# Analytic Pontryagin Duality

## TL;DR

This paper develops a geometric model for a specific K-theory group on manifolds, establishing a duality pairing involving analytic eta-invariants and topological data, enriching the understanding of $R/Z$-valued invariants.

## Contribution

It introduces a new geometric model for $K^0(X,R/Z)$ and constructs a non-degenerate duality pairing with Baum-Douglas K-homology, combining analytic and topological methods.

## Key findings

- Established a non-degenerate duality pairing involving eta-invariants.
- Provided explicit formulas for the pairing in special cohomology cases.
- Developed a robust $R/Z$-valued invariant for smooth compact manifolds.

## Abstract

Let $X$ be a smooth compact manifold. We propose a geometric model for the group $K^0(X,\mathbb{R}/\mathbb{Z}).$ We study a well-defined and non-degenerate analytic duality pairing between $K^0(X,\mathbb{R}/\mathbb{Z})$ and its Pontryagin dual group, the Baum-Douglas geometric $K$-homology $K_0(X),$ whose pairing formula comprises of an analytic term involving the Dai-Zhang eta-invariant associated to a twisted Dirac-type operator and a topological term involving a differential form and some characteristic forms. This yields a robust $\mathbb{R}/\mathbb{Z}$-valued invariant. We also study two special cases of the analytic pairing of this form in the cohomology group $H^1(X,\mathbb{R}/\mathbb{Z})$ and $H^2(X,\mathbb{R}/\mathbb{Z}).$