The Geometry of Loop Spaces II: Corrections

TL;DR

This paper corrects previous results on the geometry of loop spaces by replacing findings about the fundamental group of the diffeomorphism group with those about the isometry group, refining the mathematical understanding.

Contribution

It provides necessary corrections to earlier work, specifically updating the focus from diffeomorphism groups to isometry groups in the context of loop space geometry.

Findings

Results about $oldsymbol{\pi_1( ext{Diff}(M)))}$ are replaced with results about $oldsymbol{\pi_1( ext{Isom}(M)))}$

Clarifies the relationship between loop space geometry and isometry groups

Ensures mathematical accuracy in the characterization of fundamental groups

Abstract

This paper contains corrections to Madea, Rosenberg, Torres-Ardila, "The Geometry of Loop Spaces II: Characteristic Classes," Advances in Math. (287), 2016, 485-518. The main change is that results about $\pi_1({\rm Diff}(M))$ are replaced by results about $\pi_1({\rm Isom}(M))$, where Diff$(M)$, Isom$(M)$ refer to the diffeomorphism and isometry group of the manifold $M$.