information Science
plac Politechniki 1, 00-661 Warszawa, Poland
e-mail: [email protected]
12. Eichhorn, Jüergen, Greifswald, GermanyInstitut für Mathematik, Greifswald UniversityGreifswald, D�17489, Germanye-mail: [email protected]
13. Ermolitski, Alexander, Minsk, BelarusMinsk 220004, Romanovskaya Str. 24-19, Belaruse-mail: [email protected]
14. Frégier, Yaël,Luxembourg, Grand-Duchy of LuxembourgThe Institute of Mathematics , University of LuxembourgCampus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
15. Gancarzewicz, Jacek, Cracow, PolandInstitute of Mathematics, Jagiellonian University30-059 Kraków, ul. Reymonta 4, Polande-mail: [email protected]
16. Grabowska, Katarzyna, Warsaw, PolandDepartment of Physics, University of WarsawHo·za 69, 00-680 Warsaw, Polande-mail: [email protected]
17. Grabowski, Janusz, Warsaw, PolandInstitute of Mathematics of the Polish Academy of SciencesWarsaw, Polande-mail: [email protected]
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18. Gutik, Oleg, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
19. Hajduk, Bogus÷aw, Wroc÷aw, PolandMathematical Institute, University of Wroc÷awpl. Grunwaldzki 2/4, 50�384 Wroc÷aw, Polande-mail: [email protected]
20. Hall, Graham, Aberdeen, UKDepartment of Mathematical Sciences, University of AberdeenAberdeen, AB24 3UE, Scotland, UKe-mail: [email protected]
21. Hausmann, Jean-Claude, Geneve, SwitzerlandSection de mathematiques, Universite de GeneveC.P. 64, CH-1211 Geneve 4, Switzerlande-mail: [email protected]
22. Hryniv, Olena, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
23. Huebschmann, Johannes, L�viv, UkraineUniversité des Sciences et Technologies de Lille, Francee-mail: [email protected]
24. Iwase, Norio, Kyushu, JapanFaculty of Mathematics, Kyushu Universitye-mail: [email protected]
25. Jaworowski, Jan, Bloomingon, Indiana, USADepartment of Mathematics, Bloomingon, IndianaIN 47405-5701, USAe-mail: [email protected]
26. Józefowicz, Ma÷gorzata, Cracow, PolandInstitute of Mathematics, Jagiellonian University30-059 Kraków, ul. Reymonta 4, Polande-mail: [email protected]
27. Kass, Guy, Luxembourg, Grand-Duchy of LuxembourgThe Institute of Mathematics , University of LuxembourgCampus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
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28. Kotov, Alexei, Luxembourg, Grand-Duchy of LuxembourgThe Institute of Mathematics , University of LuxembourgCampus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
29. Kowalik, Agnieszka, Cracow, PolandAGH University of Science and Technologyal. Mickiewicza 30, 30�059 Cracow, Polande-mail: [email protected]
30. Krantz, Thomas, Luxembourg, Grand-Duchy of LuxembourgThe Institute of Mathematics , University of LuxembourgCampus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
31. Kravchuk, Olga, Khmelnytskyi, UkraineThe Khmelnytskyi National UniversityZarichanska St., 36/1, apt. 63, Khmelnytskyi, Ukraine, 29000e-mail: [email protected]
32. Krot-Sieniawska, Ewa, Bia÷ystok, PolandInstitut of Computer Science, University in Bia÷ystokul. Sosnowa 64, PL�15�887 Bia÷ystok, Polande-mail: [email protected]
33. Kubarski, Jan, ×ódz, PolandInstitute of Mathematics, Technical University of Lodzul. Wólczanska 215, 90�924 ×ódz, Polande-mail: [email protected]
34. Kwasniewski, Andrzej Krzysztof, Bia÷ystok, PolandInstitute of Theoretical Physics, University of Bialystokul. Lipowa 41, 15�424 Bia÷ystok, Polande-mail: [email protected]
35. Langerock, Bavo, Gent, BelgiumDepartment of Architecture, Sint-Lucas, Institute for Higher Education in the Sciences &the Arts (W&K)Hoogstraat 51, B-9000 Gent, Belgiume-mail: [email protected]
36. Léandre, Rémi, Dijon, FranceInstitut de Mathématiques, Université de BourgogneBd Alain Savary, Dijon 21078, Francee-mail: [email protected]
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37. Lech, Jacek, Cracow, PolandAGH University of Science and Technologyal. Mickiewicza 30, 30�059 Cracow, Polande-mail: [email protected]
38. Lyaskovska Nadya, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, 1 Ukrainee-mail: [email protected]
39. Maakestad, Helge, Trondheim, NorwayDepartment of Mathematics, NTNUTrondheim 7031, Neufeldtsgt 1, Norwaye-mail: [email protected]
40. Maksymenko, Sergiy, Kyiv, UkraineTopology Department, Institute of Mathematics NAS of UkraineTereshchenkivska St., 3, Kyiv 01601, Ukrainee-mail: [email protected]
41. Matsyuk, Roman, L�viv, UkraineInstitute for Applied Problems of Mechanics and Mathematics ofNational Academy of Sciences of UkraineDudajewa 15, 79005 L�viv, Ukrainee-mail: [email protected]
42. Mazurenko, Nataliia, Ivano-Frankivsk, UkrainePre-Carpathian National University UkraineShevchenka Str., 67, Ivano-Frankivsk 76005, Ukrainee-mail: [email protected]
43. Michalik, Ilona, Cracow, PolandAGH University of Science and Technologyal. Mickiewicza 30, 30�059 Cracow, Polande-mail: [email protected]
44. Mishchenko, Alexandr, Moscow, RussiaDepartment of Mechanics and Mathematics, Moscow State UniversityLeninskije Gory 119992, Moscow, Russiae-mail: [email protected]
45. Mishchenko, Tatiana, Moscow, RussiaRussian Academy of Education, Moscow, Russia
46. Mormul, Piotr, Warsaw, PolandInstitute of Mathematics, Warsaw UniversityBanach St. 2, 02�097 Warsaw, Polande-mail: [email protected]
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47. Muranov, Yury Vladimirovich, Vitebsk, BelarusVitebsk State University210026 Vitebsk, Moskovskii pr. 33, Belaruse-mail: [email protected]
48. Mykytyuk, Ihor, Rzeszów, PolandInstitute of Mathematics, Rzeszow State UniversityRejtana St., 16A, 35�310, Rzeszów, Polande-mail: [email protected]
49. Ngui¤o Boyom, Michel, Montpellier, FranceUniversity Montpellier 23 rue du Dahomey, 34090 Montpellier, Francee-mail: [email protected]
50. Olszak, Karina, Wroc÷aw, PolandInstitute of Mathematics, Wroclaw University of TechnologyWybrze·ze Wyspianskiego 27, 50�370 Wroc÷awe-mail: [email protected]
51. Olszak, Zbigniew, Wroc÷aw, PolandInstitute of Mathematics, Wroclaw University of TechnologyWybrze·ze Wyspianskiego 27, 50�370 Wroc÷awe-mail: [email protected]
52. Opozda, Barbara, Cracow, PolandInstitute of Mathematics, Jagiellonian University30-059 Kraków, ul. Reymonta 4, Polande-mail: [email protected]
53. Ovsienko, Valentin, Lyon, FranceCNRS, Institut Camille Jordan, University Claude Bernard Lyon 143 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, Francee-mail: [email protected]
54. Oziewicz, Zbigniew, Mexico, MexicoUniversidad Nacional Autonoma de Mexico, Facultad de Estudios Superiores Cuautitlan,Apartado Postal # 25, C.P. Cuautitlan Izcalli, Estado de Mexicoe-mail: [email protected]
55. Panasyuk, Andriy, Warsaw, Poland; L�viv, UkraineKMMF, Department of Physics, University of WarsawHo·za St. 74, 00�682 Warszawa, Poland;Institute for Applied Problems of Mechanics and Mathematics of National Academy ofSciences of Ukraine, L�viv, Ukrainee-mail: [email protected]
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56. Pavlov, Alexander, Moscow, RussiaDepartment of Mechanics and Mathematics, Moscow State UniversityLeninskije Gory 119992 Moscow, Russiae-mail: [email protected]
57. Pelykh, Wolodymyr, L�viv, UkraineInstitute for Applied Problems of Mechanics and Mathematics of National Academy ofSciences of UkraineNaukova 3B, 79060 L�viv, Ukrainee-mail: [email protected]
58. Petrenko, O., Kyiv, UkraineDepartment of Cybernetics,Kyiv National UniversityVolodymyrska 64, Kyiv, 01033, Ukrainee-mail: [email protected]
59. Poncin, Norbert, Luxembourg, Grand-Duchy of LuxembourgInstitute of Mathematics, University of Luxembourg,Campus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
60. Popescu, Paul, Craiova, RomaniaUniversity of Craiova, Romaniae-mail: [email protected]
61. Pradines, Jean, Toulouse, FranceUniversite Toulouse III26 rue Alexandre Ducos, Toulouse, F31500 Francee-mail: [email protected]
62. Protasov, Igor, Kyiv, UkraineKyiv National UniversityDavydova 7/101, Kyiv, Ukrainee-mail: [email protected]
63. Pyrch, Nazar, L�viv, Ukraine,UUkrainian Academy of PrintingL�viv 79066, Ukraine,e-mail: [email protected]
64. Radoux, Fabian, Luxembourg, Grand-Duchy of LuxembourgInstitute of Mathematics, University of Luxembourg,Campus Limpertsberg, avenue de la Faiencerie, 162AL-1511 Luxembourg City, Grand-Duchy of Luxembourge-mail: [email protected]
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65. Rodrigues, Alexandre, São Paulo, BazilUniversity of São Paulo, Brazile-mail: [email protected]
66. Rotkiewicz, Miko÷aj, Warsaw, PolandWarsaw University, Banach St. 2, 02�097 Warsaw, Polande-mail: [email protected]
67. Rubin, Matatyahu, Beer Sheva, IsraelDeptartment of Mathematics, Ben Gurion UniversityBeer Sheva, Israel 85105e-mail: [email protected]
68. Rybicki, Tomasz, Cracow, PolandAGH University of Science and Technologyal. Mickiewicza 30, 30�059 Cracow, Polande-mail: [email protected]
69. Shabat, Oryslava, L�viv, Ukraine,Ukrainian Academy of PrintingPidholosko St., 19, L�viv 79066, Ukraine,e-mail: [email protected]
70. Samoilenko, Anatoly, K�iev, UkraineInstitite of Mathematics, National Academy of Sciences of Ukraine3, Tereshchenkivska st, 01601 K�iev, Ukrainee-mail: [email protected]
71. Sarlet, Willy, Gent, NederlandsThe Department of Mathematical Physics and Astronomy, Ghent UniversityKrijgslaan 281 �S 9, B-9000 Gent, Nederlandse-mail: [email protected]
72. Sharko, Vladimir, K�iev, UkraineInstitite of Mathematics, National Academy of Sciences of Ukraine3, Tereshchenkivska st, 01601 K�iev, Ukrainee-mail: [email protected]
73. Sharkovsky, Aleksandr Nikolayevich, K�iev, UkraineInstitite of Mathematics, National Academy of Sciences of Ukraine3, Tereshchenkivska st, 01601 K�iev, Ukrainee-mail:[email protected]
74. Shukel, Oksana, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
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75. Szajewska, Marzena, Bia÷ystok, PolandInstitite of Mathematics, University of Bia÷ystokul. Akademicka 2, 15�267 Bia÷ystok, Polande-mail: [email protected]
76. Szeghy, David, Budapest, HungaryDepartment of Geometry, Eötvös UniversityPazmany Peter stny. 1/C, H�1117 Budapest, Hungarye-mail: [email protected]
77. Szenthe, János, Budapest, HungaryDepartment of Geometry, Eötvös UniversityPazmany Peter stny. 1/C, H�1117 Budapest, Hungarye-mail: [email protected]
78. Tadeyev, Petro, Rivne, UkraineThe International University of Economics and Humanitiesafter acad. Stepan Demianchuk, Rivne 33027, Ukrainee-mail: [email protected]
79. Teleman, Nicolae, Ancona, ItalyDipartimento di Scienze Matematiche, Università Politecnica delle Marche60161 �Ancona, Italiae-mail: [email protected]
80. Tosun, Murat, Sakarya, TurkeyDepartment of Mathematics, Sakarya University54187 Sakarya, Turkeye-mail: [email protected]
81. Tulczyjew, W÷odzimierz, Monte Cavallo, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di NapoliValle San Benedetto, 2, 62030 Monte Cavallo, Italye-mail: [email protected]
82. Urbanski, Pawe÷, Warsaw, PolandIKMMF, Faculty of Physics, University of WarsawHo·za 74, 00�682 Warszawa, Polande-mail: [email protected]
83. Voytsitskyy, Rostislav, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National UniversityL�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
84. Wolak, Robert, Cracow, PolandInstitute of Mathematics, Jagiellonian University30-059 Kraków, ul. Reymonta 4, Polande-mail: [email protected]
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85. Yurchuk, Irina, Kyiv, UkraineTaras Shevchenko National University of Kyiv47 Lomonosova St., Kiyv 03022, Ukrainee-mail: [email protected]
86. Yusenko, Kostyantyn, Kiev, UkraineInstitute of Mathematics NAS of Ukraine3, Tereshchenkivska St. Kiev, Ukrainee-mail: [email protected]
87. Zarichny, Michael, Rzeszów, Poland; L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National University,L�viv 79000, Universitetska 1, UkraineInstitute of Mathematics, Rzeszow Universityul. Rejtana 16 A, 35-959 Rzeszów.e-mail: [email protected]
88. Zarichny, Ihor, L�viv, UkraineFaculty of Mechanics and Mathematics of L�viv Ivan Franko National University,L�viv 79000, Universitetska 1, Ukrainee-mail: [email protected]
89. Zung, Nguyen Tien, Toulouse, FranceDépartement de Mathématiques, Université Toulouse 3,118 Route de Narbonne, 31062 Toulouse, Francee-mail: [email protected]
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TITLES OF LECTURES
1. Kojun Abe, On equivariant Lipschitz homeomorphisms of G-manifolds withcodimension one orbit (joint work with Kazuhiko Fukui)
2. Yuriy Aminov, Electronic movements and ruled surfaces
3. Taras Banakh, Homeomorphism groups of non-compact manifolds (joint work withK.Sakai, K.Mine)
4. T. Banakh, K. Sakai, M. Yaguchi, I. Zarichnyi, Topological classi�cation of thehyperspaces of closed convex subsets of a Banach space
5. Lidia Bazylevych, Hyperspaces of max-plus convex sets which are in�nite-dimensionalmanifolds
6. Paolo Casati, Vertex Operator construction of coupled soliton hierarchies
