continua

WEAK SEMICOMPLEXE$ AND THE FIXED POINT THEORY OF TREE-LIKE CONTINUA

RICHARD B. THOMPSON

In this paper we consider a regularity condition for tree-like continua andshow that this condition completely characterizes those tree-like continuawhich admit weak semicomplex structures. This provides a purely topologicaldescription of the largest class of such continua which can be shown to havethe fixed point property by existing algebraic methods.

1. Introduction and definitions. The term continuum will always stand for acompact, connected, Hausdorff space, and we shall use (X) for the collectionof all finite covers of a space X by open sets. A tree is a one-dimensional,acyclic, finite simplicial complex. If X is a continuum and is a finite collec-tion of open subsets of X, then is called a simple chain or a tree chain, ac-cording as the nerve of , N, is an arc or a tree. A continuum X is calledchainable, or tree-like, if there is a cofinal subfamily 0 (X) of (X) suchthat each member of o (X) is, respectively, a simple chain or a tree chain.These spaces have been discussed by Bing in [1], where metric chainable con-tinua are referred to as snake-like.

Recall that a space X is said to have the fixed point property if every mapof X into itself leaves some point fixed. Bing has called the question of whetheror not all tree-like continua have the fixed point property one of the mostinteresting unsolved problems in geometric topology [3; 122]. A discussion ofthe history of this question and a guide to its literature can be found in [16,Chapter II].Our interest in the present paper is in the algebraic approach of applying the

Lefschetz fixed point theorem to tree-like continua. We will say that a spacesatisfies this theorem if every fixed point free self-map has a Lefschetz numberof zero. Since tree-like continua are acyclic, all of their self-maps have Lefschetznumber of one. Hence such spaces have the fixed point property if they satisfythe Lefschetz fixed point theorem. Dyer established this fact for chainablecontinua in [5] by showing that such spaces are quasi-complexes, as defined in[11; 322]. However, an example given by Chamberlin in [4] shows that not alltree-like continua are quasi-complexes.

Since other general settings for the Lefschetz fixed point theorem have beendeveloped, it is natural to ask which tree-like continua can now be treated inthis fashion. In particular, we note the weak semicomplexes defined by thisauthor in [12] and [13], and the Q-simplicial spaces of Knill in [10]. Surprisingly,

Received July 5, 1968; in revised form August 7, 1969. This research was partially supportedby the National Science Foundation under Grants GP-8709 and GP-13623.

211

212 RICHARD B. THOMPSON

although these concepts differ in general, tree-like continua are simple enoughthat the properties of being a quasi-complex, Q-simplicial or admitting a weaksemicomplex structure are all equivalent for such spaces. (In the case of thefirst two properties, this is clear from their definitions and Theorem (1) in[4; 512]. The equivalence of a quasi-complex and a weak semicomplex struc-ture for these continua is established in Proposition (1.3) at the end of thissection.) This leads one to speculate that these theories may be, in a sense,the best possible results in this direction, i.e., if a tree-like continuum satisfiesthe Lefschetz fixed point theorem, it can be shown to do so via one--and henceany one--of these theories.

In the remainder of this section we will define the notion of a regular familyof covers for a tree-like continuum. The existence of such a family will beshown in 2 to be both necessary and sufficient for such a continuum to admita weak semicomplex structure. The concluding section of the paper applies thismain result to the question of the fixed point property of tree-like continua.

Let T be a tree-like continuum and denote by o (T) the family of all treechains in (T). Recall that (T) is given a quasi-order by the relation ofrefinement, where we use > to denote that refines . A set V e o (T)will be called an end set if V intersects at most one other set of and will becalled a branch set if it intersects at least three other sets of . The termspecial set will refer to either an end or a branch set and two special sets willbe called adiacent if they are the ends of a simple chain which contains noother special sets.

If (T), then an ordered list, o (V1, V.), of members of iscalled a subchain of if V, V+I and V V+ .; for 1

_i s 1.

This is called a simple subchain if all of the V’s are distinct and its nerve is anarc.

