Higher Connectivity and Stiefel-Whitney Classes

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. 

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. 

Proposition 8.25. Let X and Y be two regular CW complexes with a free Z2-action such that for some k 0, we have dimX k and Y is k — 1)-connected. Assume further that we have a Z2-map if X — Y, for some d —1. Then there exists a Z2-map y X — X such that extends if. 

Please note the following convention used in the formulation of Proposition 8.25 d = —1 means that we have no map if (in other words, X = 0), hence no additional conditions on the map p. 

By assumption, the cellular structure on X is Z2-invariant. We construct inductively on the i-skeleton of X, for i d-h 1. If d = — 1, we start by defining on the 0-skeleton as follows for each orbit a, b consisting of two vertices of X, simply map a to an arbitrary point y Y, and then map b to 7(2/), where 7 is the free involution of Y. 

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