Borsuk-Ulam theorem

For example, by the Borsuk-Ulam theorem it is impossible to map a higher dimensional sphere to a lower dimensional one in a way that commutes with the antipodal maps. Or here, if X G) is connected and X Ks) consists of several connected components and no single component is mapped to itself by the involution, then again no such involution-preserving map is possible. Note that to use this statement for different graphs, we would need each time to worry only about the connectivity of X G), since the complex X Kz) could be analyzed and understood once and for all. 

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

The main topological tool that Lovasz employed was the Borsuk-Ulam theorem. Since then, topological equivariant methods have gained ground and become part of the standard repertoire in combinatorics. 

This, however, contradicts the Borsuk-Ulam theorem (Theorem 8.22). ... 

Wa83] J.W. Walker, A homology version of the Borsuk-Ulam theorem, Amer. Math. Monthly 90, (1983), no. 7, 466-468. 

---