A strong homotopy axiom for Alexander cohomology

Abstract:

It is shown that the following form of the homotopy axiom holds for Alexander-Čech cohomology. Suppose that $X$ and $Y$ are any spaces, that $T$ is a compact, connected space, and that $G$ is an abelian group which either admits the structure of a compact topological group or is the additive group of a finite-dimensional vector space. Then for any continuous function $F:X \times T \to Y$, one has $F_r^\ast = F_s^\ast :{H^\ast }(Y;G) \to {H^\ast }(X;G)$ for all $r,s \in T$, where ${F_t}:X \to Y$ is defined by ${F_t}(x) = F(x,t)$.