7. Jüergen Eichhorn, Hilbert di¤eomorphism groups and their geometry for openmanifolds
8. Alexander Ermolitski, The Poincaré conjecture and a geometric model of "blackhole"
9. Yaël Frégier, S-cohomologie vs Diagram cohomology
10. Janusz Grabowski, Higher vector bundles and multigraded manifolds (joint workwith M. Rotkiewicz)
11. Bogus÷aw Hajduk, Singular symplectic forms and round functions
12. Graham Hall, The Equivalence Principle and Projective Structure in 4-dimensionalLorentz manifolds
13. Jean-Claude Hausmann, The topology and geometry of polygon spaces
14. Jan Jaworowski, Maps of G-manifolds
15. Guy Kass, Examples of multiple solutions for the Yamabe problem on scalar curvature
16. Alexei, Kotov, On some generalizations of the Poisson sigma model
17. Thomas Krantz, Holonomy representations admitting two pairs of supplementaryinvariant subspaces
24
18. Olga Kravchuk, To Invariant Normales of Vector Field in Four-Dimmentional A¢ neSpace
19. Ewa Krot-Sieniawska, Remarks on Extended Operator Calculus
20. Jan Kubarski, Flat connections with singularities in some Lie algebroids
21. Andrzej Krzysztof Kwasniewski, On Ivan Bernoulli Series Universalissima
22. Rémi Léandre, Deformation quantization in in�nite dimensional analysis
23. Jacek Lech, Fragmentations of the second kind in some di¤eomorphism groups (jointlywith Tomasz Rybicki)
24. Helge Maakestad, Jetbundles on �agvarieties
25. Sergiy Maksymenko, Presentation for the fundamental groups of the orbits of Morsefunctions on surfaces
26. Roman Ya. Matsyuk, A note on the Lagrange di¤erential in bigraded module ofvertical tangent bundle valued forms
27. Nataliia Mazurenko, Hyperspaces of Riemannian manifolds related to the Hausdor¤dimension
28. Ilona Michalik, On the reconstruction problem for some nontransitive homeomor-phism groups (joint work with Agnieszka Kowalik and Tomasz Rybicki)
29. Piotr Mormul, The least Underdetermined systems of ODEs - geometric approachof E.Cartan
30. Yury Vladimirovich Muranov, Surgery on strati�ed manifolds
31. Ihor Mykytyuk, Orbits structure of co-adjoint action and bystages hypothesis
32. Michel Ngui¤o Boyom, Di¤erential of superorder following Koszul and KV coho-mology following Nijenhuis
33. Barbara Opozda, A¢ ne special Lagrangian submanifolds of Cn
34. Valentin Ovsienko, Projective structures on smooth manifolds: homotopy andinvariant di¤erential operators
35. Alexander Pavlov, An analogue of the Kazhdanâ�s property (T) for operator algebras
36. Wolodymyr Pelykh, Node points of Sen-Witten equations and positive energytheorem
37. O. Petrenko, Around Birkho¤ Theorem
38. Norbert Poncin, Poisson cohomology and deformation quantization (joint work withMourad Ammar)
25
39. Paul Popescu, Totally singular Lagrangians and a¢ ne hamiltonians
40. Igor Protasov, Topological groups through the looking glass
41. Nazar M. Pyrch, Generalized retracts concerned to free topoplogical groups
42. Fabian Radoux, The existence and the uniqueness of natural projectively equivariantquantizations by means of the theory of Cartan connections
43. Alexandre Rodrigues, Involutive Lie Pseudogroups and Filtered Lie Algebras
44. Matatyahu Rubin, Recovering a foliated manifold from its homeomorphism group
45. Willy Sarlet, Geometric calculus associated to second-order dynamics andapplications
46. Oryslava Shabat, Manifolds modeled on the direct limits of Tychonov cubes
47. Vladimir Sharko, Flows and L-2 invariant
48. Aleksandr Nikolayevich Sharkovsky, A little from history, a little from topologyand dynamical systems
49. David Szeghy, The order of the di¤erentablity of horizons
50. János Szenthe, On stable orbit types of isometric actions on Lorentz manifolds
51. Petro Tadeev, The Bases of Di¤erential Geometri of Vector Field in n-measure Speaseof A¢ ne Connection
52. Nicolae Teleman, Modi�ed Hochschild and Periodic Cyclic Homology
53. Murat Tosun, On the Sectional Curvatures of Time Like Generalized Ruled Surfacesin IRn1
54. Rostislav Voytsitskyy, ANR-property of hyperspaces with the Attouch-Wetstopology
55. Irina Yurchuk, About topological equivalence of some functions
56. Michael Zarichnyi, Universal maps of in�nite-dimensional manifolds
26
A B S T R A C T S
On equivariant Lipschitz homeomorphisms of G-manifolds withcodimension one orbit
Kojun Abe and Kazuhiko Fukui (Kyoto Sangyou University)
In this talk we shall investigate the group of equivariant Lipschitz homeomorphisms of asmooth G-manifold M with codimension one orbit.Let DG(M) denote the group of equivariant di¤eomorphisms of M which are isotopic to
the identity through equivariant di¤eomorphisms, with C1-topology. Then DG(M) is notperfect and in [A-F] we calculated the the �rst homology of DG(M).Let LG(M) denote the group of equivariant Lipschitz homeomorphisms ofM . Let LG(M)
(HLIP;G(M)) be the the connected component of the identity of LG(M) with compact opentopology (with compact open Lipschitz topology). We shall prove that the groupHLIP;G(M)is perfect. In the case of the complex plane C with the canonical U(1)-action, the �rsthomology H1(LU(1)(C)) admits continuos moduli ( [A-F-M]). Thefore the �rst homology isquiet di¤erent between HLIP;U(1)(C) and LU(1)(C).
References
[A-F] K. Abe and K. Fukui, On the structure of the group of equivariant di¤eomorphismsof G-manifolds with codimension one orbit, Topology, 40 (2001), 1325�1337.
[A-F-M] K. Abe, K. Fukui and T. Miura, On the �rst homology of the group of equivariantLipschitz homeomorphisms, J. Math. Soc. Japan, 58 (2006), 1�15.
28
Electron motion and ruled surfaces
Yuriy Aminov
The motion of an electron (point charge) and its emission of an electromagnetic �eldhave been considered keeping within the bounds of classical electrodynamics. Owing to thespeci�c properties of the emitted �eld, we were able to �nd a natural correlation betweenthe emission and the ruled surfaces characterizing this emission in di¤erent directions. Inclassical electrodynamics the emission is described by two vector �elds - electric Eemit andmagnetic Bemit. They are orthogonal to each other and to the direction of emission. The�elds Eemit and Bemit have di¤erent intensities and vectors in di¤erent directions and aredescribed by the formulas used in our study (see [1], Eq.(19.1)). Besides, polarization is alsoan important property of the emission.We consider the simplest case of the motion of a charge in a constant magnetic �eld. It
is well known that under the in�uence of the constant magnetic �eld and the Lorentz force,the charge moves along a straight line, or a circle, or a helix.Here we consider a helical motion. O.A.Goncharova [2] introduced standard ruled surfaces
�i in n-dimensional Euclidean space. In these surfaces the directing curve has constantnon-zero curvatures ki (i = 1; :::; n � 1) and the straight -line generator is along the basicvector �i of the natural -frame. For n=3 these surfaces are �1 (developable helicoid), �2(helicoid) and �3 (a new surface).Let us consider the emission along a certain basic vector �l at a distance � from the
electron. The vectors Eemit and Bemit are parallel to the other basic vectors �i; �j; l 6=i 6= j 6= l: Translate the vectors Eemit(��l) in such a way their beginnings are at the pointof electron. Then their end points will be situated in the straight line directed along thebasic vector �i. On varying �, the end of the vector Eemit will move along the straight lineadvancing its or half length.In the curse of the electron motion this straight line describes the surface �i. A similar
procedure can be used to construct a surface with the help of Bemit. Omitting the notationemit we can write down
E(��1) = ��1�2�
; E(��2) = ��2�1�c
; E(��3) = ��3�3�
;
B(��1) = ��1�3�c
; B(��2) =�2�3�c2
; B(��3) =�3�1�c
:
Here �i is the constant determined by the universal physical constants, the electrontrajectory curvature and the electron velocity; the constant c is the velocity of light. Corre-spondingly, we obtain the following standard ruled surfaces:
�2; �1; �2;
�3; �3; �1:
The natural ruled surfaces appear while considering the directions along which the emis-sion is zero. These directions are found in the osculating plane. They are symmetrical with
29
respect to the vector �1 and make an constant angle � with it in the course of the helicalmotion of the electron. We call � a zero-emission angle. The electron under considerationcan be denoted as Q1. If we draw a straight line through each point of the electron trajec-tory towards zero emission, we can obtain a certain ruled surface, which we identify as " azero-emission surface". The are two surface of this typei; i = 1; 2: The directions of thezero emission are in the plane of the vectors �1 and �2. For this reason the trajectory of theelectron motion on the surface �i is an asymptotic curve. This conclusion is true not onlyfor electron motion in the constant magnetic �eld, but in the general case as well.Let us consider the question whether there exists another electronQ2 whose zero-emission
surface coincides with that of the electron Q1. In our study [3] we proved the followingtheoremTheorem. For an electron moving in a constant magnetic �eld on each surface of zero
emission there exists one and only one second electron so that during the motion they fallsimultaneously on the common straight line of zero emission.The second electron can be called "conjugate" to the �rst electron. Note the following
interesting property of the surface of zero emission: its striction curve is geodesic and takesthe medial position between the speci�ed asymptotic curves - the trajectories of the electronsQ1 and Q2. This curve is also helical.
References
[1] W.K.H. Panofsky, M.Phyllips, "Classical electricity and magnetism", Cambridge,Addison-Wesley.
[2] O.A.Goncharova, "Standard ruled surfaces in En", Dopovidi of NAS of Ukraine,2006, N 3, p. 7-12.
[3] Y.A.Aminov, "On physical interpretation of some ruled surfaces in E3 with thehelp of motion of point charge", Matem. Sbornik, 2006, v. 197, N 12, p. 3�10.
30
Poisson cohomology in arbitrary dimension
Mourad Ammar and Norbert Poncin
Poisson Geometry is the natural frame for Deformation Quantization. Kontsevich�s for-mality theorem provides complete understanding of the emergence of Poisson cohomologyin deformation quantization of Poisson manifolds. In recent years many papers on Pois-son homology and primarily on Poisson cohomology have been published. Cohomology ofregular Poisson manifolds, (co)homology and resolutions, duality, cohomology in low dimen-sions or for speci�c cases, extensions of Poisson cohomology, e.g. Lie algebroid cohomology,Jacobi cohomology, Nambu-Poisson cohomology, double Poisson cohomology, have been in-vestigated. However, no appropriate conceptual approach is available so far. In this talk wepropose a general approach to the cohomology of the Poisson tensors of the Dufour-Harakiclassi�cation. We also show that our cohomological technique for strongly classical r-matrixinduced three-dimensional Poisson structures can be extended to an arbitrary dimensionalspace. In the main, Poisson cohomology reduces to a Koszul cohomology and a relativecohomology.
Homeomorphism groups of non-compact manifolds
T. Banakh, K. Sakai, K. Mine (Tsukuba University, Japan)
For a paracompact spaceX byH(X) we denote the homeomorphism group ofX endowedwith the Whitney (or else graph) topology generated by the base consisting of the sets �U =fh 2 H(X) : �h � Ug where U runs over open subsets of X and �h = f(x; h(x)) : x 2 Xgstands for the graph of h. Let Hc(X) be the connected component of H(X) containing theidentity homeomorphism.
Theorem 1. For any non-compact connected 2-manifold M the homeomorphism groupHc(M) is homeomorphic to l2 � R1.Here R1 is the linear space with countable Hamel basis and the strongest liner topol-
ogy. For higher-dimensional Euclidean spaces Rn we have the following characterizationconnecting the topology of the group Hc(Rn) with the topology of the group H@(I
n) ofhomeomorphisms of the n-cube In that do not move the points of the boundary of In.
Theorem 2. The homeomorphism group Hc(Rn) is an l2 � R1-manifold if and only if thegroup H@(I
n) is an ANR. For n � 2 the group Hc(Rn) is homeomorphic to l2 � R1.It is an old open problem if the homeomorphism group H@(I
n) is an AR for n � 3.However it is known (and easily seen) that this group is contractible.
31
Topological classi�cation of the hyperspaces of closed convex subsets ofa Banach space
T. Banakh, K. Sakai, M. Yaguchi (Tsukuba University, Japan), I. Zarichnyi, (LvivNational University, Ukraine)
In the talk we shall classify topologically the spaces ConvH(X) of non-empty closedconvex subsets of a Banach space X, endowed with Hausdor¤ (in�nite-valued) metric
dH(A;B) = maxfsupa2A
dist(a;B); supb2B
dist(b; A)g 2 [0;1]
where dist(a;B) = infb2B ka � bk is the distance from a point a to a subset B in X. Thespace ConvH(X) is locally connected. The connectd component of ConvH(X) containing agiven closed convex set C � X coincides with the set fA 2 ConvH(X) : dH(A;C) <1g.
