Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

on

the

axio

matiz

atio

nof

double

pse

udolin

earrangements

Mich

elP

occhiola,

Lau

sanne,

29Sep

tember

2010

Universite

Pierre

etM

arieC

urie,

Paris

1

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

Fig

ure

1:

Scu

lptu

red’A

ngel

DU

ART

E,Lausa

nne,

Suisse.

2

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

summary

3

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

pse

udolin

es

and

double

pse

udolin

es

P≈

D2/{x

∼−

x|x∈

∂D

2}

4

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

Hil

bert,D

.[1

899]G

rundlagen

derG

eometrie.

«It

was

Hilb

ert’saim

togive

asim

ple

axiomatic

characterization

ofreal(E

uclid

ean)geom

etries.H

e

expressed

the

necessary

continuity

assum

ption

sin

terms

ofprop

ertiesof

anord

er.In

deed

,th

ereal

pro

jectiveplan

eis

the

only

desargu

esianord

eredpro

jectiveplan

ew

here

everym

onoton

esequ

ence

ofpoints

has

alim

it,see

the

elegantexp

ositionof

Coxeter

[61].»

Hilb

ert1899,

Kolm

ogoroff1932,

Koth

e1939,

Skorn

jakov1954,

Salzm

ann

1955,Freu

denth

al1957,

and

others

[1]H

elmut

Salzm

ann,

Dieter

Betten

,T

heo

Gru

ndhofer,

Herm

ann

Hah

l,R

ainer

Low

en,

and

Marku

sStrop

pel.

Com

pact

pro

jectiveplan

es.N

um

ber

21in

De

Gru

yterexp

ositions

inm

ath-

ematics.

Walter

de

Gru

yter,1995.

5

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

real

tw

o-d

imensio

nal

proje

ctiv

egeometrie

s

u∗

ℓ

ℓ

u

point

space

line

space

DF

1(H

ilbert

etal.).

Areal

two-dim

ension

alprojective

geometry

isa

topologicalpoin

t-line

projectivein

cidence

geometry

whose

pointspace

isa

projectiveplan

ean

dwhose

line

spaceis

asu

bspaceof

thespace

ofpseu

dolines

ofthe

pointspace.

TH

1(H

ilbert

et

al.).

The

line

spaceof

areal

two-dim

ension

alprojective

geometry

isa

projectiveplan

ean

dthe

pencil

oflin

esthrou

gha

pointis

apseu

doline

ofthe

line

space

(P,L

)→

(L,P

∗)→

(P∗,L

∗)≈

(P,L

)

6

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

arrangements

of

pse

udolin

es

Levi

1926,R

ingel

1956,G

runb

aum

1972,etc.

DF

2.Let

Pbe

aprojective

plane.

An

arrangem

entof

pseudolin

esin

Pis

afinite

family

of

pseudolin

esin

Pwith

theproperty

thatan

ytw

oin

tersectexactly

once.

TH

2(G

oodm

ann,Polla

ck,W

enger

&R

am

fire

scu94).

Every

arrangem

entof

pseu-

dolines

canbe

extended

toa

(realtw

o-dimen

sional)

projectivegeom

etry.

7

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

arrangements

of

double

pse

udolin

es

γ

M(γ

)

DF

3.Let

Pbe

aprojective

plane.

An

arrangem

entof

double

pseudolin

esin

Pis

afinite

family

ofdou

blepseu

dolines

inP

,with

theproperty

thatan

ytw

oin

tersect(tran

sversally)

inexactly

four

points

and

indu

cea

cellstru

cture

onP

,that

is,the

connected

compon

ents

of

thecom

plemen

tof

theunion

ofthe

double

pseudolin

esare

two-cells.

8

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

iso.

classe

ssim

ple

arrang.

three

double

pse

udolin

es

04

24

07

618

437

615

243

2

22

433

232

225

225∗

136

12

64

24

9

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

connectedness

under

mutatio

ns

movin

gcu

rve

TH

3(H

abert

&P.06).

The

spaceof

arrangem

ents

ofn

double

pseudolin

esis

connected

under

mutation

s.

TH

4(F

erte

,P

ilaud

&P.

08).

The

one-exten

sionspace

ofan

arrangem

ent

ofdou

ble

pseudolin

esis

connected

under

mutation

s.

