Stiefel-Whitney class

We will expand on this line of argument in the last part of the book, where we will actually look at applications of somewhat more sophisticated invariants, which are routinely used in obstruction theory. Namely, we explore the use of Stiefel-Whitney classes associated to involutions (or to free Z-actions, or to line bundles) for this sort of question. However, before an application of such complexity can be properly put in context, it is important to develop the toolbox of Combinatorial Algebraic Topology. 

The Stiefel-Whitney classes can be used to determine the nonexistence of certain Z2-maps. The following theorem is an example of such a situation. 

Properties of Stiefel-Whitney Classes 123 8.3.2 Stiefel Whitney Height... 

Many results giving topological obstructions to graph colorings had the k-connectivity of some space as the crucial assumption. We describe here an important connection between this condition and nonnullity of powers of Stiefel-Whitney classes. 

Proof. Since is fc-dimensional, the statement follows immediately from Proposition 8.25. To see that Wi(X) 0, recall that the Stiefel-Whitney classes are functorial therefore we have ppITiq) poXiX ) ) = Ti7f( ), and the latter has been verified to be nontrivial. ... 

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 us describe how the construction used in the proof of Proposition 8.25 can be employed to obtain an explicit combinatorial description of the Stiefel-Whitney classes. 

To describe the powers of the Stiefel-Whitney classes, o7j(X), we need to recall how the cohomology multiplication is done simplicially. In fact, to evaluate vj X) on a A -simplex (wq,v, . .., Vk), we need to evaluate wi(X) on each of the edges vi, r i+i), for i = 0,..., A — 1, and then multiply the results. Thus, the only A -simplices on which the power zu X) evaluates nontrivially are those whose ordered set of vertices has alternating elements from A and from B. We call these simplices multicolored. We summarize with... 

In principle, all sorts of characteristic classes carry obstructions to graph colorings. Here we shall look at the applications of Stiefel-Whitney classes associated to free involutions. 

Powers of Stiefel-Whitney Classes and Chromatic Numbers of Graphs... 

We can now use Theorem 19.10 to give lower bounds for chromatic numbers of graphs in terms of Stiefel-Whitney classes of complexes of graph ho-momorphisms from complete graphs. 

Stiefel-Whitney Characteristic Classes and Test Graphs.327... 

As another example, we refer to a combinatorial computation of some concrete Stiefel-Whitney characteristic classes in Theorem 19.13 in Subsection 19.2.2. There, once the combinatorial description of the characteristic classes has been found, the entire calculation hinges on one combinatorial lemma, namely Lemma 19.14. 

This space will be of use once we look at the classifying spaces of finite groups and Stiefel-Whitney characteristic classes in Chapter 8. 

Principal P-Bundles and Stiefel—Whitney Characteristic Classes... 

It is then a positive surprise that the tests based on the Stiefel-Whitney characteristic classes are pol3momially computable if we fix the test graph and the tested dimension and consider the computational complexity with respect to the number of vertices. 

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. 

Since the Stiefel-Whitney characteristic classes are functorial and we assumed that vj Eom T,Km)) = 0, the existence of the Z2-map p implies that also ujf (Horn (T, G)) =0, thus yielding a contradiction to the assumption of the theorem. ... 

We describe an important extension property of the class of Stiefel-Whitney test graphs. 

Proof. Let n be an arbitrary positive integer. By the functoriality of Stiefel-Whitney characteristic classes, we have... 

We find the cochain, whose coboundary equals the representative of the appropriate power of the Stiefel Whitney characteristic class by setting... 

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