Transfer maps in generalized group homology via submanifolds

Abstract

Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. 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 $E_\ast(\text{B}\pi_1M) \to E_{\ast-k}(\text{B}\pi_1N)$ 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 $k \leq 3$. For $k \leq 2$, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.