Codimension 2 transfer of higher index invariants

Abstract

This paper is devoted to the study of the higher index theory of codimension 2 submanifolds originated by Gromov–Lawson and Hanke–Pape–Schick. The first main result is to construct the ‘codimension 2 transfer’ map from the Higson–Roe analytic surgery exact sequence of a manifold M to that of its codimension 2 submanifold N under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of M to those of N. The second is to establish that the codimension 2 transfer map is adjoint to the co-transfer map in cyclic cohomology, defined by the cup product with a group cocycle. This relates the Connes–Moscovici higher index pairing and Lott’s higher $$\rho $$ -number of M with those of N.