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Codimension 1 transfer maps of K theoretic indexes

Yuetong Luo

TL;DR

This work provides an explicit operator-algebra realization of Zeidler’s codimension-1 transfer for K-theory by constructing a $* $-homomorphism $\rho: C^*\Gamma\to\mathcal{Q}_{C^*\pi}$ into a Calkin algebra, so that the induced map on $K$-theory, $\rho_*$, matches the transfer $\rho_{M,N}$ sending the Rosenberg index $\alpha(M)$ to $\alpha(N)$ and, in the oriented case, maps higher signatures up to a power of two: $\rho_{M,N}(\mathrm{sgn}(M,f^*\nu_M)) = 2^{\epsilon}\mathrm{sgn}(N,f^*\nu_N)$. The authors treat separating and non-separating codimension-1 cases, using Mayer–Vietoris sequences and Kasparov products to relate global indices on $M$ to those on $N$, and provide a detailed proof that the transfer is independent of auxiliary choices. They also extend the construction to reduced group C*-algebras, and discuss extensions to L-theory, highlighting KK-theoretic interpretations and potential broader applications. Overall, the paper delivers a hands-on, explicit bridge between geometric transfer phenomena and concrete C*-algebraic maps, enriching the toolkit for obstructions to positive scalar curvature and higher index theory.

Abstract

Let $M$ be a closed spin manifold and $N$ be a codimension 1 submanifold of it. Given certain homotopy conditions, Zeidler shows that the Rosenberg index of $N$ is an obstruction to the existence of positive scalar curvature on $M$. He further gives a transfer map between the K groups of the group $C^*$ algebras of the foundemental group. The transfer map maps the Rosenberg index of $M$ to the one of $N$. In this note, we present an alternative formulation of the transfer map using maps between $C^*$ algebras, and give an analogus result for the codimension 1 transfer of higher K theoretic signatures.

Codimension 1 transfer maps of K theoretic indexes

TL;DR

This work provides an explicit operator-algebra realization of Zeidler’s codimension-1 transfer for K-theory by constructing a -homomorphism into a Calkin algebra, so that the induced map on -theory, , matches the transfer sending the Rosenberg index to and, in the oriented case, maps higher signatures up to a power of two: . The authors treat separating and non-separating codimension-1 cases, using Mayer–Vietoris sequences and Kasparov products to relate global indices on to those on , and provide a detailed proof that the transfer is independent of auxiliary choices. They also extend the construction to reduced group C*-algebras, and discuss extensions to L-theory, highlighting KK-theoretic interpretations and potential broader applications. Overall, the paper delivers a hands-on, explicit bridge between geometric transfer phenomena and concrete C*-algebraic maps, enriching the toolkit for obstructions to positive scalar curvature and higher index theory.

Abstract

Let be a closed spin manifold and be a codimension 1 submanifold of it. Given certain homotopy conditions, Zeidler shows that the Rosenberg index of is an obstruction to the existence of positive scalar curvature on . He further gives a transfer map between the K groups of the group algebras of the foundemental group. The transfer map maps the Rosenberg index of to the one of . In this note, we present an alternative formulation of the transfer map using maps between algebras, and give an analogus result for the codimension 1 transfer of higher K theoretic signatures.

Paper Structure

This paper contains 9 sections, 14 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Geometric setting
  3. Transfer map
  4. Proof of theorem \ref{['main thm']}
  5. Separating case
  6. Non-separating case
  7. Proof of Lemma \ref{['indelem']}
  8. Extension to reduced $C^*$ algebra
  9. Further directions

Key Result

Theorem 1.1

Let $M$ be a closed connected spin manifold of dimension $m$ and $N \subset M$ a connected submanifold of codimension one with trivial normal bundle. Assume that the inclusion map $N \hookrightarrow M$ induces an injection on the fundamental group. Then there is a homomorphism (called the transfer m s.t. $\rho_{M,N}$ maps the Rosenberg index of $M$ to the one of $N$, i.e. $\rho_{M,N}(\alpha(M))=\a

Theorems & Definitions (23)

  • Theorem 1.1: Theorem 1.7 in Zeidlercodim1
  • Theorem 1.2
  • Definition 3.1: Hilbert module
  • Example 3.2: Standard Hilbert module
  • Definition 3.3: bounded adjointable operators
  • Lemma 3.4
  • proof
  • Lemma 4.1: Lemma 2.4 in Kubotacodim2
  • Lemma 4.2: Lemma 2.6 in Kubotacodim2
  • Theorem 4.3
  • ...and 13 more