Transfer maps in generalized group homology via submanifolds

Martin Nitsche   ; Rudolf Zeidler   ; Thomas Schick

Documenta mathematica, Tome 26 (2021), pp. 947-979

Résumé

Let N⊂M be a submanifold embedding of spin manifolds of some codimension k≥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⊂M are satisfied. In the cases k=1 and k=2, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer KO∗​(C∗π1​M)→KO∗−k​(C∗π1​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∗​(Bπ1​M)→E∗−k​(Bπ1​N) for different generalized homology theories E and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer E∗​(M)→E∗−k​(N) induced by the inclusion N⊂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≤3. For k≤2, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Détail

DOI : 10.4171/dm/833

Classification : 19K35, 55N20, 55N22, 55N91
Mots-clés : group cohomology, transfer maps, geneneralized cohomology, codimension 2 submanifold obstruction to positive scalar curvature