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 $N$.