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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The cofibre of the transfer map


Author: Larry W. Cusick
Journal: Proc. Amer. Math. Soc. 93 (1985), 561-566
MSC: Primary 55R20; Secondary 55R12, 57S17
DOI: https://doi.org/10.1090/S0002-9939-1985-0774023-9
MathSciNet review: 774023
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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_ + }\mathop \to \limits^{{\text{tr}}} {\Sigma ^0}{X_ + } \to \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.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0774023-9
Article copyright: © Copyright 1985 American Mathematical Society

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