Theorem 1. Let X be a Banach space. Each connected component H of the space ConvH(X)is homeomorphic to f0g, R, R � R+, Q � R+, or l2(�) for an in�nite cardinal �. Moreprecisely, H is homeomorphic to:
1. f0g i¤ H contains the whole space X;
2. R i¤ H contains a half-space;
3. R� R+ i¤ H contains a linear subspace of X of codimension 1;
4. Q� R+ i¤ H contains a linear subspace of X of �nite codimension � 2;
5. l2(�) for an in�nite cardinal � i¤ H contains no half-space and no linear subspace of�nite codimension.
Here R= = [0;1) is the half-line, Q = [0; 1]! is the Hilbert cube and l2(�) is the Hilbertspace of density �.A closed convex subset C of a Banach space X is called
� a half-space if C = f�1[a;+1) for some real number a and some non-trivial linearcontinuous functional f : X ! R;
� a polyhedral set if C =Tni=1Hi is a �nite intersection of half-spaces.
Theorem 2. Let X be a �nite-dimensional Banach space. For a connected component H ofConvH(X) the following conditions are equivalent:
1. H has density dens(H) < c;
2. H is separable;
3. H contains a polyhedral convex set.
32
Combining Theorems 1 and 2 we obtain the topological classi�cation of connected com-ponents of the spaces ConvH(Rn).
Corollary 3. Let X be a �nite-dimensional Banach space. Each connected component H ofthe space ConvH(X) is homeomorphic to f0g, R, R�R+, Q�R+, l2 or l1. More precisely,H is homeomorphic to:
1. f0g i¤ H contains the whole space Rn;
2. R i¤ H contains a half-space;
3. R� R+ i¤ H contains a linear subspace of X of codimension 1;
4. Q� R+ i¤ H contains a linear subspace of X of codimension � 2;
5. l2 i¤ H contains a polyhedral convex set but contains no linear subspace and no half-space;
6. l1 i¤ H does not contain a polyhedral convex set.
Corollary 4. Let X be a �nite-dimensional Banach space. The space ConvH(X) is homeo-morphic to the topological sum:
1. f0g � R� R� (R� R+) i¤ dim(X) = 1;
2. f0g � Q� R+ � c� (R� R� R+ � l2 � l1) i¤ dim(X) = 2;
3. f0g � c� (R� R� R+ � Q� R+ � l2 � l1) i¤ dim(X) � 3.
33
Hyperspaces of max-plus convex sets which are in�nite-dimensional manifolds
Lidia Bazylevych
The classical result by Nadler, Quinn and Stavrokas [1] asserts that the hyperspace ofconvex compact subsets in Rn, n � 2, is a contractible Q-manifold. Recall that a Q-manifoldis a manifold modeled on the Hilbert cube Q = [0; 1]!.Let Rmax = R[f�1g. Given x; y 2 Rn and � 2 R, we denote by x�y the coordinatewise
maximum of x and y and by � � x the vector obtained from x by adding � to every itscoordinate. A subset A in Rn is said to be tropically convex if � � a � � � b 2 A for alla; b 2 A and �; � 2 Rmax with �� � = 0. The tropical convexity (or max-plus convexity, inanother terminology) was introduced in [2].The main result states that the hyperspace of compact max-plus convex sets in Rn, n � 2,
is a contractible Q-manifold Qnf�g. This is a max-plus counterpart of the mentioned resultfrom [1].We conjecture that, for a nonempty open subset U of Rn, n � 2, the hyperspace of
compact max-plus convex sets contained in U is homeomorphic to the Q-manifold Q �[0; 1)�U . The corresponding result for the hyperspace of compact convex sets is proved byL. Montejano [3].
[1] S.B. Nadler, Jr., J. Quinn , N.M. Stavrokas, Hyperspace of compact convex sets, Pacif.J. Math. 1979. V. 83. P. 441�462.
[2] G. L. Litvinov, V. P. Maslov, and G. B. Shpiz. Idempotent functional analysis: Analgebraic approach., Translated from Matematicheskie Zametki, vol. 69, no. 5, 2001,758�797.
[3] L. Montejano, The hyperspace of compact convex subsets of an open subset of Rn, Bull.Pol. Acad. Sci. Math. V.35, No 11-12 (1987), 793�799.
34
Vertex Operator construction of coupled soliton hierarchies
Paolo Casati
The aim of this talk is present the paper [1] written together with Giovanni Ortenziwhere we give a representation�theoretic interpretation of recent discovered coupled solitonequations [2],[6] [3] [5] using vertex operators construction of a¢ nization of not simple butquadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupledto each Drinfeld�Sokolov of A, B, C, D hierarchies and to construct their soliton solutions.The not simple Lie algebras considered in this talk are given by the tensor product
g(n) = g C(n)(�) where g is any Lie algebra and C(n)(�) = C[�]=(�)n+1. This algebramay be identi�ed with the Lie algebra of polynomial maps from C[�]=(�n+1) in g, hence anelement X(�) in g(�) can be viewed as the mapping X : C! g, X(�) =
Pnk=0Xk�
k whereXk 2 g. In this setting the Lie bracket of two elements in g(�), X(�) =
Pnk=0Xk�
k andY (�) =
Pnk=0 Yk�
k can be written explicitly as
[X(�); Y (�)] =nXk=0
(kXj=0
[Xj; Yk�j]g)�k:
The most important peculiarity of this obviously not simple Lie algebras is that if g admitsa symmetric ad�invariant non�degenerated bilinear form then roughly speaking this bilinearform is inherited by the whole Lie algebra g(n). Using this fact we will able to a¢ nizesuch algebras, obtaining in�nite dimensional Lie algebras with a multidimensional centralextensions. The key point to link these in�nite dimensional Lie algebras with hierarchies ofsoliton equations is that, despite to the fact that they are not a¢ ne Kac�Moody Lie algebras,they still admit representations trough Vertex Operators Algebras. These representationsmay be lifted to representations of in�nite dimensional Lie groups explicitly de�ned duringthe talk. This fact �nally allows us to implement the theory developed by Kac, Petersonand Wakimoto to construct corresponding generalized Hirota bilinear equation and theirmultisoliton solution in term of ��functions. Explicit examples of such construction will bepresented during the talk, namely those of the case of the of the coupled AKP BKP andtheir reduction to Lie algebras generalizing the algebras A(1)1 , A
(1)2 , A
(2)1 , and B
(1)2 making
the direct connection whit the hierarchies of such type already presented in the literature.Finally it will be brie�y sketched how we intend together with professor Johan van de Leurproceed further in this direction [4] .AMS Subject Classi�cation: Primary 17B69, 37K10, Secondary 37K30
References
[1] P. Casati, G. Ortenzi, New Integrable Hierachies from Vertex Operator Representa-tions of Polynomial Lie AlgebrasJournal of Geometry and Physics vol. 56, 418�449(2006)
35
[2] R. Hirota, X. Hu, X. Tang, A vector potential KdV equation and vector Ito equation:soliton solutions, bilinear Bäcklund transformations and Lax pairs J. Math. Anal.Appl. 288 (2003), no. 1, 326�348.
[3] S. Kakei, Dressing method and the coupled KP hierarchy. Phys. Lett. A 264 (2000),no. 6, 449�458.
[4] J. van de Leur, Bäcklund transformations for new integrable hierarchies related tothe polynomial Lie algebra gl(n)1 , J. Geom. Phys. 57 (2007), no. 2, 435�447. 37K10
[5] S. Yu. Sakovich, A note in the Painlev é property of coupled KdV equation
[6] W. Ma , X. Xum, Y. Yufeng Zhang, Semi-direct sums of Lie algebras and discreteintegrable couplings arXiv:nlin.SI/0603064
36
Hilbert di¤eomorphism groups and their geometry for open manifolds
Jürgen Eichhorn
Let Mn be closed, D(M) � Di�(M) the group of smooth di¤eomorphisms. For manyapplications in PDE theory one needs a completed version Dp;r(M), 1 � p; r Sobolev index.For r > n
pand Mn closed, this easily can be done. One de�nes Dp;r by means of a �nite
cover U = fU�; ��g1���m and imposes Euclidean Sobolev conditions,
� � f � ��1� : ��(U�) �! IRn is in W p;r(�(U); IRn):
It easily follows that Dp;r is independent of the choice of the �nite cover. For open manifolds,this is totally wrong. We de�ned for open manifolds (Mn; g) of bounded geometry completeddi¤eomorphism groups Dp;r(Mn; g) satisfying the following conditions1) Dp;r is a Banach manifold, for p = 2 it is a Hilbert manifold,2) it depends only on the component comp(g) �Mp;r(I; Bk), the completed space of metricsof bounded geometry,3) if (Mn; g) is compact then our de�nition coincides with all other de�nitions and is com-pletely independent of g.In a second step, we construct Hilbert submanifolds of volume (element) preserving, sym-
plectic, contact and gauge di¤eomorphisms, de�ne for them a (weak) Riemannian structureand calculate their curvature. All this has many applications in mathematical physics.
37
The Poincaré conjecture and a geometric model of "black hole"
Alexander Ermolitski
Let Mn be a compact, closed, smooth manifold. Then there exists a Riemannian metricg on Mn and a smooth triangulation of Mn. Both the structures make possible to prove thefollowing
Theorem 1. The manifold Mn has a decomposition Mn = Cn [Kn�1, Cn \Kn�1 = ;,where Cn is a n � dimensional cell and Kn�1 is a connected union of some (n � 1)simplexes of the triangulation.
Kn�1 is called a spine of Mn.
Using the theorem 1 we have obtained
Theorem 2. (Poincaré). Let M3 be a compact; closed, smooth simply connected manifoldof dimension 3. Then M3 is diffeomorphic to S3; where S3 is the sphere of dimension 3:
For any point z 2 Kn�1 we can consider the closed geodesic ball B (z; ") of a small radius" > 0. A set Tb(Kn�1) =
Sz2Kn�1
B(z; ") is called "black hole". It is clear thatMnn Tb(Kn�1)
is a cell for some small ".
Theorem 3. Let � be a pseudoriemannian metric onMn. Then there exists a deformation� of � on M with the following properties.
a) � is continuous and sectionally smooth.
b) If a point x 2 Mnn Tb(Kn�1) and � is smooth at the point x , then the curvaturetensor Rx of � vanishes. In other words, all the curvature of � is concentrated inTb(Kn�1).
S-cohomologie vs Diagram cohomology
Yael Fregier
In our thesis we have introduced a cohomology associated to simultaneous deformationsof two Lie algebras and a Homomorphism between them. M. Gerstenhaber and S. Schack hadalready introduced such a cohomology, in the associative setting, in the context of diagramsof algebra. We will in this talk recall the main features of the two theories and show in whichsense they coincide.
38
Higher vector bundles and multi-graded manifolds (joint work withM. Rotkiewicz)
Janusz Grabowski
A natural condition is given assuring that an action of the multiplicative monoid ofnon-negative reals on a manifold F comes from homoteties of a vector bundle structure onF , or, equivalently, from an Euler vector �eld. This is used in showing that double (orhigher) vector bundles known in the literature can be equivalently de�ned as manifolds witha family of commuting Euler vector �elds. Higher vector bundles can be therefore de�nedas manifolds admitting certain Nn-gradation in the structure sheaf.The n-vector bundles F admit canonical lifts to the tangent and to the cotangent bundles
TF and T �F . In particular, the iterated tangent and cotangent bundles are canonical exam-ples of higher vector bundles. The cotangent bundle T �F is of particular interest, since it iscanonically �bred not only over F but also over all duals F �(k) of F with respect to all its vec-tor bundle structures F ! F[k]. The side bundles F[k] are canonically (n� 1)-vector bundlesthemselves. We prove the existence of a canonical identi�cations T �F ' T �F �(k) ' T �F �(l)which are additionally symplectomorphisms. They can be viewed as a generalization ofthe celebrated "universal Legendre transformation" T �TM ' T �T �M . Moreover, the setof higher vector bundles fF; F �(1); : : : ; F �(n)g is closed with respect to duality (under naturalidenti�cations) for all vector bundle structures on them. This is a phenomenon observed �rstfor double and triple vector bundles by K. Konieczna, P. Urbanski and K. C. H. Mackenzie.Next, we prove that symplectic n-vector bundles, i.e. n-vector bundles with a symplectic
form which is linear (1-homogeneous) with respect to all vector bundle structures, are alwaysof the form T �F for certain (n � 1)-vector bundle F . This, in turn, generalizes the knownfact that any vector bundle equipped with a linear symplectic form is, in fact, T �M .Consequently, multi-graded (super)manifolds are canonically associated with higher vec-
tor bundles which is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. The symplecticmulti-graded manifolds, equipped with certain homological Hamiltonian vector �elds, lead toan alternative to D. Roytenberg�s picture generalization of Lie bialgebroids, Courant brack-ets, Drinfeld doubles and can be viewed as higher BRST and Batalin-Vilkovisky formalisms.
39
Singular symplectic forms and round functions
Bogus÷aw Hajduk
This is a report on a joint work with V. Sharko and M. Lewkowicz on singular sym-plectic forms and Morse-Bott 1-forms with round singularities. Singular symplectic formsare di¤erential 2-forms we consider a generic representatives of a given cohomology classa 2 H2(M;R) and vanish only on a set of dimension smaller or equal 1. If ! is such a formis invariant with respect to an action of the circle, then we get a 1-form � = �X!, where Xis the vector �eld which generates the action. Now � has the same singularities as !: In thecase of an action without �xed points singularity set is a collection of circles. So we come toa problem what is the minimal number of such circles, or equivalently what is the minimalnumber of singular circles of a function with values in the circle and round (consisting ofcircles) singularity set. The aim of the present project is to extend existence theorems forsingular symplectic representatives given by LeBrun, Honda and others, calculate minimalnumber of singular circles for some classes of 1-forms as well as to study mutual applicationsof these two subjects.
