Z2-Spaces and the Definition of Stiefel-Whitney Classes

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. 

Clearly, the same combinatorial data describes all possible Z2-principal bundles. Therefore, we can freely switch between these two families of bundles. Starting from the line bundle, the associated Z2-principal bundle is obtained by replacing each line by the unit sphere S°. Conversely, starting from a Z2-principal bundle, the associated line bundle is obtained by sticking a line into each fiber, with this line oriented away from the identity element toward the nontrivial element of Z2. 

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 

Let us remark that the infinite-dimensional projective space RP°° appearing in (8.4) could also be defined as a colimit of the embedding sequence of the finite-dimensional projective spaces 

The crucial fact now is that in this particular case, the induced Z2-algebra homomorphism 

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