10

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

04

07

18

181

37

15

151

43

431

22

221

33

331

32

321

322

25

251

25∗

25∗1

25∗2

36

64

11

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

04

07

18

181

37

15

151

43

431

251

33

331

25

25∗

25∗1

25∗2

32

321

322

22

221

36

64

12

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

numbers

of

arrangements

n2

34

5

aSn

113

6570181

403533

an

146

153528

nc

bSn

116

11502238

834187

bn

159

245351

nc

J.Ferte,

V.P

ilaud

and

P.2010

–tw

odou

ble

corew

orkstations

at2G

Hz,

three

weeks

-

13

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

sub-a

rrangements

of

size

three

replacem

en

1

11

1

2

22

2

3

33

3

4

44

4

TH

5(H

abert

&P.06).

The

isomorphism

classof

anin

dexedarran

gemen

tof

oriented

double

pseudolin

esdepen

dson

lyon

itschirotope,

i.e.,on

them

apthat

assigns

toecah

triple

ofdistin

ctin

dicesthe

isomorphism

classesof

thesu

barrangem

entin

dexedby

thistriple.

14

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

examples

of

double

pse

udolin

es

U

U∗

u∗

ℓ

ℓ

u

DF

4(H

abert

&P.09).

Acon

vexbody

ofa

projectivegeom

etryis

aclosed

subset

ofpoin

ts

with

non

empty

interior

whose

intersection

with

any

line

isan

interval

ofthat

line.

TH

6(H

abert

&P.09).

Let

Ube

acon

vexbody

ofa

projectivegeom

etry(P

,L).

Then

the

boundary

ofU

isa

double

pseudolin

ein

Pan

dthe

dual

ofU

isa

double

pseudolin

ein

L.

15

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

examples

of

arrangements

of

double

pse

udolin

es

U

V

U∗

V∗

u∗

v∗

u

v

TH

7(H

abert

&P.06-0

9).

The

dual

family

ofa

finite

family

ofpairw

isedisjoin

tcon

vex

bodiesof

a(real

two

dimen

sional)

projectivegeom

etryis

anarran

gemen

tof

double

pseudo-

lines.

Con

versely,an

yarran

gemen

tof

double

pseudolin

esis

isomorphic

tothe

dual

family

ofa

finite

family

ofdisjoin

tcon

vexbodies

ofa

projectivegeom

etry.

16

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

cocycles

of

afa

mily

of

bodie

s

thrcd

ℓ

12

3

4

{2,4,13•

3}∼

{2,4,13•

3}

DF

5.Let

∆be

afinite

indexed

family

ofpairw

isedisjoin

torien

tedcon

vexbodies

ofa

projectivegeom

etry.The

cocyclesof

∆are

theisom

orphismclasses

ofthe

arrangem

ents

∆∪{ℓ}

asℓ

ranges

overthe

spaceof

lines

ofthe

projecctivegeom

etry.

17

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

example

AB

CD

A

B

C

D

12

31

1

11

2

2

2

2

3

3

3

3

{{1•

2•3•},{

1•3•

2•},{1•

2•3•},{

1•3•

2•}}

18

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

dualit

y

AB

CD

A

A

B

B

CC

D

D1

2

31

2

3

11

11

2

2

2

2

3

3

3

3

TH

8(H

abert

&P.09).

Two

finite

indexed

families

ofpairw

isedisjoin

torien

tedcon

vex

bodieshave

isomorphic

dual

arrangem

ents

ifan

don

lyif

theyhave

thesam

eset

ofcocycles

iffthey

havethe

same

setof

extremal

cocyclesiff

theyhave

thesam

echirotope.

19

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

chir

otopes

and

vis.

graph

algorit

hms

TH

9(H

abert

&P.07,A

ngelie

r&

P.03,P.&

Vegte

r96).

The

kedges

ofthe

visibility

graphof

aplan

arfam

ilyof

npairw

isedisjoin

tcon

vexbodies

presented

byits

chirotopeis

compu

tablein

time

O(k

+n

logn)

and

linear

workin

gspace.

20

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

axio

matiz

atio

ntheorem

Let

Ibe

afin

itein

dexin

gset.

DF

6.A

k-chirotope

(ofdou

blepseu

doline

arrangem

ents)

isa

map

χdefi

ned

onthe

setof

triplesof

Isu

chthat

foran

ysu

bsetJ

ofI

ofsize

atm

ostk

therestriction

ofχ

tothe

setof

triplesof

Jis

thechirotope

ofa

double

pseudolin

earran

gemen

tin

dexedby

J.W

eden

oteby

Ck

theset

ofk-chirotopes.

TH

10

(Habert

&P.06).

C3

)C

4)

C5

=C

6=C

7=···

.