40
The Equivalence Principle and Projective Structure in 4-dimensionalLorentz manifolds
Graham Hall
The purpose of this lecture is to try and show how far one can proceed in the �ndingof the metric on a 4-dimensional Lorentz manifold if the projective structure of its Levi-Civita connection is given. This work has applications to the general theory of relativity. Itrepresents joint work with Dr David Lonie in Aberdeen.Let M be a smooth 4-dimensional manifold and let g and g0 be Lorentz metrics on M .
Let r and r0 be the Levi-Civita connections associated with g and g0, respectively. Supposethat, for each p 2 M , there exists an open subset Gp of the tangent space TpM to M atp such that (i) each member of Gp is timelike with respect to g and g0 and (ii) for eachu 2 Gp the (unparametrised) r-geodesic starting at p and with initial tangent u is also an(unparametrised) geodesic for r0. The restriction to the subsets Gp simulates the restrictedexperimental evidence of the principle of equivalence, that the paths of �free�particles willfollow such geodesics. It can be shown from this that r and r0 in fact share their geodesicsand are thus projectively related. In showing this, the assumption (i), introduced for obviousphysical reasons, is not used. Thus, r and r0 determine a global 1-form on M (theprojective 1-form) and, because they are metric connections, is a closed 1-form [1]. Thereis also a convenient relationship between the curvature tensors R and R0 from r and r0,respectively, in terms of the 1-form . There are two convenient equivalent statements of thisprojective equivalence between r and r0 [1, 2], one relating these connections themselvesand a second one which relates the r covariant derivative of g0 to g0 and .To obtain information relating g and g0 under the above conditions, an approach has
been devised in [3, 4] which uses the algebraic nature of the curvature tensor, together withthe equivalent projective conditions given above. Another method may also be applied whenone of the metrics admits certain symmetries [5] because one can then take advantage ofthe nice relation between the symmetries of projectively related metrics. An example of the�rst method arises in the important situation for general relativity when one of the metrics,say g, is a vacuum (Ricci-�at) metric. (The physical assumption that (M; g) is non-�atis also imposed, that is, the curvature tensor R does not vanish over any non-empty opensubset ofM). In this case, one may appeal to Petrov�s algebraic classi�cation of the vacuumcurvature tensor R [6]. In fact, one may topologically decompose M disjointly into �veopen subsets, one for each Petrov type, (such that each subset contains only points of thatparticular Petrov type) together with a closed nowhere dense set F . It can then be shown,using the convenient algebraic properties of the curvature tensor for each of the Petrov types,that r and r0 agree on the open dense subset, M n F , of M and hence on M . It followsfrom this that g0 is also a vacuum metric on M . Use of holonomy theory and the factthat M is a connected manifold then shows, with one highly specialised case excepted, thatg and g0 are conformally related on M by a constant conformal factor. (The special caseconcerns a subclass of the so-called pp-waves and can be easily handled separately.) Thus,from the physical viewpoint, knowledge of the geodesics (as described above) essentiallyuniquely determines the metric up to �units of measurement�. As an example of the secondmethod, let g be a member of the important class of Friedmann-Robertson-Walker-Lemaitre
41
(FRWL) cosmological metrics. In this case the high degree of symmetry possessed by gis useful. Here, the known result that, with the somewhat unphysical Einstein static andde-Sitter type metrics excluded, the dimension of the Lie algebra of projective vector �eldsfor (M; g) is at most seven, is useful [7, 8]. From this, and the fact that projectively relatedconnections have the same Lie algebras of projective vector �elds, one can show that if g0 isprojectively related to g, in the sense de�ned above, then g0 is also an FRWL metric sharingthe same space slices of constant cosmic time and having the same Killing algebra as g andis conveniently related to g (but not so tightly as in the vacuum case).
References
[1] L. P. Eisenhart, "Riemannian geometry", Princeton, 1966.
[2] T. Y. Thomas, "Di¤erential Invariants of Generalised Spaces", Cambridge, 1934.
[3] G. S. Hall, to appear in the conferences proceedings "Extending Einstein�s Ideas",Cairo, 2005.
[4] G. S. Hall and D. P. Lonie, preprint, University of Aberdeen, 2007.
[5] G. S. Hall and D. P. Lonie, preprint, University of Aberdeen, 2007.
[6] A. Z. Petrov, "Einstein Spaces" Pergamon, 1969.
[7] G. S. Hall, Class. Quant. Grav. 17, (2000), 4637.
[8] G. S. Hall, "Symmetries and Curvature Structure in General Relativity", WorldScienti�c, 2004.
42
The topology and geometry of polygon spaces
Jean-Claude Hausmann
Polygon spaces in Rd occur in connection with statistical shape theory and robotic. Ford = 3, they became also a chapter of Hamiltonian geometry, as a good source of examples,closely related to toric manifolds. This talk will be a survey of these various aspects ofpolygon spaces.
The degree of maps of manifolds with actions of compact Lie groups
Jan Jaworowski
Suppose that G is a compact Lie group, M and N are orientable, connected, smooth,free G-manifolds. We show that for certain class of maps f :M ! N , including equivariantmaps, the degree of f satis�es a formula involving data given by the classifying maps of theorbit spacesM=G and N=G. In particular, if f is equivariant, and if the generator of the topdimensional cohomology of M=G with integer coe¢ cients is in the image of the cohomologymap induced by the classifying map for M , then the degree of f is one. We also study thedegree of maps f :M ! N that are �equivariant up to an exponent�, or equivariant �up toa homomorphism�. The degrees mod p, for actions of p-groups, where p is a prime greaterthan 2, are also studied. Several results of this paper have been obtained jointly with NeµzaMramor-Kosta.
43
Examples of multiple solutions for the Yamabe problem on scalar curvature
Guy Kass
In the conformal class of a Riemannian metric on a compact connected manifold, thereexists at least one metric with constant scalar curvature. In the case with positive scalarcurvature, there may be many (non-homothetic) metrics with constant scalar curvature in aconformal class. R. Schoen gave a beautiful example of that phenomenon for a one-parameterfamily of metrics on S1 � Sn�1. This construction may be generalized on products S1 �Nis a compact connected Riemannian manifold with positive constant scalar curvature. Aunique (ordinary) di¤erential equation, depending only on the dimension, is the key to thatconstruction. We will give some details on the solutions of that equation and study theirbehaviour in a one-parameter family.
On some generalizations of the Poisson sigma model(joint with T. Strobl and, partially, with P. Schaller)
Alexei Kotov
The Poisson sigma model (PSM), invented in early ninetieth, together with the Wess-Zumino-Witten (WZW) and Chern-Simons (CS) sigma models, inspired a great source ofmathematical results, the most known of which is the deformation quantization of arbitraryPoisson manifolds. We provide certain natural generalizations of the PSM, linking it with theGauged WZW model and higher order analogues of Poisson manifolds (Courant algebroidsand their extended versions). We compute the classical equations of motion and gaugesymmetries of the corresponding sigma models, showing the geometrical meaning of them.
44
On the reconstruction problem for some nontransitive homeomorphism groups
Agnieszka Kowalik, Ilona Michalik and Tomasz Rybicki
Let G(Mi) be a group of homeomorphisms of a manifold Mi, i = 1; 2. Usually we willassume that Mi is smooth and G(Mi) is the automorphism group of a geometric structure.Given a group isomorphism ' between G(M1) and G(M2), the reconstruction problem con-sists in the question whether there exists a homeomorphism � of M1 onto M2 such that 'is induced by � . In this talk we will consider this problem in case of some nontransitivegeometric structures.Our study has been inspired by recent papers of Matatyahu Rubin and his collaborators.
Bibliography[1] M. Rubin, Y. Yomdin: Reconstruction of manifolds and subsets of normed spaces fromsubgroups of their homeomorphism groups, Warsaw, IM PAN, 2005.[2] E.B. Ami, M. Rubin, On the reconstruction problem of factorizable homeomorphismgroups and foliated manifolds, Preprint.
45
Remarks on Extended Operator Calculus
Ewa Krot-Sieniawska
In mathematics, before the 1970s, the term umbral calculus (or Finite Operator Calculus)was understood to mean the surprising similarities between otherwise unrelated polynomialequations, and certain shadowy techniques that can be used to �prove�them. These tech-niques were introduced in the 19th century and are sometimes called Blissard�s symbolicmethod, and sometimes attributed to James Joseph Sylvester, or to Edouard Lucas. In the1970s, Gian-Carlo Rota, and others [1-5] developed the umbral calculus as the Finite Op-erator Calculus by means of linear operators on spaces of polynomials. Currently, umbralcalculus is understood primarily to mean the study of She¤er sequences, including polynomialsequences of binomial type and Appell sequences. Already since thirties of XX-th centuryit had been realized that operator methods might be extend to the use of any polynomialsequences instead of these of binomial type only. The very foundations of such an extensionswere laid in 1937 by M.Ward, [6] . The next major contributions we owe to Viskov [7, 8] andMarkowsky [9]. The main statements of - extended Rota�s �nite operator calculus weregiven by A.K.Kwasniewski [10-12]. I�ll present some topics related to this theory.
References
[1] G.-C.Rota and R.Mullin: On the foundations of combinatorial theory III. Theoryof binomial enumeration. In B. Harris, editor, Graph theory and its applications,pages 213. Academic Press, 1970.
[2] G.-C. Rota, D. Kahaner, and A. Odlyzko: On the foundations of combinatorialtheory VII. Finite operator calculus. J. Math. Anal. Appl., 42:684;760, 1973.
[3] G.-C. Rota: Finite operator calculus. Academic Press, New York, 1975.
[4] S.M. Roman and G.-C. Rota :The umbral calculus. Adv. Math., 27:95;188, 1978.
[5] S.M. Roman: The Umbral Calculus. Academic Press, 1984.
[6] M. Ward: A calculus of sequences. Amer. J. Math.58, 255 (1936).
[7] O. V. Viskov: Operator characterization of generalized Appell polynomials SovietMath. Dokl. 16, 1521 (1975).
[8] O. V. Viskov: On bases in the space of polynomials, Soviet Math. Dokl.19, 250(1978).
[9] G. Markowsky: Di¤erential operators and the theory of binomial enumeration. J.Math. Anal. Appl. 63, 145 (1978).
[10] A. K. Kwasniewski: Towards -extension of Finite Operator Calculus of Rota Rep.Math. Phys. 47, 305 (2001).
46
[11] A. K. Kwasniewski:On Simple Charactrrisation of She¤er -polynomials and Re-lated Proposition of the Calculus of Sequences Bulletin de la Soc. des Sciences etde Lettres de Lodz ; 52, Ser. Rech. Deform. 36 (2002) pp.45-65 .
[12] A.K.Kwasniewski: Main theorems of extended �nite operator calculus IntegralTransforms and Special Functions Vol. 14, No 6, pp. 499-516, 2003 Taylor & FrancisOnline Journal
[13] E.Krot: -extensions of q-Hermite and q-Laguerre Polynomials - properties andprincipal statements Czech. J. Phys. Vol.51(2001) No12, p. 1362-1367.
[14] E.Krot: An Introduction to Finite Fibonomial Calculus, CEJM 2(5)2005 p.754-766,ArXiv: math.CO/0503218
[15] E.Krot: The �rst Ascent into the incidence algebra of the Fibonacci cobweb posetAdv.Stud. in Cont. Math. 11(2005), No. 2, p.179-184
47
Holonomy representations admitting two pairs of supplementary invariant sub-spaces
Thomas Krantz
We show that if a representation admits two pairs of supplementary invariant subspaces,or one pair and an invariant re�exive form, then it is a direct sum of three canonical subrepre-sentations which we characterize. We then focus on holonomy representations with the sameproperty. The work is a contribution to the classi�cation of the holonomy representations oftorsion-free connections and for which the non semi-simple case is open.
To Invariant Normales of Vector Field in Four-Dimmentional A¢ ne Space
Olga Kravchuk
In four- dimensional a¢ ne space invariant points, straights, hypersur�ces and two-dimensionalspaces associated with vector �eld have been built. The received data are transformed intospaces of inde�nite size.
Flat connections with singularities in some Lie algebroids
Jan Kubarski
We study �at connections in some Lie algebroidsA de�ned everywhere but a �nite numberof points. Under some assumptions ( H top (A) 6= 0; the structural Lie algebras g = R; sk (3)or sl (2;R) ; dimM = dim g + 1 ) with any such isolated singularity we join a real numbercalled an index. We prove the index theorem saying that the index sum is independent ofthe choice of a connection (forM compact and oriented). Multiplying this index sum by theorientation class of M; we get the Euler class of this Lie algebroid. Some integral formulaefor indexes are given.
48
Ivan Bernoulli Series Universalissima
A. Krzysztof Kwasniewski
One states, recalls here that the textbooks formula named as celebrated Taylor formulapertains historically to Johann Bernoulli (1667 - 1748). When Ivan Bernoulli�s Series Uni-versalisima was published in Acta Eruditorum in Leipzig in 1694 Brook Taylor was nineyears old. As on the 01.01.2006 one a¢ rms 258-th anniversary of death of Johann Bernoullithis calendarium-article is Pro-posed Pro hac vice, Pro memoria, Pro nunc and Proopportunitate - as circumstances allow.
KEY WORDS: Bernoulli-Taylor formula, Graves-Heisenberg-Weyl (GHW) algebra, umbralcalculus.
AMS S.C. (2000): 01A45, 01A50, 05A40, 81S99.
Deformation quantization in in�nite dimensional analysis
Rémi Léandre
This talk is divided in 3 parts:
� In the �rst part in a commun work with G. Dito, we de�ne a Moyal product in theframework of the Malliavin Calculus.
� In the second part, we de�ne a Moyal product in the framework of white noise analysis.
