Central extensions of mapping class groups from characteristic classes

TL;DR

This paper develops a framework for constructing higher group extensions of mapping class groups using characteristic classes, generalizing known surface results to higher-dimensional manifolds and TQFTs.

Contribution

It introduces a method to produce higher extensions of automorphism stacks and mapping class groups from tangential structures, extending Segal's approach to higher dimensions.

Findings

Provides conditions for extensions to be central

Reconstructs Segal's $Z$-extensions for surfaces

Generalizes to higher-dimensional manifolds and TQFTs

Abstract

We characterize, for every higher smooth stack equipped with "tangential structure", the induced higher group extension of the geometric realization of its higher automorphism stack. We show that when restricted to smooth manifolds equipped with higher degree topological structures, this produces higher extensions of homotopy types of diffeomorphism groups. Passing to the groups of connected components, we obtain abelian extensions of mapping class groups and we derive sufficient conditions for these being central. We show as a special case that this provides an elegant re-construction of Segal's approach to $\mathbb{Z}$-extensions of mapping class groups of surfaces that provides the anomaly cancellation of the modular functor in Chern-Simons theory. Our construction generalizes Segal's approach to higher central extensions of mapping class groups of higher dimensional manifolds with higher tangential structures, expected to provide the analogous anomaly cancellation for higher dimensional TQFTs.