21

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

axio

matiz

atio

ntheorem

Let

Ibe

afin

itein

dexin

gset.

DF

6.A

k-chirotope

(ofdou

blepseu

doline

arrangem

ents)

isa

map

χdefi

ned

onthe

setof

triplesof

Isu

chthat

foran

ysu

bsetJ

ofI

ofsize

atm

ostk

therestriction

ofχ

tothe

setof

triplesof

Jis

thechirotope

ofa

double

pseudolin

earran

gemen

tin

dexedby

J.W

eden

oteby

Ck

theset

ofk-chirotopes.

TH

10

(Habert

&P.06).

C3

)C

4)

C5

=C

6=C

7=···

.

n0

12

34

5

aSn

11

113

6570

180403

533

ρSn

11

1214

2415

112nc

2nn

!aSn

6242

822580

692749

566720

bSn

11

116

11502238

834187

τSn

11

1118

541820

nc

2n!b

Sn192

552096

57320

204880

22

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

a4-c

hir

otope

2

22

2

3

33

3

44

4

4

5

55

5

1

11

1

12341245

1253

13 542354

23

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

k-c

hir

otopes

–k-a

rrangements

DF

7.A

k-arran

gemen

tof

double

pseudolin

esis

afinite

indexed

byI

family

τof

simple

closed

oriented

curves

embedded

ina

compact

surface

Sτ

with

theproperties

that(1)

τin

duces

a

regular

celldecom

positionX

τof

Sτ ;

and

(2)an

ysu

bfamily

νof

τof

sizeat

least2

and

at

most

k,con

sideredas

embedded

not

inS

τbu

tin

thecom

pactsu

rfaceS

νdefi

ned

bythe

local

embeddin

gof

νin

Sτ ,

isan

arrangem

entof

double

pseudolin

es.W

eden

oteby

Ak

theset

of

k-arran

gemen

tsan

dby

Ak→

Ck

them

apthat

assigns

toa

k-arran

gemen

tits

chirotope.

TH

11.The

graphof

mutation

son

A5

iscon

nected.

TH

12.A

5→

C5

ison

e-to-one

and

onto.

24

Conferen

ceon

Geo

metric

Gra

ph

Theo

ryM

ichel

Pocch

iola

Cen

treIn

terfacu

ltaire

Bern

oulli

short

bib

lio

graphy

[1]Ju

lienFerte,

Vin

centP

ilaud,

and

Mich

elP

occhiola.

On

the

num

ber

ofsim

ple

arrangem

ents

offive

dou

ble

pseu

dolin

es.D

iscreteC

omput.

Geom

.(to

appear),

2010.Special

issue

devoted

toth

eW

orkshop

onTran

sversalan

dH

elly-type

Theorem

sin

Geom

etry,C

ombin

atoricsan

d

Top

ology,B

anff

Station

,Sep

tember

2009.A

prelim

inary

versionap

peared

inth

eab

stracts

18thfall

Worksh

opC

omput.

Geom

.,Troy,

2008(F

WC

G2008).

[2]L.H

abert

and

M.P

occhiola.

Arran

gements

ofdou

ble

pseu

dolin

es.Subm

ittedto

Disc.C

omput.

Geom

.A

bbreviated

versionin

Proc.

25thA

nnu

.A

CM

Sym

pos.

Com

put.

Geom

.(S

CG

09),

pages

314–323,Ju

ne

2009,A

ahru

s,D

enm

ark.A

partial

abbreviated

versionap

pears

inth

e

Abstracts

12thE

urop

eanW

orkshop

Com

ut.G

eom.p

ages211–214,2006,D

elphes,an

da

poster

versionw

aspresented

atth

eW

orkshop

onG

eometric

and

Top

ologicalC

ombin

atorics(satellite

conferen

ceof

ICM

2006),Sep

tember

2006,A

lcalade

Hen

ares,Spain

.,octob

er2006.

[3]J.

E.

Good

man

,R

.P

ollack,R

.W

enger,

and

T.

Zam

firescu.

Arran

gements

and

topological

plan

es.A

mer.

Math

.M

onthly,

101(9):866–878,N

ovember

1994.

[4]H

elmut

Salzm

ann,

Dieter

Betten

,T

heo

Gru

ndhofer,

Herm

ann

Hah

l,R

ainer

Low

en,

and

Marku

sStrop

pel.

Com

pact

pro

jectiveplan

es.N

um

ber

21in

De

Gru

yterexp

ositions

inm

ath-

ematics.

Walter

de

Gru

yter,1995.

25