� In the third part, we de�ne Fedosov quantization for Taubes limit model.
49
Fragmentations of the second kind in some di¤eomorphism groups
Jacek Lech, Tomasz Rybicki
Let G(M) � Di� r(M) be a di¤eomorphism group, and Gc(M)0 be its compactly sup-ported identity component. Suppose that for an isotopy ft in Gc(M)0 with supp(ft) �Ski=1 Ui, Ui are open, there exist isotopies fj;t in Gc(M)0, j = 1; : : : ; l, with ft = f1;t � : : :�fl;t
such that supp(fj;t) � Ui(j) for all j. Then G(M) is said to satisfy a fragmentation propertyof the �rst kind.Our idea is to introduce also the second kind of fragmentations. Such fragmentations are
considered only for isotopies and di¤eomorphisms in a su¢ ciently small C1-neighborhoodof the identity in the group Gc(Rn)0. On the other hand, we claim that the factors of thefragmentation are uniquely de�ned by the initial di¤eomorphism and that the Cr-norms ofthe factors are controlled in a convenient way by the Cr-norms of the initial di¤eomorphism.In proofs of some theorems on the simplicity and perfectness of di¤eomorphism groups a
clue role is played by both kinds of fragmentation properties. Here we consider fragmenta-tions of both kinds for symplectomorphisms and contactomorphisms.
Jetbundles on �agvarieties
Helge Maakestad
Ill discuss some results on the structure of the jetbundles J(L) of a linebundle L onprojective space and grassmannians as bimodule.
50
Presentation for the fundamental groups of orbits of Morse functinos on surfaces
Sergiy Maksymenko
Let M be a compact surface, P either a real line or a circle, and f : M ! P a Morsemapping. Denote by Of the right orbit of f , i.e. the orbit of f with respect to the action ofthe group of di¤eomorphisms Diff(M) of M de�ned by the formula:
h � f = f � h�1; h 2 Diff(M):
(1) We show that Of is homotopy equivalent to some covering space of the n-th con�g-uration space of M , where n is a total number of critical points of f .It was proved by the author earlie, that for aspherical surfaces, i.e. all surfaces except
2-sphere and projective plane, the orbit Of is aspherical. Then (1) implies that �1Of is asubgroup of the n-th braid group of M .(2) We also construct a �nite presentation for the group �1Of for the case when M is
orientable of genus g � 2.It turns out that this presentation has similarity with the presentation for Artin groups.
In particular, analogues of K(�; 1) conjecture and Tits conjecture on squares of generatorsin Artin group can easily be formulated and proved for �1Of .
51
A note on the Lagrange di¤erential in bigraded module of vertical tangentbundle valued forms
Roman Matsyuk
Within the �nite order approach to the split calculus of variations on �bred manifoldswe cast the set of Euler�Lagrange expressions in the guise of a volume semi-basic di¤erentialform with values in the dual to the bundle V of vertical tangent vectors on some �brationY ! B. Although this treatment in not new, it helps to establish a simple relation betweenthe operators d� of �ber derivative andD of semi-basic derivative in the appropriate bigradedtensor algebra of cross-sections of the vector bundle ^�V �
r Yr^�T �B on one hand, and the
operators dV and dH previously introduced by Tulczyjew in the total bigraded algebra ofexterior di¤erential forms on the corresponding jet bundle prolongation �r : Yr ! B, onthe other hand. The split bigraded algebra Sec
�^� V �
r Yr^�T �B
�may be converted
into the total algebra ^�T �Yr+1 applying the dual of the Cartan contact form morphismT �Yr+1 ! Vr together with subsequent alternation. By means of this mapping operators dVand dH correspond to the operators d� and D respectively.Let y�N denote standard coordinates in Yr with multiindex N of order kNk � r. Let
�idy�N = dy�N�1i be the order-reducing operators similar to those introduced by Tulczyjew,and let Di =< @i; D >. The split Lagrange di¤erential � is �rst de�ned on the cross-sectionsof ^�V �
r as
�' = deg(d�') +XkNk�0
(�1)kNkN!
DN�Nd�' ;
and then extended to the total algebra Sec�^� V �
r Yr^�T �B
�by trivial action on the
subalgebra of scalar di¤erential forms on B.As conventionally assumed, the variational derivative of the Lagrange density � 2 Sec
�^dimB
T �B�at the extremal section � of Y is an rth order di¤erential operator E� acting from the
space of cross-sections of the induced vertical tangent bundle ��1V to the space Sec�^dimB
T �B�. Then the transpose operator tE� is given by the formula
t�E���1�=�j2r�
���� ;
where �� 2 V � Y2r^dimBT �B, and the pull-back of V ��valued di¤erential form
�j2r�
����
is a cross-section of the vector bundle ��1V � ^dimBT �B.
52
Hyperspaces of Riemannian manifolds related to the Hausdor¤ dimension
Nataliia Mazurenko
Di¤erent authors (see, e.g. [1�3]) considered the hyperspaces of compact subsets andsubcontinua of given Lebesgue dimension. In particular, in [2] the topology of the system ofhyperspaces of given Lebesgue dimension in the Hilbert cube is described. In [3] this resultis extended over the case of hyperspaces of countable in�nite products of nondegeneratedPeano continua and in [1] over the case of a Peano continuum in which every open setcontains sets of arbitrary �nite dimension.The author [4, 5] obtained counterparts of the above mentioned results for the hyper-
spaces of compacta and continua of given Hausdor¤dimension in the cube [0; 1]n. The aim ofthe talk is to extend these results over the case of the hyperspaces of Riemannian manifolds.ByQ we denote the Hilbert cube [�1; 1]!, and byB(Q) = f(xi) 2 Q j xi 2 f�1; 1g for some ig
the pseudoboundary of Q. Further, HD> (X)(HDc> (X)) is the hyperspace of compacta
(continua) X of the Hausdor¤ dimension > .The following is the main result of the talk.
Theorem. Let n 2 N, X be an n-dimensional compact connected Riemannian manifold,and � be some countable ordered set.(1) If � � [0; n) then there is a homeomorphism � : exp(X)! Q� such that for every 2 �
�[HD> (X )] =[ 0�
Y 00 6= 0
Q 00 � B(Q) 0!:
(2) If n � 2 and � � [1; n) then there is a homeomorphism � : expc(X)! Q� such that forevery 2 �
�[HDc> (X)] =
[ 0�
Y 00 6= 0
Q 00 �B(Q) 0
!:
[1] R. Cauty Suites F�-absorbantes en theorie de la dimension // Fundamenta Mathemat-icae �1999 �Vol. 159 �N 194 2.P.115-126.
[2] J.J. Dijkstra, J. van Mill and J. Mogilski. The space of in�nite-dimensional compactaand other topological copies of (l2f )
! //Paci�c J. Math. - 152(1992). - P. 255 - 273.
[3] H. Gladdines, Absorbing systems in in�nite-dimensional manifolds and applications. -Amsterdam: Vrije Universiteit. - 1994. - 117p.
[4] N. Mazurenko, Absorbing sets related to Hausdor¤ dimension// Visnyk Lviv Univ.,Ser. Mech-Math. - 2003. - Vol.61. - P.121-128.
[5] N. Mazurenko, Topology of the hyperspaces of continua of given Hausdor¤ dimensionin a �nite-dimensional cube//Nauk. visn. Cherniv. univ. Matematyka. - 2004. - 228. -P.60 �65.
53
Underdetermined systems of ODEs �the geometric approach of E.Cartan
Piotr Mormul
The equationz0 = (y00)
2 (1)
(derivatives are with respect to an independent variable x) canNOT be solved by givingparametric expressions x(t) = ', y(t) = , z(t) = �, where '; ; � are certain �xedfunctions of t; f(t); f 0(t); : : : ; f (n)(t) and f is a free function of one variable t. (The ordern of the highest derivative involved is also �xed, not depending on f .) This was shown in1912 by D.Hilbert, [H]; the equation itself was very popular in the 1910s. Other researchersactive around that time in the �eld of underdetermined systems of ODEs were E.Cartan,E.Goursat, P. Zervos. The �rst of them published in 1914, [C], a theorem giving a localcharacterization of the underdetermined systems, of degree of underdeterminacy 1, havingthe mentioned property of parametrization of solutions. And Hilbert�s example (1) simplydid not satisfy that characterization. (Very close topics had also been addressed by Zervosin [Z].)
That is to say, Cartan proposed a theorem subsuming Hilbert�s result. Th equation (1)turns out to have a too rich geometry (so-called special Cartan �2, 5�geometry) precludingsuch parametrizations of solutions. The accent in the talk will be precisely on the geometricside of the problem, partially following a modern approach of A.Kumpera in [K]. Also anopen question will be formulated, falling close to Cartan�s theorem.
References
[C] E.Cartan; Sur l�équivalence absolue de certains systèmes d�équations di¤érentielleset sur certaines familles de courbes, Bull. Soc. Math. France 42 (1914), 12 �48.
[H] D.Hilbert; Uber den Begri¤ der Klasse von Di¤erentialgleichungen, Fest-schriftHeinrich Weber (1912), 130 �146.
[K] A.Kumpera; Flag systems and ordinary di¤erential equations, Annali di Matemat-ica Pura ed applicata CLXXVII (1999), 315 �329.
[Z] P. Zervos; Sur l�integration de certains systemes indetermines d�equations di¤eren-tielles, Journal de Crelle (J. reine angew. Math.) 143 (1913), 300 �312.
54
Surgery on strati�ed manifolds
Yuri V. Muranov
The question of whether an element x of the Wall group Ln(�) is the surgery obstructionof a degree-one normal map of closed manifolds with �1(X) = � is one of the basic problemsin surgery theory. This question is equivalent to studying if x belongs to the image of theassembly map A : Hn(B�;L�)! Ln(�). Note that any element x 2 Ln(�) is represented bya normal map of closed manifolds with boundaries.Let X be a strati�ed manifold in the sense of Browder and Quinn. A pair Y n�q � Xn of
closed manifolds in the sense of Ranicki is the simplest case of a strati�ed manifold. In thiscase, the splitting obstruction groups LSn�q(F ) are de�ned, where
F =
0BB@�1(@U) ! �1(X n Y )# #
�1(U) ! �1(X)
1CCAis the push-out square of fundamental groups for the splitting problem. All elements of thegroups LS�(F ) are realized by maps of manifolds with boundaries. The problem of realiz-ability splitting obstructions by homotopy equivalences of closed manifolds closely relates tocomputing of the Assembly map.We give a generalization of the splitting theory to the case of strati�ed manifolds. The
structure of a strati�ed manifold provides new algebraic approach to computing of the As-sembly map. In the particular case of a Browder-Livesay �ltration this approach closelyrelates to the approach that is based on conception of iterated Browder-Livesay invariantsdeveloped by Hambleton and Kharshiladze. We describe new results in surgery on �lteredmanifolds and relations to classical surgery theory. We consider several examples and givean application of our results to the closed manifolds surgery problem and to the problem ofrealization of splitting obstructions by simple homotopy equivalences of closed manifolds.
55
Orbits structure of co-adjoint action and bystages hypothesis
Ihor V. Mykytyuk
We shall describe the structure of orbits of the coadjoint action of a connected Lie groupon the dual space to its Lie algebra. Classical results of Kostant give a fairly completeinvariant-theoretic picture of the (co)adjoint action in the case of a reductive Lie group. Ourresult is a generalization of the results of Rawnsley [1], Guillemin and Sternberg [2] and Raïs[3] obtained for semi-direct product of a Lie group and a linear space. This linear space isa normal commutative subgroup. We shall consider the general case of a non-commutativenormal subgroup.Let G be a connected real (or complex) Lie group with the Lie algebra g. Consider the
coadjoint representation Ad� of the Lie group G on the dual space g�. Let O� = Ad�(G)�be an orbit in g� through an arbitrary covector � 2 g�.Suppose that G is not a simple Lie group. Let A � G be a normal closed subgroup of G
with the Lie algebra a � g. Since the subalgebra a is an ideal of g, the adjoint representationsof G in g induces the representation � of G in a, and consequently the action of G in a�
(associated with the dual representation ��). Denote by O0� the corresponding orbit in a
�
through the element (restriction) � = �ja. Then O0� = G=G� , where G� is the isotropy group
of �.We shall prove that the G-orbits O� in g� are �bered over the G-orbits O0
� in a�; its �bre
type is a direct product of some vector space V� and a coadjoint orbit of some (not necessaryconnected) Lie group Q� , which is a factor group of the isotropy group G� . There is a oneto one correspondence between the set of all G-orbits O� with �ja = � and the set of allcoadjoint Q�-orbits in the dual space of the Lie algebra of Q� . The space V� is the same forall these orbits O� (with �ja = �) and has a dimension equal to the codimension of the Liegroup G� � A in G.Let g� be the Lie algebra with the Lie group G� . As a corollary, from the proof of the
above described result, we obtain the following statement ("bystages hypotesis" [4]):
(BH)Let �1, �2 be covectors from g� such that for its restrictions to subspace a and g� wehave �1ja = �2ja = � and �1jg� = �2jg� = � . Then �2 = Ad
�g �1 for some g 2 (G�)� ,
where (G�)� is the isotropy group of � 2 g�� for the coadjoint action of G� on g�� .[1] J. H. Rawnsley, Representation of a semi-direct product by quantization, Math. Proc.
Cambr. Phil. Soc., 78 (1975), 345�350.
[2] V. Guillemin and S. Sternberg, The moment mat and collective motion, Annals ofPhysics, 127 (1980), 220�253.
[3] M. Raïs, L�indice des produits semi-directs E�� g, C. R. Acad. Sci. Paris Ser. A, 287(1978), 195�197.
[4] J. Marsden, G. Misio÷ek, M. Perlmutter, and T. Ratiu, Symplectic reduction for semi-direct products and central extensions, Di¤. Geom. and its Appl. 9 (1998), 173�212.
