Transfer maps in generalized group homology via submanifolds

Abstract

Let $N\subset M$ be a submanifold embedding of spin manifolds of some codimension $k\ge 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k=2$ and the Dirac operator of $N$ has non-trivial index, provided that suitable geometric conditions on $N\subset M$ are satisfied. In the cases $k=1$ and $k=2$, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer ${\text{KO}}_{\ast }({\text{C}}^{\ast }{\pi }_{1}M)\to {\text{KO}}_{\ast -k}({\text{C}}^{\ast }{\pi }_{1}N)$ which maps the index class of $M$ to the index class of $N$. The main goal of this article is to construct analogous transfer maps $E_{\ast }(\text{B}{\pi }_{1}M)\to E_{\ast -k}(\text{B}{\pi }_{1}N)$ for different generalized homology theories $E$ and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer $E_{\ast }(M)\to E_{\ast -k}(N)$ induced by the inclusion $N\subset M$ for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for $k\le 3$. For $k\le 2$, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.