Chern Simons invariants in $KK$ theory

TL;DR

This paper introduces a new KK-theoretic element associated with unitary representations of fundamental groups, connecting Chern-Simons invariants with KK-theory using traces on C*-algebras, without relying on Chern characters.

Contribution

It defines an intrinsic KK-theoretic element linked to alpha-invariants, expanding the understanding of Chern-Simons invariants in KK-theory without using Chern characters.

Findings

Defines a KK-theoretic element with real coefficients

Connects Chern-Simons invariants to KK-theory intrinsically

Provides a new perspective on alpha-invariants

Abstract

For a unitary representation of the fundamental group of a compact smooth manifold, Atiyah, Patodi, Singer defined the so called alpha-invariant of the representation using Chern-Simons invariants. In this article using traces on C*-algebras, we define intrinsically(i.e without using Chern character) an element in KK-theory with real coefficients theory whose pullback by the representation is the alpha-invariant.