Thin compactifications and relative fundamental classes

TL;DR

This paper introduces a new concept of relative fundamental class applicable to moduli spaces in gauge theory and symplectic Gromov-Witten theory, especially when the compactification boundary has high codimension.

Contribution

It defines the notion of relative fundamental class for moduli spaces with thin compactifications, extending the applicability of fundamental classes in gauge and symplectic theories.

Findings

Defines relative fundamental class for moduli spaces with thin compactifications

Ensures the class exists if the boundary has codimension at least two

Provides a homological framework for universal moduli spaces

Abstract

We define a notion of relative fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov-Witten theory. For universal moduli spaces over a parameter space, the relative fundamental class specifies an element of the Cech homology of the compactification of each fiber; it is defined if the compactification is "thin" in the sense that its boundary has homological codimension at least two.