Let e o (T), X e (T) and suppose that o (V, Vs), is a sub-chain of . A subchain ho (U, Ur) of h will be called a X-neighborhoodof o if there exist integers 1 il _<: i; < i2

_z’ < < ir

_i’ swith

i+ i’+lforl_n_r-- lsuchthatV.,.-.,V,. Uforl n_ r.Suppose that o is as above and that o (W1, W,) is a subchain ofo (T). We say that o is X-close to o if there is a h-neighborhood, ho

(U, U,), of o such that the sets in ho can be used, with possible repeti-tions, to form a h-neighborhood, ?,o (U,, U.), of o such that Ui,U, Ui U, and ]_ :t= 1 for 2 _< m __< u.

Finally, we extend the notion of X-closeness in a pieeewise fashion to entiretree chains. If , e o (T) and X (T), then is X-close to. if for eachspecial set V in there is a set (not necessarily special), W(V), in b such thateach simple subehain o connecting two adjacent special sets V and V’ of hasa h-close subehain, o, of b that runs from W(V) to W(V’),We can now define the main condition on tree covers of a continuum which

we will study.

TREE-LIKE CONTINUA 213

DEFINITION (1.1). Given a tree-like continuum T, a cofinal subset 4 of0 (T) is called regular if for each k e ’ (T) there exists (k) suchthat any e which refines is k-close to .Some examples of regular and nonregular sets of covers will help clarify this

concept. If C is a chainable continuum and is the subfamily of (C) con-sisting of all covers that are simple chains, then is regular. This is easilyverified since each member of has only two special sets. Also, if T is a tree,then the open stars of the vertices in its successive barycentric subdivisionsform a regular family of covers. On the other hand, Chamberlin gives anexample in [4; 514] of a tree-like plane continuum X formed by a "T" with aray spiraling around it. It is easily seen geometrically that the cofinal set ofcovers defined by Chamberlin fails to be regular. In fact, although it is phrasedin terms of chain maps, this is essentially what he proves in his discussion ofthis example.We now establish the necessary algebraic concepts for the introduction of

weak semicomplexes. All chain complexes and homology groups are takenwith rational coefficients, and the notation of [13] will be followed. If X is aspace with , (X) and a > , then v: C(N) --, C(N) will denote theusual chain map induced by a vertex transformation based on set inclusion.The support of a chain c C(N) is written as sup(c) and is the union of all setsin a which appear in simplexes of N that occur with non-zero coeificients in c.The following definition is given in [13; 9].

DEFINITION (1.2). /k weak semicomplex (WSC), S(X) IX, 2, C}, is atriple where X is a compact Hausdorff space; 2 is a function assigning to eachk (X) a cofinal subset 2x of (X) which has a designated coarsest elementCo(k) such that so(k) > X; and C is a function assigning to each k (X) afamily, Cx, of chain maps consisting of one or more chain maps c’C(N) --C(N) for every pair a, 2x such that a > > So(k). Each c Cx has theproperty that if N, then there is a set U k with sup () k sup (c()) U.These chain maps are called antiprojections and are assumed to satisfy thefollowing axioms.

is chain homo-(i) If a > / > , > Co(X), a, f, . 2x and c. c. Cx, then c.topic (-) to cr,"a.

(ii) If a > fl > / > So(X), a, f, . ftx and c, c] Cx, then c rc.(iii) If a > So(X), a ax and c: e Cx, then c. H(N.) -- H(N.) is an idem-

potent endomorphism whose image is exactly the image of the projection homo-morphism p.:H(X) ---> H(N.) where H(X) denotes (eeh homology withrational coefficients.

We will say that each chain map c given above is X-small to indicate thecondition that sup () k/sup (c()) U X.A weak semicomplex, {X, 2, C}, is called simple (SWSC) if for each X (X),

a 2x and c: Cx, c: lcN:C(N) C(N).


The existence of a WSC structure on an arbitrary space X appears to be aless stringent condition than requiring that X be a quasi-complex. However,for tree-like continua we have the following result.

PROPOSITION (1.3). A tree-like continuum T admits a weak semicomplexstructure i] and only i]T is a quasi-complex.

Pro@ By Theorem (4.4) of [13; 19], T has an SWSC structure if and onlyif it is a quasi-complex. Since an SWSC is obviously a WSC, we need onlyshow that the existence of a WSC, S(T) {T, t, C}, implies the existence ofan SWSC structure on T.

Since T is connected, N is a connected complex for each cover . Hencethe Kronecker index (KI) has a constant value on the images of vertices of Nunder each chain map c,# in S(T) and some multiple of c,# will actuMly preservethe KI. If all of the chain maps ca are multiplied by this common factor, theresult will be a new WSC, S’(T), whose chain maps preserve the KI. Byproposition (1.3) of [13; 10], S’(T) is an SWSC.

