Pairs of pants, Pochhammer curves and $L^2$-invariants

TL;DR

This paper offers an intuitive geometric interpretation of nontrivial $L^2$-Betti numbers for compact Riemann surfaces using loops in pairs of pants, connecting topology with quantum physics applications.

Contribution

It introduces a new interpretation of $L^2$-Betti numbers via twisted homology and sewing properties in pair-of-pants decompositions, linking topology and physics.

Findings

Provides a geometric interpretation of $L^2$-Betti numbers.

Establishes a sewing property for pair-of-pants decompositions.

Discusses applications to supersymmetric quantum mechanics with Aharonov-Bohm phases.

Abstract

We propose an intuitive interpretation for nontrivial $L^2$-Betti numbers of compact Riemann surfaces in terms of certain loops in embedded pairs of pants. This description uses twisted homology associated to the Hurewicz map of the surface, and it satisfies a sewing property with respect to a large class of pair-of-pants decompositions. Applications to supersymmetric quantum mechanics incorporating Aharonov-Bohm phases are briefly discussed, for both point particles and topological solitons (abelian and non-abelian vortices) in two dimensions.