The Geometry of Loop Spaces II: Characteristic Classes

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 $p\ddot{I}‰,|p|\gg 0$, of the Kähler form $\ddot{I}‰$ over an integral Kähler surface.

The previous version incorrectly claimed that these 5-manifolds had diffeomorphism groups with infinite fundamental group. The corrections which give the main results for the isometry groups of $M$ are in "The Geometry of Loop Spaces II: Corrections," arXiv:2405.00651. arXiv admin note: text overlap with arXiv:0705.1008