2. Main theorems. The first result shows that the algebraic condition of theexistence of a WSC structure implies the purely topological condition of havingregular sets of covers.

PROPOSITION (2.1). I] a tree-like continuum T admits a weat semi-complexstructure, then every cofinal subJamily o] o (T) is regular.

Proof. Suppose that {T, ft, C} is a WSC and that (b is cofinal in 0 (T).Let X e (T) and pick X’ e o (T) to be a star refinement of X. Select a cover

(X)suchthat > ao ao(X’)ftx,. Ifwith >,thentake# 2x, such that f > and define a chain map d’C(N) --> C(N) by d c-".owhere c Cx,. Since c is a non-trivial X’-chain map, d is a non-trivial X-chain map. Moreover, since N, is a connected complex, the I(roneeker indexof d(V) is the same for all vertices V of 4. Thus some multiple of d preservesthe KI, and we can assume without loss of generality that KI (d(V)) 1for all V .We will now show that the existence of d implies that b is X-close to . Totally

order the special sets of as (V, V) in such a way that each V is adjacentto some V;() with j(i) < i. Since N, is aeyelic, the set V;() is uniquely de-termined by V.

Let be the simple subehain of that runs from V to V and pick a setW(V) that occurs with positive coefficient in d(V). Since d is a chain mapand N,, is connected, there exists a subehain of such that , begins withW(V) and ends with some set W(W) that occurs with positive coefficient ind(V). One can now verify easily that d, being a X-chain map, implies thatbl is X-close to .

Suppose, as an induction hypothesis, that we have picked some set W(V)with positive coefficient in d(V) for 1 <_ i < n and have found subchains


of that run from W(Vi()) to W(V) and are X-close to the simple subchainsof that connect V ") to V. The same argument used above to find W(V1)

and 1 starting with W(V), can now be applied to start with W(V(’) andfind W(V(’) and a subchain , that is X-close to the simple subchain con-necting V() to V". Thus, by induction, we can show that the conditions ofDefinition (1.1) are satisfied and hence that is regular.

Before showing what is essentially the converse of the last proposition, weneed to establish a technical lemma.

LEMMA (2.2). Suppose that T is a tree-like continuum, and are in o(T), X and ’ are in (T), and X is a star refinement oJ X’. IJ a subchain o(W Wt) oJ is X-close to a simple subchain o (V V,) oJ , thenthere exists a X’-chain map " C(N,.) C(N.) with (V) W and w(V.) Wt

ProoJ. Let o (U U) be a X-neighborhood of 4o whose memberscan be used to form a X-neighborhood Xo (U,, U.) of o as in thedefinition of X-closeness in 1. Also, suppose the integers i, i, andi[, i are as in the definition of X-neighborhood of o. Since Xo is a X-neighborhood of o there are integers 1 k k[ < k k < < k, Nk =twithk+ =k+lforl mu- lsuchthatW,... ,W, Ufor 1 < m < u.

Recall that one of the conditions of the X-closeness of o to o is that j]_ 1 for 2 N m N u. In this range, define the signature of m to be v, where]a - + v, and denote tNs integer by s(m). This is augmented to includem 1 by setting s(1) 1.We are now ready to define the desired chain map " C(N,o) C(No). If

i, N h N i and 1 N n < r, let (V,) ,W , W,,+, whereP {m 1 n and s(m) s(m + 1) 1} and Q {m j, n and s(m)s(m+l) -1}. To cover the case of n randihNi,weset(V)W,. W,. Suppose that 1 N n < r 1 and that [V,V+.] istheoriented simplex in N,. running from V, to V,+, Let

mRn

+ + + +where R {m[i n, and s(m) -1 or s(m) s(m + 1) 1} and S,{mja=n,s(m) -lands(m- 1) 1}. If n= r- landz V,,_,V,I,then we define (z) to be the expression given by the formula above plus[W,W,+ + + [W,,_W,]. Finally, we set (z) equal to zero if is anysimplex in N. other than those considered above.The above definition of on the free generators of C(No) can now be ex-

tended linearly to all of C(N.). Using the fact that is a star refinement of’, one can verify that is a h’-chain map with the required properties.

PROPOSITION (2.3). I] a tree-like continuum T has a regular set o] covers,then T admits a weak semicomplex structure.

