Using Brouwer’s Fixed Point Theorem

Abstract

Brouwer’s fixed point theorem from 1911 is a basic result in topology—with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem.

Appendix: Fixed Point Theorems and Homology

1.1 Lefschetz’ Theorem

Fixed point theorems are “global–local tools”: From global information about a space (such as its homology) they derive local effects, such as the existence of special points where “something happens.” Of course, in applications to combinatorial problems we need to combine them with suitable “continuous–discrete tools”: From continous effects, such as topological information about continuous maps of simplicial complexes, we have to find our way back to combinatorial information. In this Appendix we assume familiarity with more Algebra and Algebraic Topology than in other parts of these lecture notes, including some basic finite group theory, chain complexes, etc. As this is meant to be a reference and survey section, no detailed proofs will be given. A main result we head for is Oliver’s Theorem A.7, which is needed in Sect. 4. On the way to this, skim or skip, depending on your tastes and familiarity⁴ with these notions.

A powerful tool on our agenda (which yields a classical proof for Brouwer’s fixed point theorem and some of its extensions) is Hopf’s trace theorem. For this let V be any finite-dimensional vector space, or a free abelian group of finite rank. When we consider an endomorphism g:  V  ⟶  V then the trace trace(g) is the sum of the diagonal elements of the matrix that represents g. The trace is independent of the basis chosen for V. In the case when V is a free abelian group, then trace(g) is an integer.

Theorem A.1 (The Hopf trace theorem)

Let $\Delta$ be a finite simplicial complex, let $f: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$ be a self-map, and denote by f _#i resp. f _∗i the maps that f induces on i-dimensional chain groups resp. homology groups.

Using an arbitrary field of coefficients k , one has

$$\displaystyle{\sum _{i}(-1)^{i}\mathit{\mbox{ trace}}(f_{ \#i})\ \ =\ \ \sum _{i}(-1)^{i}\mathit{\mbox{ trace}}(f_{ {\ast}i}).}$$

The same identity holds if we use integer coefficients, and compute the traces for homology in the quotients $H_{i}(\Delta, \mathbb{Z})/T_{i}(\Delta, \mathbb{Z})$ of the homology groups modulo their torsion subgroups; these quotients are free abelian groups.

This theorem is remarkable as it allows to compute a topological invariant that depends solely on the homotopy class of f, by means of a simple combinatorial counting. The proof for this uses the definition of simplicial homology, and simple linear algebra; we refer to Munkres [58, Thm. 22.1] or Bredon [19, Sect. IV.23].

For an arbitrary coefficient field k, the Lefschetz number of the map $f: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$ is defined as

$$\displaystyle{L_{\mathbf{k}}(f)\:=\ \sum _{i}(-1)^{i}\mbox{ trace}(f_{ {\ast}i})\ \ \in \mathbf{k}.}$$

Similarly, taking integral homology modulo torsion, the integral Lefschetz number is defined as

$$\displaystyle{L(f)\:=\ \sum _{i}(-1)^{i}\mbox{ trace}(f_{ {\ast}i})\ \ \in \mathbb{Z}.}$$

The universal coefficient theorems imply that one always has $L_{\mathbb{Q}}(f) = L(f)$: Thus the integral Lefschetz number L(f) can be computed in rational homology, but it is an integer.

The Euler characteristic of a complex $\Delta$ coincides with the Lefschetz number of the identity map $\mbox{ id}_{\Delta }: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$,

$$\displaystyle{\chi (\Delta ) = L(\mbox{ id}_{\Delta }),\quad \text{where }\mbox{ trace}((\mbox{ id}_{\Delta })_{{\ast}i}) =\beta _{i}(\Delta ).}$$

Thus the Hopf trace theorem yields that the Euler characteristic of a finite simplicial complex $\Delta$ can be defined resp. computed without a reference to homology, simply as the alternating sum of the face numbers of the complex $\Delta$, where $f_{i} = f_{i}(\Delta )$ denotes the number of i-dimensional faces of $\Delta$:

$$\displaystyle{\chi (\Delta )\:=\ f_{0}(\Delta ) - f_{1}(\Delta ) + f_{2}(\Delta ) -\cdots \,.}$$

This is then a finite sum that ends with $(-1)^{d}f_{d}(\Delta )$ if $\Delta$ has dimension d. Thus the Hopf trace theorem applied to the identity map just reproduces the Euler–Poincaré formula. This proves, for example, the d-dimensional Euler polyhedron formula, not only for polytopes, but also for general spheres, shellable or not (as discussed in Ziegler [77]). The Hopf trace formula also has powerful combinatorial applications, see Ziegler [78]. However, for us its main consequence is the following theorem, which is a vast generalization of the Brouwer fixed point theorem.

Theorem A.2 (The Lefschetz fixed point theorem)

Let $\Delta$ be a finite simplicial complex, and k an arbitrary field. If a self-map $f: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$ has Lefschetz number L _k (f) ≠ 0, then f and every map homotopic to f have a fixed point.

In particular, if $\Delta$ is $\mathbb{Z}_{p}$ -acyclic for some prime p, then every continuous map $f: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$ has a fixed point.

