Nitsche, Martin; Schick, Thomas; Zeidler, Rudolf

Transfer Maps in Generalized Group Homology via Submanifolds

Doc. Math. 26, 947-979 (2021)
DOI: 10.25537/dm.2021v26.947-979
Communicated by Mike Hill


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.

Mathematics Subject Classification

55N20, 55N22, 55N91, 19K35


transfer maps, geneneralized cohomology, group cohomology, codimension 2 submanifold obstruction to positive scalar curvature


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Nitsche, Martin
Institut für Geometrie, TU Dresden, Germany
Schick, Thomas
Mathematisches Institut, Universität Göttingen, Germany
Zeidler, Rudolf
Mathematisches Institut, WWU Münster, Germany