216 RICHARD B. TH01IPSON

Proo]. Suppose is a regular cofinal subset of o (T). Given X (T),pick covers k’, )," (T) such that ’ is a star refinement of X and )," is, inturn, a star refinement of ),r. Now select a cover ao a0(X) such that ao isa common refinement of k" and (X’), where is as in definition (1.1) ofthe regularity of . Let 2x { 1 > no(X)} which is cofinal in (T).If/ 2, we will use the fact that is X’-close to to construct a chain map

"(N,) --, C(N).As in the proof of Proposition (2.1), totally order the special sets of as

(V, V) in such a way that each V is adjacent to some V () with j(i) < iand let be the simple subchain of that runs from V; () to V. Since/ ish"-close to , there exist, as in the definition of this property, subchains f of fwith being "-close to , and running from W(V ()) to W(V).By Lemma (2.2) there are h-chain maps C(N,) C(N,) with the prop-

erty that o,(V")) W(V() and 0,(Vi) W(V). Any simplex N, iseither a special set V of or occurs in one and only one of the complexes N,.In the first case, we define 0() to be W(V) while in the latter case, we set(a) o(a). A linear extension of o to all of C(N) gives a ’-chain map intoC(Na). Moreover, since KI (0(V)) 1 for each i, and N is connected, wesee that must preserve the Kronecker index.Assume that a 2x with/ > a and define a chain map c"" C(N.) C(N)

by setting c cot/. Because ’ is a star refinement of , c is a ,-chain mapwhich preserves the Kronecker index. Finally, let C be the function thatassigns to each (T) the set Cx of the chain maps c with f > a and ,x that are defined above.We will now verify that S(T) {T, , C} satisfies the axioms of Definition

(1.2) for a weak semicomplex. If a, and , are as in axiom (i) of that defini-tion, then c. and c. and c. both preserve the KI and have N. as a commonacyclic carrier. Hence, as shown in [7; 111], these chain maps are chain homo-topic. Similar remarks show that axiom (ii) holds and that each c.".I(v.):H(N.) H(N.). Thus c.". is clearly idempotent and, since eachprojection p.’H(T) ---> H(N.) is epimorphic in this case, we have the image ofc,". equal to the image of p.. This completes the proof that S(T) is a WSCstructure on T.The preceding proposition is actually stronger than a converse of Proposition

(2.1). Probably the most useful form in which to summarize these results isgiven in. Theorem (2.4).

THEOREM (2.4).tinuum T.

The following are equivalent conditions on a tree-litce con-

(a) T has a regular set o] covers.(b) Every cofinal collection o] tree chains on T is regular.(c) T admits a weat semicomplex structure.

3. Applications to fixed point theory. The primury use of Theorem (2.4) is to


give a complete geometric topological characterization of those tree-like con-tinua which can be shown to have the fixed point property by known algebraicmethods.

THEOREM (3.1). I] a tree-like continuum T has a regular ]amily o] covers,then T has the fixed point property.

Proo]. By Theorem (2.4), T has a WSC structure and hence satisfies theLefschetz fixed point theorem as shown in [13; 15]. Since T is acyclic, anyself-map has non-zero Lefschetz number and, therefore, leaves some point ofT fixed.One advantage of using algebraic methods in fixed point theory is that several

standard constructions preserve the applicability of these methods. Thefollowing two results illustrate this observation.

PROPOSITION (3.2). The Cartesian product o] the members o] any ]amiIy o]tree-like continua, each o] which has a regular set o] covers, has the fixed pointproperty.

Proof. Suppose {T. Is e A} is such a family of continua. If B is a finitesubset of A, then IX,,, T, admits a WSC structure as shown in [13; 64]. Onecan easily check that l-, T, is acyclic and hence must have the fixed pointproperty.Now, we follow the lead of Dyer [5; 665] and invoke the theorem that if

is a family of compact Hausdorff spaces such that the product of the membersof any finite subcollection of has the fixed point property, then the productof all members of has that property. This gives us the fixed point propertyfor II,, T, as claimed.

PROPOSITION (3.3). I1 T is a tree-lilce continuum possessing a regular Iamilyo] covers, then any retract o] T has the fixed point property.

Proof. Let X be a retract of T. Note that X is acyclic and, as shown in[14], must admit a WSC structure. This implies that X has the fixed pointproperty.