(A complex is $\mathbb{Z}_{p}$-acyclic if its reduced homology with $\mathbb{Z}_{p}$-coefficients vanishes. That is, in terms of homology it looks like a contractible space, say a d-ball.)

Proof (Sketch)

For a finite simplicial complex $\Delta$, the polyhedron $\|\Delta \|$ is compact. So if f does not have a fixed point, there is some ɛ > 0 such that | f(x) − x | > ɛ for all $x \in \Delta$. Now take a subdivision $\Delta '$ of $\Delta$ into simplices of diameter smaller than ɛ, and a simplicial approximation of error smaller than ɛ∕2, so that the simplicial approximation $f': \Delta ' \rightarrow \Delta '$, which is homotopic to f, does not have a fixed point, either.

Now apply the trace theorem to see that L _k (f) is zero, contrary to the assumption, where the induced map f _∗0 ^′ = f _∗0 in 0-dimensional homology is the identity. □

Note that Brouwer’s fixed point Theorem 2.4 is the special case of Theorem A.2 when $\Delta$ triangulates a ball.

For a reasonably large class of spaces, a converse to the Lefschetz fixed point theorem is also true: If L(f) = 0, then f is homotopic to a map without fixed points. See Brown [21, Chap. VIII].

1.2 The Theorems of Smith and Oliver

In addition to the usual game of connections between graphs, posets, complexes and spaces, we will now add groups. Namely we will discuss some useful topological effects caused by symmetry, that is, by finite group actions.

A (finite) group G acts on a (finite) simplicial complex $\Delta$ if each group element corresponds to a permutation of the vertices of $\Delta$, where composition of group elements corresponds to composition of permutations, in such a way that g(A) : =  {gv: v ∈ A} is a face of $\Delta$ for all g ∈ G and for all $A \in \Delta$. This action on the vertices is extended to the geometric realization of the complex $\Delta$, so that G acts as a group of simplicial homeomorphisms $g: \ \|\Delta \|\ \longrightarrow \ \|\Delta \|$.

The action is faithful if only the identity element in G acts as the identity permutation. In general, the set $G_{0}\:=\ \{g \in G:\, gv = v\mbox{ for all }v \in \mbox{ vert}(\Delta )\}$ is a normal subgroup of G. Hence we get that the quotient group G∕G ₀ acts faithfully on $\Delta$, and we usually only consider faithful actions. In this case, we can interpret G as a subgroup of the symmetry group of the complex $\Delta$. The action is vertex transitive if for any two vertices v, w of $\Delta$ there is a group element g ∈ G with gv = w.

A fixed point (also known as stable point) of a group action is a point $x \in \|\Delta \|$ that satisfies gx = x for all g ∈ G. We denote the set of all fixed points by $\Delta ^{G}$. Note that $\Delta ^{G}$ is in general not a subcomplex of $\Delta$.

Example A.3

Let $\Delta = 2^{[3]}$ be the complex of a triangle, and let $G = \mathbb{Z}_{3}$ be the cyclic group (a proper subgroup of the symmetry group $\mathfrak{S}_{3}$), acting such that a generator cyclically permutes the vertices, 1 ↦  2 ↦  3 ↦  1.

This is a faithful action; its fixed point set consists of the center of the triangle only—this is not a subcomplex of $\Delta$, although it corresponds to a subcomplex of the barycentric subdivision $\mbox{ sd}(\Delta )$.

Lemma A.4 (Two barycentric subdivisions)

 

• (1) After replacing $\Delta$ by its barycentric subdivision (informally, let $\Delta:=\mathrm{ sd}(\Delta )$ ), we get that the fixed point set $\Delta ^{G}$ is a subcomplex of  $\Delta$ .

• (2) After replacing $\Delta$ once again by its barycentric subdivision (so now $\Delta:=\mathrm{ sd}^{2}(\Delta ))$ , we even get that the quotient space $\|\Delta \|/G$ can be constructed from $\Delta$ by identifying all faces with their images under the action of G. That is, the equivalence classes of faces of  $\Delta$ , with the induced partial order, form a simplicial complex that is homeomorphic to the quotient space $\|\Delta \|/G$ .

We leave the proof as an exercise. It is not difficult; for details and further discussion see Bredon [18, Sect. III.1].

“Smith Theory” was started by P. A. Smith [69] in the thirties. It analyzes finite group actions on compact spaces (such as finite simplicial complexes), providing relations between the structure of the group to its possible fixed point sets. Here is one key result.

Theorem A.5 (Smith [68])

If P is a p-group (that is, a finite group of order | P | = p ^t for a prime p and some t > 0), acting on a complex $\Delta$ that is $\mathbb{Z}_{p}$ -acyclic, then the fixed point set $\Delta ^{P}$ is $\mathbb{Z}_{p}$ -acyclic as well. In particular, it is not empty.

