The topology of Kuranishi atlases

TL;DR

This paper develops a topological theory of Kuranishi atlases to address algebraic and topological challenges in virtual regularization of moduli spaces, applicable in various complex settings.

Contribution

It introduces a transparent topological framework for Kuranishi atlases and cobordisms, extending their applicability beyond smooth structures.

Findings

Provides a topological approach to Kuranishi structures

Resolves algebraic and topological challenges in virtual regularization

Applicable to settings with isotropy, boundaries, and corners

Abstract

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Starting from the same core idea (patching local finite dimensional reductions) we develop a theory of topological Kuranishi atlases and cobordisms that transparently resolves algebraic and topological challenges in this virtual regularization approach. It applies to any Kuranishi-type setting, e.g. atlases with isotropy, boundary and corners, or lack of differentiable structure.