Remark (3.4). Since we now have a complete characterization of thosetree-like continua which admit WSC structures and hence have the fixed pointproperty, it is of interest to see what implications this has about the fixedpoint property for tree-like continua that do not have WSC structures. Thefact that all known contexts for applying the Lefschetz fixed point theorem areequivalent for tree-like continua and that these spaces are particularly suitedin their manner of definition to the algebraic approach gives some hope for anaffirmative answer to the following question which contains Theorem (3.1) andits converse.

Question (3.5). Does a tree-like continuum T have the fixed point propertyif and only if it has, a regular set of covers?


Another factor which indicates the possible truth of this question is thatseveral examples, including those given by Kinoshita in [8] and Bing in [2],show that there do exist acyclic continua that fail to have the fixed point prop-erty and therefore admit no WSC structures. These are, however, not tree-like.On the other hand, if we allow continua of dimension greater than one, the

following situation can occur.

PROPOSlTIO (3.6). There exists a contractible, 2-dimensional continuum Bthat admits no wealc semicomplex structure but still has the fixed point property.

Proo]. Let B be the continuum described by Knill in [9; 42]. He showsthere that B is contractible, 2-dimensional and has the fixed point property.However, he proves that B [0, 1] does not have the fixed point property.Assume for the moment that B has a WSC structure. Since [0, 1] is a polyhe-dron, it has a WSC structure and hence, by Theorem (2.1) in [13; 64], so doesB [0, 1] have the fixed point property since it is acyclic and satisfies theLefschetz fixed point theorem. This contradiction shows that B could nothave had a WSC structure.The general implications of this example for algebraic fixed point theories

are discussed in [15].Finally, we note that a tree-like continuum can fail to have a WSC structure

but still possess the fixed point property for homeomorphisms. The space Xformed by a "T" with a ray spiraling down to it provides such an example.As we commented at the end of 1 of this paper, X has a non-regular cofinalset of covers and thus has no WSC structure. It also contains no indecomposablesubcontinua and hence is shown to have the fixed point property for homeo-morphisms by a result of Hamilton in [6].Whether it is true or false, it does seem that Question (3.5) may provide a

solvable line of attack on the problem of the fixed point property for tree-likecontinua.

REFERENCES

1. R. H. BING, Snake-like continua, Duke Math. J., vol. 8(1951), pp. 653-663.2. t. H. BING, Challenging conjectures, Amer. Math. Monthly, voh 74(1967), pp. 56-64.3. R. I-I. BING, The elusive fixed point property, Amer. Math. Monthly, vol. 76(1969), pp. 119-

132.4. R. CHAMBERLIN, Tree-like continua and quasi-complexes, Duke Math. J., vol. 26(1959),

pp. 511-517.5. E. DYER, A fixed point theorem, Proc. Amer. Math. Soc., vol. 7(1956), pp. 662-672.6. O. HAMILTON, Fixed points under transformations of continua which are not connected in

kleinen, Trans. Amer. Math. Soc., vol. 44(1938), pp. 18-24.7. P. HILTON AND S. WYLIE, Homology Theory, Cambridge, Great Britain, 1960.8. S. KINOSHITA, On some contractible continua without the fixed point property, Fund. Math.,

voh 40(1953), pp. 96-98.9. R. KNILL, Cones, products and fixed points, Fund. Math., vol. 60(1967), pp. 35-46.

10. R. KNILL, Q-simplicial spaces, Illinois J. Math., vol. 14(1970), pp. 40-51.

TREE-LIKE CONTIN 219

11. S. LEFSCHETZ, Algebraic Topology, Amer. Math. Soc. Colloq. Publ., vol. 27, Amer. Math.Soc., Providence, R. I., 1942.

12. R. THOMPSON, On the semicomplexes of F. Browder, Bull. Amer. Math. Soc., vol. 73(1967),pp. 531-536.

13. R. THOMPSON, A unified approach to local and globalfixed point indices, Advances in Mathe-matics, vol. 3(1969) 1-71.

14. R. THOMPSON, Retracts of semicomplexes, (to appear in Illinois Journal of Math. 1971).15. R. THOMPSON, A metatheorem for fixed point theories, Comm. Math. Univ. Carolinae, vol.

11, 4(1970).16. T. vAN DER WALT, Fixed and Almost Fixed Points, Amsterdam Mathematisch Centrum,

Amsterdam, Netherlands, 1963.

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