56
Di¤erential of superorder following Koszul and KV cohomologyfollowing Nijenhuis
Michel Ngui¤o Boyom
In [Homologie des formes di¤érentielles d�ordre supérieur, Ann. Sci. Ec. Norm. Sup.,1974] J.-L. Koszul pointed out some (canonical) non vanishing cohomology classes such asthe divergence class. At anoter side [Sur quelques propriétés communes r di¤érents typesd�algµcbres, Enseignement Mathem, 1986] by A. Nijenhuis is the pionering work on the co-homology theory of Koszul-Vinberg algrbras. The relationships between these two works isthe subject of this talk.
A¢ ne special Lagrangian submanifolds of Cn
Barbara Opozda
The theory of special Lagrangian submanifolds or, in other words, minimal totally realsubmanifolds of complex space forms is very rich and has large literature. An a¢ ne versionof the theory is proposed. It essentially extends the Riemannian case as there are manyexamples of a¢ ne Lagrangian submanifolds which cannot be Lagrangian in the Riemanniansense even locally. Minimality is studied from various viewpoints and a few interpretationsof the notion is given. In particular induced volume forms (complex and real) and integralformulas, the second variational formula, an interpretation by a¢ ne calibrations and phasesare discussed. Examples are provided.
57
Projective structures and invariant di¤erential operators
Valentin Ovsienko
A (�at) projective structure on a smooth manifold M is a local identi�cation of M withthe real projective space. We will discuss the problem of existence and classi�cation ofprojective structures. In particular, we give a simple proof of the following theorem: twoprojective structures which are homotopic in the class of projective structures with the samemonodromy, are equivalent. This is a multi-dimensional analog of the classical result in thecase of the circle (Kuiper-Kirillov-Segal-...).
An analogue of the Kazhdan�s property (T) for operator algebras(joint with Evgenij Troitsky)
Alexander Pavlov
In the talk the property (T) at least at one point of the spectrum of a C�-algebra will bediscussed. It is an analogue of the Kazhdan�s property (T). This property was de�ned forW �-algebras by A. Connes and V. Jones [2] and for C�-algebras by B. Bekka [1]. But ourapproach is di¤erent from their one. Namely we consider this problem from the topologicalpoint of view and it allows to solve the following rather subtle problem in the theory ofC�-Hilbert modules.We prove that a separable unital C�-algebra A has property MI (module-in�nite, i. e.
any C�-Hilbert module over A is self-dual if and only if it is �nitely generated projective) ifand only if it does not have the property (T) at least at one point of the spectrum. The talkis based on the results of [3, 4].
References
[1] B. Bekka, A property (T) for C�-algebras, arXiv:math.OA/0505189, 2005.
[2] A. Connes and V. Jones, Property T for von Neumann algebras, Bull. London Math.Soc., 17, 1985, N 1, 57�62.
[3] A.A. Pavlov, E.V. Troitsky A C�-analogue of Kazhdan�s property (T), Max-Plank-Institut f�ur Mathematik, Preprint Series 2006 (82).
[4] A.A. Pavlov, E.V. Troitsky A C�-analogue of Kazhdan�s property (T), arXiv:math.OA/0606724, 2006.
58
Node points of Sen-Witten equations and positive energy theorem
Wolodymyr Pelykh
The existence of the tensor method for the positive energy theorem (PET) proof ingeneral relativity was unanimously declared impossible owing to existence of node points forDirac equation in R3 or Sen-Witten equation on asymptotically �at manifolds. We give thecorrect tensor proof of the PET on the base developed by us a new approach for establishingthe conditions of the Dirichlet problem solvability and zeros absence for general-covariantand locally SU(2)-covariant elliptic system of equations, which contains in particular caseDirac and Sen-Witten equation.We obtain the new condition of the Dirichlet problem solvability and the condition of
zeros absence for solutions of this general-covariant and locally SU(2)-covariant system
1p�h
@
@x�
�p�hh�� @
@x�uA
�+ CA
BuB = 0; (1)
where h�� � components of the metric tensor in V 3; they are arbitrary real functions ofindependent real variables x�, continuous in and the quadratic form h������ is negativelyde�ned. The unknown functions uA of independent variables x� are the elements of complexvector space C2, in which the skew symmetric tensor "AB is de�ned, and the group SU(2)acts. CAB is Hermitian (1; 1) spinorial tensor.On this basis we prove further that Sen-Witten equation have not node points if ini-
tial data set is asymptotically �at, dominant energy condition is ful�lled and at least onecomponent of Sen-Witten spinor �eld asymptotically nowhere equals to zero.Our work is next substantial argument after [1] in favour of geometrical nature of the
Sen-Witten spinor �eld.
References
[1] Pelykh V. Comment on "Self-dual teleparallel formulation of general relativity andthe positive energy theorem" // Phys.Rev.D. �2005. �Vol. 72. �108502.
59
Around Birkho¤Theorem
O. Petrenko and I. V. Protasov
Let X be a topological space, f : X ! X. A point x 2 X is said to be recurrent if, forevery neighbourhood U of x and every n 2 !, there exists m > n such that fm(x) 2 U . ByBirkho¤Theorem, every continuous mapping f : X ! X of a compact space has a recurrentpoint.We say that a topological space X is totally recurrent if every mapping f : X ! X has
a recurrent point.
Theorem. A Hausdor¤ space X is totally recurrent if and only if X is either �nite ora one-point compacti�cation of an in�nite discrete space.
Problem 1. Detect all totally recurrent T1�spaces.Problem 2. Characterize all Hausdor¤ spaces in which every injective (resp. surjective,bijective) self-mapping has a recurrent point.
Problem 3. Describe all Hausdor¤ spaces in which every continuous self-mapping hasa recurrent point.
S.Kolyada suggested the following question.
Problem 4. Which compact metric spaces X admit a structure of minimal dynamicalsystem, i.e. a continuous mapping f : X ! X such that the orbit ffn(x) : n 2 !g ofeach point x 2 X is dense in X.
60
Totally singular Lagrangians and a¢ ne hamiltonians
Paul Popescu and Marcela Popescu
The duality between Lagrangians and Hamiltonians is usually associated with a dual-ity between tangent and cotangent spaces. In the case of hyperregular Lagrangians andHamiltonians, the duality is related by a Legendre transformation.Unfortunately, the class of hyperregular Lagrangians and Hamiltonians is too restrictive.
The constraints used by Dirac allow to perform a Legendre transformation in a large classof non-hyperregular Lagrangians and Hamiltonians. We do not use here Dirac�s constraints,but Lagrangians and Hamiltonians that have null vertical Hessians, i.e. the �most singular�Lagrangians and Hamiltonians; they are called here as totally singular ones. The Lagrangiancase is remarked in [1], being related to a classi�cation of singular Lagrangians. Marsdenand Ratiu are concerned in [2] with a special case, called here the regular case, when thevector �eld given by the Euler equation is uniquely determined. We consider in the paperallowed totally singular Lagrangians, i.e. totally singular Lagrangians together with a vector�eld that is a solution of its Euler equation (in [1] one says that the Lagrangian allows aglobal dynamics). This Lagrangians are in duality with some corresponding totally singularHamiltonians, so that the Legendre map sends the integral curves of the Euler equationto integral curves of the Hamilton equation. We prove that every allowed totally singularLagrangian has a dual allowed totally singular Lagrangian, but the converse is true onlylocally. Certain examples of totally singular Lagrangians and Hamiltonians are given.The second part is devoted to the case of allowed totally singular Lagrangians and Hamil-
tonians de�ned on the higher order tangent space T kM , when the vertical Hessians of k-velocities for Lagrangians and of momenta for a¢ ne Hamiltonians respectively are vanishing(see [3, 4] for a setting of Lagrangians and Hamiltonians of higher order). A duality betweentotally singular allowed Lagrangians and a¢ ne Hamiltonians is considered also in this case.Allowed totally singular Lagrangian has a dual allowed totally singular Hamiltonian, but forthe converse situation, assuming some conditions, we prove that an allowed totally singularHamiltonian of order k has a totally singular allowed Lagrangian of order k and both can berelated to ordinary dual (allowed totally singular) Lagrangians and Hamiltonians on T k�1M .In order to have consistent examples of totally singular Lagrangians and Hamiltonians
of higher order, lifting procedures are given. In this way, certain examples considered in the�rst section, or examples considered in [2] or [1] can be lifted to totally singular Lagrangiansand Hamiltonians of higher order.
References
[1] Cantrijn F., Cariñena J.F., Crampin M., Ibort A., Reduction of degenerate La-grangian systems, J. Geom. Phys, 3, (3) (1986), 353-400.
[2] Marsden J., Ratiu T., Introduction to Mechanics and Symmetry, Secon Edition,Springer-Verlag New York, Inc., 1999.
61
[3] Popescu P., Popescu Marcela, A¢ ne Hamiltonians in Higher Order Geometry, Int.J. Theor. Phys. (to appear).
[4] Popescu P., Popescu Marcela, Lagrangians and Hamiltonians on A¢ ne Bundlesand Higher Order Geometry, Proc. vol. VII International Conference Geometryand Topology of Manifolds, The Mathematical Legacy of Charles Ehresmann onthe occasion of the hundredth anniversary of his birthday (B¾edlewo, Poland, May8�15, 2005), Banach Center Publications, Editors Jan Kubarski, Jean Pradines,Tomasz Rybicki, Robert Wolak, (to appear).
62
Topological groups through the looking-glass
I. V. Protasov
A ball structure is a triplet B = (X;P;B), where X;P are non-empty sets and, for anyx 2 X and � 2 P , B(x; �) is a subset of X which is called a ball of radius � around x. Itis supposed that x 2 B(x; �) for all x 2 X, � 2 P . The set X is called the support of B,P is called the set of radii. Given any x 2 X and � 2 P , we put B�(x; �) = fy 2 X : x 2B(y; �)g.A ball structure X is called a ballean (or a coarse structure) if
� for any �; � 2 P , there exist �0; �
0 2 P such that, for every x 2 X,B(x; �) � B�(x; �
0); B�(x; �) � B(x; �
0);
� for any �; � 2 P , there exist 2 P such that, for every x 2 X,B(B(x; �); �) � B(x; ):
Let B1 = (X1; P1; B1); B2 = (X2; P2; B2) be balleans. A mapping f : X1 ! X2 is calleda �-mapping if, for every � 2 P1, there exists � 2 P2 such that, for every x 2 X1,
f(B1(x; �)) � B2(f(x); �):
The category of balleans and �-mappings can be considered (see [1], [3]) as an asymptoticre�ection of the category of uniform spaces and uniformly continuous mappings.A family I of subsets of a group G is called a Boolean group ideal if
� A;B 2 I ) A [B 2 I;
� A 2 I; A0 � A) A0 2 I;
� A;B 2 I ) AB 2 I; A�1 2 I;
� F 2 I for every �nite subset F of G.Every Boolean group ideal I determines the ballean B(G; I) = (G; I; B), whereB(g; A) =
gA. The balleans on G determined by the Boolean group ideals are the natural (see [1, Chap-ter 6]) counterparts of the group topologies on G.We show that, for every countable group G, there are 2{ distinct Boolean group ideals on
G, describe some fragments of the lattice of the Boolean group ideals on G, its interactionwith T -sequences from [2], and apply these results to the Stone-µCech compacti�cation �Gof a discrete group G.
REFERENCES
[1] I.Protasov, M.Zarichnyi, General Asymptology, Math.Stud. Monogr.Ser. 13, 2006.
[2] I.Protasov, E.Zelenyuk, Topologies on Groups Determined by Sequences, 4, 1999
[3] J.Roe, Lectures on Coarse Geometry, University Lectures Series, 31, Amer.Math.Soc.,Providence, R.I, 2003.
63
Generalized retracts concerned to free topoplogical groups
Nazar M. Pyrch
Let F (X) be free topological group over a Tychono¤ space X. A subspace Y of Xis called an F-retract of X if there exists a homomorphism h : F (X) ! F (Y ) such thath(y) = y for all y 2 Y . The following result generalizes Theorem 2.5 from [1].
Proposition 1. Let K be an F -retract of X. Then free topological groups over thespace X and (X=K) _K are topologically isomorphic.
A topological space X is called retral [2] if X is a retract of some topological group G.
Proposition 2. The following are equivalent for a topological space X:� X is a retral space;� for any space Y containing X as an F -retract there exists a retraction Y ! X.
REFERENCES
[1] Okunev O.G. A method for constructing examples of M-equivalent spaces. Top. Appl.�1990. �V.36. �P. 157�171;
[2] Gartside P.M., Reznichenko E.A., Sipacheva O.V. Mal�tsev and retral spaces. Top. Appl.�1997. �V.80. �P. 115�129.
64
Natural and projectively equivariant quantization
Fabian Radoux
One deals in this work with the existence and the uniqueness of natural projectivelyequivariant quantizations by means of the theory of Cartan connections.One shows that a natural projectively equivariant quantization exists for di¤erential
operators acting between � and �-densities if and only if the corresponding sl(m + 1;R)-equivariant quantization on Rm exists. With this end in view, one writes the quantizationby means of a formula in terms of the normal Cartan connection associated to the projectivestructure of a connection.One deduces next an explicit formula for the natural projectively equivariant quantiza-
tion.One shows the non-uniqueness of such a quantization by means of the curvature of the
normal Cartan connection.Finally, one shows the existence of natural and projectively equivariant quantizations for
di¤erential operators acting between sections of other natural �ber bundles transposing themethod used in Rm to analyse the existence of sl(m + 1;R)-equivariant quantizations, thismethod being linked to the Casimir operator.
