The Geometry of Loop Spaces II: Characteristic Classes

TL;DR

This paper introduces Wodzicki-Chern-Simons classes on loop spaces using the Wodzicki residue, revealing new topological invariants that detect certain 5-manifolds with infinite fundamental groups.

Contribution

It constructs novel Wodzicki-Chern-Simons classes on loop spaces and demonstrates their ability to detect specific 5-manifolds with infinite fundamental groups.

Findings

WCS classes are associated with the residue Chern character on loop spaces.

These classes detect families of 5-manifolds with infinite fundamental groups.

Application to circle bundles over Kähler surfaces with multiple forms.

Abstract

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect several families of 5-manifolds whose isometry group has infinite fundamental group. These manifolds are the total spaces of the circle bundles associated to a multiple $p\omega, |p|\gg 0$, of the K\"ahler form $\omega$ over an integral K\"ahler surface.