Z2-space

Specifying F = Z2 in Definition 8.12, we see that a Z2-space X is a CW complex equipped with a fixed-point-free involution, that is, with a continuous map 7 X —> X such that 7 is an identity map. Correspondingly, we have Z2-maps as continuous maps that commute with these involutions, and a category of Z2-spaces as objects and Z2-maps as morphisms, which is called Z2-Sp. 

There are many classical examples of Z2-spaces. A central one is that of an n-dimensional sphere, with the antipodal map x —x playing the role of... 

The Borsuk-Ulam theorem makes the following terminology useful for formulating further obstructions to maps between Z2-spaces. 

The following number is a standard benchmark for comparing Z2-spaces with each other. 

Definition 8.24. Let X be an arbitrary nonempty Z2-space. The Stiefel-Whitney height of X (or simply the height of X), denoted by h(X), is defined to be the maximal nonnegative integer h such that Wi X) 0. If no such h exists, then the space X is said to have infinite height. 

For example, we have h(S ) = n, for all nonnegative integers n. We remark that for a nonempty Z2-space X we have wi X) = 0 if and only if no connected component of X is mapped onto itself by the structural Z2-action in other words, the structural involution must swap the connected components of X. In this case, consistent with Definition 8.24, we will say that the height of X is equal to 0. 

First, it is trivial that if X is a nonempty Z2-space, then one can equi-variantly map S° to X. It is possible to extend this construction inductively to an arbitrary Z2-space, in a way analogous to our proof of Proposition 8.16. 

Corollary 8.26 explains the rule of thumb that in dealing with Z2-spaces, the condition of Ac-connectivity can be replaced by the weaker condition that the (A - -l)th power of the appropriate Stiefel-Whitney class is different from 0. 

Let X be a regular CW complex and a Z2-space, and denote the fixed point free involution on X by 7. As mentioned above, one can choose a simplicial structure on X such that 7 is a simplicial map. We define a Z2-map p X ... 

As an important example we single out the case in which 7 actually flips some edge of the graph T. In this situation, Horn (T, G) is actually a Z2-space, meaning a topological space with a free Z2-action. The reflection actions on K2, G2r+i, and C r (flipping two edges) are special cases of that. 

The following theorem describes the standard way to use the nonnullity of the powers of Stiefel-Whitney characteristic classes associated to Z2-spaces as tests for graph colorings. 

Proof. By Proposition 18.12, we know that under the assumptions of the theorem, the prodsimplicial complex Horn (T, H) is a Z2-space for any loop-free graph H. 

Note that by Corollary 8.26, we have h(X) > connX + 1 for an arbitrary Z2-space X. Therefore, comparing equations (19.2) and (19.1), we see that if a graph T is a Stiefel-Whitney test graph, then, it is also a homotopy test graph. 

Let us stress again that in analogy to the fact that the height is defined for Z2-spaces, the term Stiefel-Whitney test graph actually refers to a pair (T, 7), where T is a graph and 7 is an involution of T that flips an edge. The following question arises naturally in this context. 

An easy but important observation for us is that for an arbitrary Z2-space... 

Theorem 19.16. Let X be a nonempty Z2-space with finite Stiefel-Whitney height. Then we have H X Z2) 0. 

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