65
On in�nite Lie pseudo-groups and �ltered Lie algebras
Alexandre A. Martins Rodrigues
All di¤erentiable manifolds, vector �elds and maps are assumed to be of class C1.LetM be a manifold and � a sheaf of germs of local vector �elds de�ned onM . We shall
denote by Jk� and JkT (M), k � 0, respectively the set of all k-jets of local sections of �and the vector bundle of k-jets of local sections of the tangent bundle T (M) of M . Given apoint a 2 M , �a; Jka�; JkaT (M) denote respectively the stalk of � at a, the set of all k-jetsjka� at the point a of a local section � of �, de�ned in a neigborhood of a and the �ber ofJkT (M) over a.De�nition. � is an in�nitesimal Lie pseudo-group de�ned on M (ILPG) if,
1. For every a 2 M , �a is a Lie algebra over R under the Lie bracket of germs of vector�elds.
2. There exists an integer k0 satisfying following conditions:
(a) For all k � k0, Jk� is a di¤erentiable vector sub-bundle of JkT (M)
(b) A local vector �eld � is a section of � de�ned on the open set U �M if and onlyif jk0a 2 Jk0a , for all a 2 U .
� is a transitive ILPG on M if J0� = T (M).Let La be the transitive �ltered Lie algebra of in�nite jets of local vectors de�ned in
a neighborhood of a 2 M [5]. Endowed with the topology de�ned by the �ltration, La isalso a topological Lie algebra. We denote by L(�; a) the closure in La of J1a � � La. If �is transitive on M , L(�; a) is a transitive �ltered subalgebra of La and also a topologicalsubalgebra of La.Let �1 be an ILPG de�ned on M1. An ILPG �2 de�ned on M2 is a homeomorphic
prolongation of �1 if there exists a submersion � : M2 ! M1 and for every a1 2 M1 anda2 2 M2 with �(a2) = a1, (�2)a2 is projectable by � onto (�1)a1 . By de�nition, �2 is anisomorphic prolongation of �1 if, moreover, for every germ �1 2 (�1)a1, there exists only one�2 2 (�2)a2 whose projection is �1.The ILPGs �1 and �2 are equivalent in the sense of E. Cartan, [1], [3], [4], if there exists
an ILPG �3 de�ned on a manifold M3 which is an isomorphic prolongation of M1 and M2.We say that �1 and �2 are locally equivalent at points a1 2 M1 and a2 2 M2 if there areopen neighborhoods U1 and U2 of a1 and a2 for which the restrictions �1jU1 and �2jU2 areequivalent.In the following theorem, we assume that the manifolds M1 and M2 and the ILPGs �1
and �2 are real analytic.
Theorem [2]. Let �1 and �2 be two transitive ILPGs de�ned on M1 and M2. �1 and �2are locally equivalent in the sense of E. Cartan, at points a1 2 M1 and a2 2 M2 if and onlyif L(�1; a1) and L(�2; a2) are isomorphic topological Lie algebras.
66
References
[1] A. A. M. Rodrigues, On in�nite Lie groups, Ann. Inst. Fourier, 31 (1981), 245-274.
[2] A. A. M. Rodrigues, A. Petitjean, Correspondance entre algèbres de Lie abstraiteset pseudo-groups de Lie transitifs, Ann. of Math., 101 (1975), 268-279.
[3] E. Cartan, La structure des groups in�nis, Oeuvres Complètes II, vol. 2, 1335-1384.
[4] M. Kuranishi, On the local theory of continuous in�nite pseudo-groups, I, II,Nagoya Math. J., 15, (1959), 225-260; 19, (1961), 55-91.
[5] I. M. Singer, S. Sternberg, The in�nite groups of Lie and Cartan, J. D�AnalyseMath., 15, (1965), 1-114.
67
A note on the reconstruction problem for factorizable homeomorphism groupsand foliated manifolds
Matatyahu Rubin
Let G be a group of homeomorphisms of a topological space X, and H be a group ofhomeomorphisms of a topological space Y . I shall explain some general theorem whichstates that under suitable assumptions on X; Y;G and H, every isomorphism ' between Gand H is induced by some homeomorphism � between X and Y . That is,
'(g) = � �g ���1; for every g 2 G:
The general method is used in order to prove the theorems described below. For a group Gof homeomorphisms of a regular topological space X and an open U � X, set GU := fg 2G j g j(X n U) = Idg. We say that G is a factorizable group of homeomorphisms, if forevery open cover U of X,
SU2U GU generates G.
Theorem A Let G;H be factorizable groups of homeomorphisms of X and Y respectively,and suppose that G;H do not have �xed points. Let ' be an isomorphism between G andH. Then there is a homeomorphism � between X and Y such that '(g) = � �g ���1 forevery g 2 G.Theorem A strengthens known theorems in which the existence of � is concluded from theassumption of factorizability and some additional assumptions.Theorem B For ` = 1; 2 let (X`;�`) be countably paracompact foliated (not necessarilysmooth) manifolds and G` be either the group of leaf-preserving homeomorphisms of (X`;�`)or the group of homeomorphisms of (X`;�`) which take every leaf to itself. Let ' be anisomorphism between G1 and G2. Then there is a leaf-preserving homeomorphism � betweenX1 and X2 such that '(g) = � �g ���1 for every g 2 G1.
68
Geometric calculus associated to second-order dynamics and applications
Willy Sarlet
The main part of the lecture is a review of the geometric calculus I have been using forabout 15 years in the study of (primarily) second-order ordinary di¤erential equations, plusa sketch of its most successful applications.First, we recall the notion of forms and vector �elds along the tangent bundle projection
� : TM !M , the general concept of derivation of such forms and the need for a connectionin the classi�cation problem. A second-order system provides such a connection in a canoni-cal way (Sode-connection). We further motivate the calculus along � by showing how mostgeometrical objects of interest in applications arise from tensor �elds along � via suitablelifting operations, and conversely, the decomposition of tensor �elds on the full tangent bun-dle into horizontal and vertical components, identi�es the essential ingredients of the theoryas being sections of an appropriate pullback bundle. An important class of derivations arethe self-dual derivations of degree zero. Such derivations play a key role in the linearizationof the Sode-connection, which gives rise to a so-called connection of Berwald type on thepullback bundle � �TM ! TM . Other important ingredients of the geometric calculus arethe dynamical covariant derivative r (another self-dual derivation of degree zero) and theJacobi endomorphism � (a type (1,1) tensor �eld along �). They are the main tools, forexample, in the �most economical�description of symmetries and adjoint symmetries of thegiven dynamical system, which is a generalization of the equation of geodesic deviation inRiemannian geometry, plus its dualization. Another �eld of application is the study of themultiplier problem in the inverse problem of the calculus of variations. But perhaps themost successful application so far concerns the geometric characterization of decoupling ofsecond-order equations, both into real or complex single equations, where the theory at thesame time provides algorithmic procedures for the construction of coordinates in which thedecoupling takes place.We will brie�y indicate also, how the original theory for autonomous second-order equa-
tions has been extended to time-dependent systems, to higher order di¤erential equations, tomixed �rst and second-order equations with applications to non-holonomic mechanics, andeven to what are called second-order equations in the context of dynamics on Lie algebroids.Finally, we will mention some applications in progress where the tools of our geometrical
calculus make their appearance also in studies which are as diverse as: an exterior di¤erentialsystems approach to the inverse problem of Lagrangian mechanics, the direct constructionof a hierarchy of �rst integrals for the geodesic �ow of a Finsler manifold, and the geometriccharacterization of so called �driven cofactor pair systems�.
69
Manifolds modeled on the direct limits of Tychonov cubes
Oryslava Shabat
In the paper [1], the in�nite-dimensional model space
I(�) = lim�!fI�0 �! I�0 � f0g �! I�0 � I�1 �! : : : g;
where � = (� 0; � 1; : : : ) is a sequence of ordinal numbers such that ! < � 0 � � 1 � : : : , isconsidered and characterized.The aim of the talk is to develop a theory of in�nite-dimensional manifolds modeled
on the space I(�), in particular, to prove the characterization, stability, open and closedembedding theorems in the spirit of [2].
[1] O. Shabat andM. Zarichnyi, Universal maps of k!-spaces, Matem. studii 21 (1) (2004),71�80.
[2] K. Sakai, On R1-manifolds and Q1-manifolds, Topol. Appl. 18 (1984), 69�79.t
Vector �elds on 4-manifolds
Vladimir Sharko
De�nition 1. The vector �eld X on smooth closed manifold M4 belong to the class W (T 2)if the set of non-wandering points of X consist of a disconnected union of embedded 2-toriwith have normal hyperbolic structure.
Theorem 1. On M4 exist vector �eld X from W (T 2) with Lyapunov function f , who isT 2-Bott function such that pre-image its any regular point are union 2-torus bundles overcircle,if and if M4 is semi- graph manifold.
The function f generates of the Kronrod-Reeb graph �(f). We describe combinatorialconditions of the �(f).Using classi�cation of Morse functions on surfaces obtained classi�cation L - equivalent
vector �elds from W (T 2) on semi- graph manifold M4.By de�nition two vector �elds X and Y are L - equivalent if they have topological
equivalent Lyapunov functions.
70
The order of the di¤erentablity of horizons
David Szeghy
Let L be a Lorentzian manifold. A topological hypersurface H � L is a horison, if forevery point p 2 H there is a past oriented past inextendable light-like geodesic : [0; �)! L,such that lies in H, i.e. (t) 2 H; 8t 2 [0; �) and p = (0). The geodesic is called agenerator. For example a Cauchy horizon or a black-hole event horizon is a horizon. We willshow, how can the order of the di¤erentiablity of the horizon vary along a generator, andgive some examples.
On stable orbit types of isometric actions on Lorentz manifolds
János Szenthe
The classi�cation of the orbit types of an isometric action on a Lorentz manifold involvesproblems which are not present in the Riemannian case. Namely, in the Riemannian case,assuming that the acting group is closed in the full isometry group of the manifold, theaction is proper [3] and thus the results concerning classi�cation of orbit types in case ofproper actions apply [2]. However, proper actions have compact stabilizers and thus are butexceptions in case of Lorentz manifolds.As a possible approach to the cassi�cation of orbit types of isometric actions on Lorentz
manifolds, the concepts of stable and unstable orbit types were introduced [1]. Moreover, itwas shown that if an isometric action on a Lorentz manifold has only orbits of stable andunstable types and the set of unstable orbit types is countable, then the union of orbits ofstable type is an open and dense set [1].Thus the problem arises to give conditions under which an isometric action on a Lorentz
manifold has only orbits of stable and of unstable types. The following result will be pre-sented: If a Lorentz manifold is geodesically complete and has no conjugate points then anyisometric action on this manifold has only stable and unstable orbit types.
References
[1] Alekseevsky, D. V. and Szenthe, J., Orbits of stable type in Lorentzian G-manifolds.In preparation.
[2] Palais, R., On the existence of slices for actions of non-compact Lie groups. Ann.of Math. 73(1961), 295-323.
[3] Yau, Shing Tung, Remarks on the group of isometries of a Riemannian manifold.Topology 16(1977), 239-247.
71
The Bases of Di¤erential Geometri of Vector Field in n - measure Spease ofA¢ ne Connection
Petro Tadeev
Series of invariant straights, hypersur�ces and hyperquadrics for vector �eld of n-dimensionalspace of a¢ ne connection have been built. The �splitting� of geometric models of vector�eld in transition from a¢ ne space to space of a¢ ne connection.
Modi�ed Hochschild and Periodic Cyclic Homology
Nicolae Teleman
It is known that the Hochschild and (periodic) cyclic homology of Banach algebras areeither trivial or not interesting. To correct this de�ciency, Connes had produced the entirecyclic cohomology. The entire cyclic cochains are elements of the in�nite product (b; B) coho-mology bi-complex which satisfy a certain bidegree asymptotic growth condition. The entirecyclic cohomology is a natural target for the asymptotic Chern character of ��summableFredholm modules. More recently, Puschnigg introduced the local cyclic cohomology basedon precompact subsets of the algebra in an inductive limits system setting.The main purpose of this paper is to create an analogue of the Hochschild and periodic
cyclic homology which gives the right result (i.e. the ordinary Z2-graded Alexander-Spanierco-homology of the manifold) when applied, at least, onto the algebra of continuous functionson topological manifolds and CW�complexes. This is realized by replacing the Connesperiodic bi-complex (b; B); by the bi-complex (~b; d); where the operator ~b is obtained byblending the Hochschild boundary b with the Alexander-Spanier boundary d; the operator~b anti-commutes with the operator d. The homologies of these complexes will be calledmodi�ed Hochschild, resp. modi�ed periodic cyclic homology.Our construction uses in addition to the algebraic structure solely the locality relationship
extracted from the topological structure of the algebra.The modi�ed periodic cyclic homology is invariant under continuous homotopies, while
the others are invariant at smooth homotopies (di¤eotopies) only.The modi�ed Hochschild and periodic cyclic homology are directly connected to the
Alexander-Spanier cohomology.
72
On the Sectional Curvatures of the Time-like Generalized Ruled Surface in IRn1Murat Tosun and Soley Ersoy
In IRn1 , (k + 1)�dimensional time-like ruled surface is de�ned parametrically as follows
' (t; u1 ; : : : ; uk) = � (t) +kX�=1
u�e� (t)
and denoted as M , where the base curve � of M ruled surface is time-like curve, generatingspace Ek (t) is space-like subspace. If otherwise mentioned, (k + 1)�dimensional time-likeruled surface M is supposed to have a (k �m+ 1)� dimensional central ruled surface.Two-dimensional subspace � of (k + 1)�dimensional time-like ruled surface at the point
� 2 TM (�) is called tangent section of M at point �.If ~v and ~w form a basis of the tangent section �, then Q (~v; ~w) = h~v;~vi h~w; ~wi� h~v; ~wi2 is
a non-zero quantity if and only if � is non-degenerate. This quantity represents the squareof the Lorentzian area of the parallelogram determined by ~v and ~w. Using the square of theLorentzian area of the parallelogram determined by the basis vectors f~v; ~wg, one has thefollowing classi�cation for the tangent sections of the time-like ruled surfaces:
Q (~v; ~w) = h~v;~vi h~w; ~wi � h~v; ~wi2 < 0 ; (time� like plane)
Q (~v; ~w) = h~v;~vi h~w; ~wi � h~v; ~wi2 = 0 ; (degenerate plane)
Q (~v; ~w) = h~v;~vi h~w; ~wi � h~v; ~wi2 > 0 ; (space� like plane)
For the non-degenerate tangent section � given by the basis f~v; ~wg of M at the point �,the de�nition
K (~v; ~w) =hR~v ~w~v; ~wi
h~v;~vi h~w; ~wi � h~v; ~wi2=
PRijkm�i j�k m
h~v;~vi h~w; ~wi � h~v; ~wi2
is called sectional curvature of M at the point �, where ~v =P�i
@@xi
and ~w =P j
@@xj.
