The cofibre of the transfer map

Abstract:

Suppose a finite group $G$ acts freely on a finite complex $X$ with orbit space $B$. The cofibre $\mathcal {C}$ of the transfer map, is defined by the cofibre sequence $\Sigma ^0 B_+ \stackrel {\mathrm {tr}}{\rightarrow } \Sigma ^0 X_+ \rightarrow \mathcal {C}$. We show that there is a spectral sequence $H_G^p(X;\tilde M \otimes {h^q}) \Rightarrow {h^{p + q}}(\mathcal {C})$ for any cohomology theory ${h^ * }$, where $\tilde M$ is the reduced regular ${\mathbf {Z}}$-representation for $G$. As a special case we prove that ${H^ * }(\mathcal {C};{\mathbf {Z}}_2)$ is a free ${H^ * }(B;{{\mathbf {Z}}_2})$-module on a zero-dimensional class for any two-fold cover.