Proof (Sketch)

The key is that, with the preparations of Lemma A.4, the maps that f induces on the chain groups (with $\mathbb{Z}_{p}$ coefficients) nicely restrict to the chain groups on the fixed point set $\Delta ^{P}$. Passing to traces and using the Hopf trace theorem, one can derive that $\Delta ^{P}$ is non-empty. A more detailed analysis leads to the “transfer isomorphism” in homology, which proves that $\Delta ^{P}$ must be acyclic.

See Bredon [18, Thm. III.5.2] and Oliver [60, p. 157], and also de Longueville [47, Appendix D and E]. □

On the combinatorial side, one has an Euler characteristic relation due to Floyd [25] [18, Sect. III.4]:

$$\displaystyle{\chi (\Delta )\ +\ (\,p - 1)\chi (\Delta ^{\mathbb{Z}_{p} })\ \ =\ \ p\,\chi (\Delta /\mathbb{Z}_{p}).}$$

If P is a p-group (in particular for $P = \mathbb{Z}_{p}$), then this implies that

$$\displaystyle{\chi (\Delta ^{P}) \equiv \chi (\Delta )\pmod p,}$$

using induction on t, where | P | = p ^t.

Theorem A.6 (Oliver [60, Lemma I])

If $G = \mathbb{Z}_{n}$ is a cyclic group, acting on a $\mathbb{Q}$ -acyclic complex $\Delta$ , then the action has a fixed point.

In this case the fixed point set $\Delta ^{G}$ has the Euler characteristic of a point, $\chi (\Delta ^{G}) = 1$ .

Proof

The first statement follows directly from the Lefschetz fixed point theorem: Any cyclic group is generated by a single element g, this element has a fixed point, this fixed point of g is also a fixed point of all powers of g, and hence of the whole group G.

For the second part, take p ^t to be a maximal prime power that divides n, consider the corresponding subgroup isomorphic to $\mathbb{Z}_{p^{t}}$, and use induction on t and the transfer homomorphism, as for the previous proof. □

Unfortunately, results like these may give an overly optimistic impression of the generality of fixed point theorems for acyclic complexes. There are fixed point free finite group actions on balls: Examples were constructed by Floyd and Richardson and others; see Bredon [18, Sect. I.8].

On the positive side we have the following result due to Oliver, which plays a central role in Sect. 4.5.

Theorem A.7 (Oliver’s Theorem I [60, Prop. I])

If G has a normal subgroup P ⊲ G that is a p-group, such that the quotient G∕P is cyclic, acting on a complex $\Delta$ that is $\mathbb{Z}_{p}$ -acyclic, then the fixed point set $\Delta ^{G}$ is $\mathbb{Z}_{p}$ -acyclic as well. In particular, it is not empty.

This is as much as we will need in this chapter. Oliver proved, in fact, a more general and complete theorem that includes a converse.

Theorem A.8 (Oliver’s Theorem II [60])

Let G be a finite group. Every action of G on a $\mathbb{Z}_{p}$ -acyclic complex $\Delta$ has a fixed point if and only if G has the following structure:

G has normal subgroups P ⊲ Q ⊲ G such that P is a p-group, G∕Q is a q-group (for a prime q that need not be distinct from p), and the quotient Q∕P is cyclic.

In this situation one always has $\chi (\Delta ^{G}) \equiv 1\bmod q$ .

Notes The Lefschetz–Hopf fixed point theorem was announced by Lefschetz for a restriced class of complexes in 1923, with details appearing three years later. The first proof for the general version was by Hopf in 1929. There are generalizations, for example to Absolute Neighborhood Retracts; see Bredon [19, Cor. IV.23.5] and Brown [21, Chap. IIII]. We refer to Brown’s book [21].

We refer to Bredon [18, Chapter III] for a nice textbook treatment of Smith Theory. The book by de Longueville [47, Appendix E] also has a very accessible discussion of the fixed point theorems of Smith and Oliver. The exercises concerning fixed point sets of poset maps P → P are drawn from Baclawski and Björner [6].

1.2.1 Exercises

• 1. Verify directly that if f maps ∥T∥ to ∥T∥, where T is a graph-theoretic tree, then f has a fixed point.

  How would you derive this from the Lefschetz fixed point theorem?

• 2. Let P be a poset (finite partially ordered set), and denote by $\Delta (P)$ its order complex (whose faces are the totally ordered subsets). Suppose that f:  P → P is an order-preserving mapping with fixed point set P ^f: = {x ∈ P∣f(x) = x}.

  • (a) Show that if $\Delta (P)$ is acyclic over some field, then

    $$\displaystyle{\mu (P^{f}) = 0,}$$

    where μ(P ^f) denotes the Möbius function (reduced Euler characteristic) of $\Delta (P^{f})$. In particular, P ^f is not empty.

  • (b) Does it follow also that P ^f itself is acyclic?

• 3. Suppose now that f:  P → P is order-reversing and let P _f : = {x ∈ P∣x = f ²(x) ≤ f(x)}. Show that if $\Delta (P)$ is acyclic over some field, then

  $$\displaystyle{\mu (P_{f}) = 0.}$$

  In particular, if f has no fixed edge (i.e., no x such that x = f ²(x) < f(x)) then f has a unique fixed point.