Transfers for ramified covering maps in homology and cohomology
This paper provides a neat topological description of a homology transfer for ramified covering maps. The construction is based on the identification of the reduced singular homology group \(\tilde{H}_q(X;L)\) (for any abelian group \(L\)) with the homotopy group \(\pi_q(F(X,L))\) of the abelian group \(F(X,L)\) of all functions \(X \to L\) having finite support and mapping the base-point of \(X\) to the zero element of \(L\). Given a ramified covering map \(E \to X\), a natural construction is given of a pre-transfer \(F(X,L) \to F(E,L)\) which, in turn, induces the homology transfer. As this approach to homology groups has been used in other contexts (e.g. non-commutative geometry and algebraic geometry), this approach to the transfer can be translated very directly into those contexts. This enables the authors to give slicker proofs of some established results and also to prove some new theorems.
scientific article; zbMATH DE number 5164297
scientific article