Transfer maps in generalized group homology via submanifolds

  • Martin Nitsche

    Institut für Geometrie, TU Dresden, Germany
  • Thomas Schick

    Mathematisches Institut, Universität Göttingen, Germany
  • Rudolf Zeidler

    Mathematisches Institut, WWU Münster, Germany
Transfer maps in generalized group homology via submanifolds cover
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Let NMN \subset M be a submanifold embedding of spin manifolds of some codimension k1k \geq 1. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that MM does not admit a metric of positive scalar curvature if k=2k = 2 and the Dirac operator of NN has non-trivial index, provided that suitable geometric conditions on NMN \subset M are satisfied. In the cases k=1k=1 and k=2k=2, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer KO(Cπ1M)KOk(Cπ1N)\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 MM to the index class of NN. The main goal of this article is to construct analogous transfer maps E(Bπ1M)Ek(Bπ1N)E_\ast(\text{B}\pi_1M) \to E_{\ast-k}(\text{B}\pi_1N) for different generalized homology theories EE and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer E(M)Ek(N)E_\ast(M) \to E_{\ast-k}(N) induced by the inclusion NMN \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 k3k \leq 3. For k2k \leq 2, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Cite this article

Martin Nitsche, Thomas Schick, Rudolf Zeidler, Transfer maps in generalized group homology via submanifolds. Doc. Math. 26 (2021), pp. 947–979

DOI 10.4171/DM/833