Borsuk-Ulam Theorem, Index, and Coindex

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. 

Choose representations for the cohomology algebras iJ (lRP Z2) = Z2[a] and Z2) = Z2[/ ], with the only relations on the generators be- 

Finally, since 0 + = 0 is the only relation on a, this yields the desired inequality m n.  

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

Assume that we have two Z2-spaces X and Y, and that 7 A — F is a Z2-map. Then we have the inequality 

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