Here the coordinates of the basis vectors ~v and ~w are (�0; �1; : : : ; �k) and ( 0; 1; : : : ; k),respectively.A normal tangent vector which is orthogonal to the generating space Ek (t) of M .
n =mX�=1
u��� (t) ak+� (t) + �m+1ak+m+1 (t) ;��m+1 6= 0
�is time-like or space-like vector. This means that the tangent sectional (e� ; n), 1 � � � k, atthe point 8� 2M time-like or space-like. This tangent section is called �th principal tangentsection of M . Thus, whether �th principal tangent section is time-like or space-like, we cangive following theorems and corollaries.
Theorem 1. Let M be a generalized time-like ruled surface with central ruled surface andn be non-degenerate normal tangent vector in IRn1 .Curvature of (e� ; n), 1 � � � k, non-degenerate �thprincipal sectional curvature of M , at the point 8� 2M is
K� (�) = �1
2g
@2g
@u2v+
1
4g2
�@g
@uv
�2; 1 � � � k:
73
Corollary 2. Let M be a generalized time-like ruled surface with central ruled surface andn be non-degenerate normal tangent vector in IRn1 . �th, 1 � � � m, principal sectionalcurvature and (m+ �)th, 1 � � � k � m, principal sectional curvature of M , at the point8� 2M are
K� (�) = �(��)
2
�mP�=1
(u���)2 � �2m+1 � (u���)
2
��
mP�=1
(u���)2 � �2m+1
�2 ; 1 � � � m;
Km+� (�) = 0 ; 1 � � � k �m;
respectively.
Corollary 3. Generalized time-like ruled surface with central ruled surface has no (m+ �)th,principal sectional curvature at the point 8� 2M for, 1 � � � k �m, in IRn1 .
Theorem 4. Let M be a generalized time-like ruled surface with central ruled surface andn be non-degenerate normal tangent vector in IRn1 . �th, 1 � � � m, principal sectionalcurvature and (m+ �)th, 1 � � � k �m, principal sectional curvature of M , at the centralpoint 8� 2 are
K� (�) =1P 2�
; 1 � � � m;
Km+� (�) = 0 ; 1 � � � k �m;
respectively, where P� =�m+1��
; 1 � � � m, is the �th principal distribution parameter of M .
In IRn1 , if one-dimensional generator h� = Sp fe�g, 1 � � � m, (�th principal ray)moves along the orthogonal trajectory of M , then 2�dimensional ray surface is obtained.This surface is is called �th principal ray surface and denoted by M�, 1 � � � m. Aparameterization of M� is
'� (t; u) = � (t) + ue� (t) ; 1 � � � m:
Now the theorem about non-degenerate sectional curvature of principal ray surfaces canbe given in the following.
Theorem 5. Let M be a generalized time-like ruled surface with central ruled surface andM�, 1 � � � m, be 2�dimensional time-like �th principal ray surface in IRn1 . For � 2 �M , u 2 IR the sectional curvature of M� at the point � + ue� on generator h� = Sp fe�g is
K�+ue� (e�; n) =P 2�
(u2 � P 2� )2 ; 1 � � � m (1)
where P�; 1 � � � m, is the �th principal distribution parameter of M .
Let M�, 1 � � � m, be �th principal ray surface produced by h� = Sp fe�g � Ek (t)along the orthogonal trajectory of M and P�, be the principal distribution parameter of Min IRn1 . The sectional curvature given in equation (1) is the generalized form of the Lamarleformula in IR31, which is the relationship between the Gaussian curvature and principalparameter. Thus, equation (1) is named as Generalized Lorentzian Lamarle Formula by us.
74
References
[1] Beem, J. K., Ehrlich, P. E., Global Lorentzian Geometry, Marcel Dekker Inc. NewYork, 1981.
[2] Blaschke, W., Vorlesungen über Di¤erentialgeometrie, 14. Ay�. Berlin, 1945.
[3] Frank, H., Giering, O., Zur Schnittkrümmung verallgemeinerter Regel�achen,Archiv Der Mathematik, Fasc.1, 32, 86-90, 1979.
[4] Kruppa, E., Analytische und Konstruktive Di¤erentialgeometrie, Wien Springer-Verlag, 1957.
[5] O�Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.
[6] Ratcli¤e, J. G., Foundations of Hyperbolic Manifolds, Department of Mathematics,Vanderbilt University, 1994.
[7] Tosun, M., Kuruoglu N., On (k + 1)�dimensional time-like ruled surfaces in theMinkowski space IRn1 , Institute of Mathematics and Computer Sciences, Vol.10 (3),1997.
[8] Tosun, M., Kuruoglu N., On properties of generalized time like ruled surfaces, Ant.J. Mathematics, Vol.2 (1), 2005.
75
ANR-property of hyperspaces with the Attouch-Wets topology
Rostislav Voytsitskyy
In the given report we give necessary and su¢ cient conditions on a metric space X underwhich the hyperspaces CldAW (X), BddAW (X), CompAW (X) and FinAW (X) are absoluteneighborhood retracts (brie�y ANR�s). This completes a row of results concerning the ANR-property of hyperspaces with the Attouch-Wets topology. For the Attouch-Wets topologysee [Be], [BKS].First, we give some necessary de�nitions, see [Vo], [BV]. We call a metric space (X; d):
� chain connected if for any points x; y 2 X and any � > 0 there is a sequence x =x0; x1; : : : ; xl = y of points of X such that d(xi; xi�1) < � for all i � l; such a sequencex = x0; x1; : : : ; xl = y is called an �-chain linking x and y and l is the length of thischain;
� chain equi-connected if for any � > 0 there is a number l 2 N such that any pointsx; y 2 X can be connected by an �-chain of length � l;
� uniformly locally chain equi-connected [brie�y (ulcec)] (at a subset X0 � X) if 8" >0 9� > 0 8� > 0 9l 2 N such that any points x; y in X (in X0) with d(x; y) < � can beconnected by an �-chain of diameter < " and length � l;
� continuum connected at in�nity if any bounded subset of (X; d) is contained in abounded subset complement of which has only unbounded continuum-connected com-ponents;
� path connected at in�nity if any bounded subset of (X; d) is contained in a boundedsubset complement of which has only unbounded path-connected components.
Let (X,d) be a metric space with a �xed point x0 2 X. We introduce here the followingnotion. We say that a metric space satis�es the property z if 8r > 0 9B � B(x0; r) 8R > 0:B � B(x0; R) 8� > 0 9l 2 N such that each point in X nB can be connected by an �-chainof length � l with some point in X nB(x0; R).Theorem 1. The hyperspace CldAW (X) is an ANR (an AR) if and only if X is uniformlylocally chain equi-connected at each bounded subset and satis�es the property z (and containsno bounded chain-connected components).
Theorem 2. The hyperspace BddAW (X) is an ANR if and only if CldAW (X) is an ANR ifand only if X is uniformly locally chain equi-connected at each bounded subset and satis�esthe property zFor the hyperspace of all non-empty compact subsets of a metric space we have the
following theorem.
Theorem 3. The hyperspace CompAW (X) is an ANR (an AR) if and only if X is locallycontinuum connected and continuum connected at in�nity (and contains no bounded con-nected components).
Theorem 4. The hyperspace FinAW (X) is an ANR (an AR) if and only if X is locally pathconnected and path connected at in�nity (and contains no bounded connected components).
76
References
[Be] G. Beer, Topologies on closed and closed convex sets, MIA 268, Dordrecht: KluwerAcad. Publ., 1993.
[BKS] T. Banakh, M. Kurihara, K. Sakai, Hyperspaces of normed linear spaces withAttouch-Wets topology, Set-Valued Anal. 11 (2003), 21�36.
[BV] T. Banakh and R. Voytsitskyy, Characterizing metric spaces whose hyperspaces areabsolute neighborhood retracts, Topology Appl. (to appear).
[Vo] R. Voytsitskyy, Hyperspaces with the Attouch-Wets topology which are homeomor-phic to `2, preprint.
77
About topological equivalence of some functions
Irina Yurchuk
Abstract. The number of topologically nonequivalent special functions f on the circle is cal-culated. Necessary and su¢ cient condition for topological equivalence of two pseudoharmonicfunctions F;G : D2 ! R is formulated in the terms of their combinatoric diagrams.
Keywords: pseudoharmonic functions, a combinatoric invariant, a topological equivalence.
1) Let f : S1 ! R be a continuous function with a �nite number of local extrema.
De�nition 2. An updown sequence of length n is a sequence of integers i1 < i2 > :::in, suchthat fi1; :::; ing = f1; :::; ng.
If n = 2r � 1 then the number of such sequences is equal to the tangent number Tr(see [1]).
De�nition 3. A function f is called special if arbitrary two local extrema correspond to twodi¤erent critical values.
Theorem 1. The number of topologically nonequivalent special functions f on the circlewith 2n local extrema is equal to Tn.
Denote by G(n) the number of topologically nonequivalent special functions f on thecircle with (2n+ 2) local extrema. In [4] proven that the following estimate is true:
limn!1
logG(n)
n log n= 2:
2) Let F : D2 ! R be a pseudoharmonic function, where D2 is a domain bounded bya Jordan curve. Recall that function F is pseudoharmonic if there is a homeomorphism 'of domain D2 onto itself such that F � ' is harmonic. We will consider a pseudoharmonicfunction which satis�es the following boundary condition: F jS1 is continuous with a �nitenumber of local extrema. It is known that such a function F has only saddle critical pointsin the interior of D2 (see [2], [5]).
De�nition 4. The value c of F is called regular (critical) if the connected components oflevel curves F�1(c) contain critical points (don�t contain critical points and are isomorphicto a disjoint union of segments).
De�nition 5. The value c of F is semiregular if it is neither regular nor critical.
Let us describe the combinatoric diagram of function F . At �rst we construct the(Kronrod-Reeb graph) �K�R of F j@D2. Then, we add to the graph �K�R(F j@D2) level curvesof sets bF�1(ai) and bF�1(ci), where bF�1(ai), bF�1(ci) are subsets of F�1(ai), F�1(ci) whichcontain only critical and semiregular points. Then
P (F ) = �K�R(F jS1)[i
bF�1(ai)[j
bF�1(cj);78
where every ai is critical value and every cj is semiregular value. By using the values offunction, we can put the strict partial order on vertices in the following way: v1 > v2 ()F (x1) > F (x2), where v1; v2 2 P (F ), x1; x2 are points which correspond to vertices v1; v2. IfF (x1) = F (x2) then v1 and v2 are non-comparable (see [3]).
Theorem 2. Two pseudoharmonic functions F and G are topologically equivalent if andonly if there is an isomorphism ' : P (F ) ! P (G) between their combinatoric diagramswhich preserves the strict partial order given on them.
References
[1] V.I. Arnold, Bernoulli-Euler updown numbers, associated with function singulari-ties, their combinatorics and a mathematics, Duke Math. J., 63 (1991), 537�555.
[2] M. Morse, The topology of pseudoharmonic functions, Duke Math. J., 13 (1946),21�42.
[3] I.A. Yurchuk, Topological equivalent pseudoharmonic functions. �to apper
[4] I.A. Yurchuk, The combinatorial aspects of topological classi�cation functions onthe circle �to apper in Ukranian Math. J.
[5] N. Stoilov, Lectures on topological foundations of the analytical functions theory (inRussian), Nauka, 1964.
79
Universal maps of in�nite-dimensional manifolds
Mykhailo Zarichnyi
Let R1 = lim�!Rn, Q1 = lim�!Qn, where Q denotes the Hilbert cube. The author [1]constructed a universal map ' : R1 ! Q1 and proved a characterization theorem for thismap as well as its uniqueness up to homeomorphism. A simple construction of this map isfound in [2]. Also, in [2] the local non-homogeneity of ' is established.In the paper [3], a counterpart of the universal map ' is de�ned in the category of k!-
spaces of higher weights. The aim of the talk is to establish some properties of universalmaps such as �brewise stability, �brewise open and closed embedding theorems, preservationby some functorial constructions in the category of compact k!-spaces (cf. [4]). The obtainedresults belong to the �brewise theory of R1-manifolds.We also address to the corresponding questions in the metrizable case; the universal map
of the pre-Hilbert space of �nite sequences onto the convex hull of the standard Hilbertcube in the separable Hilbert space which corresponds to the universal map ' in this case isconstructed in [5].
[1] M. M. Zarichny¼¬, Strongly countable-dimensional resolvents of sigma-compact groups.Fundam. Prikl. Mat. 4 (1998), no. 1, 101�108.
[2] T. Banakh, D. Repov�, On linear realizations and local self-similarity of the universalZarichny¼¬map. Houston J. Math. 31 (2005), no. 4, 1103�1114
[3] O. Shabat andM. Zarichnyi, Universal maps of k!-spaces, Matem. studii 21 (1) (2004),71�80.
[4] M. M. Zarichny¼¬, On universal maps and spaces of probability measures with �nitesupports. Matem. studii 2 (1993), 78�82, 108.
[5] M. Zarichny¼¬, Universal map of � onto � and absorbing sets in the classes of absoluteBorelian and projective �nite-dimensional spaces. Topology Appl. 67 (1995), no. 3,221�230.
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