Relating VFCs on thin compactifications

TL;DR

This paper extends the theory of relative fundamental classes (RFCs) for moduli spaces in geometric analysis, showing their naturality, compatibility with other invariants, and equivalence with Pardon's virtual fundamental class in various settings.

Contribution

It generalizes the concept of RFCs beyond stratified cases, proving their naturality and compatibility with pseudo-cycles, cutdown spaces, and Pardon's VFC.

Findings

RFCs agree with pseudo-cycle invariants

RFCs are compatible with cutdown moduli spaces

RFCs coincide with Pardon's virtual fundamental class

Abstract

Many moduli spaces that occur in geometric analysis admit "Fredholm-stratified thin compactifications" in the sense of [IP1] and hence admit a relative fundamental class (RFC), also as defined in [IP1]. We extend these results, emphasizing the naturality of the RFC, eliminating the need for a stratification, and proving three compatibility results: the invariants defined by the RFC agree with those defined by pseudo-cycles, the RFC is compatible with cutdown moduli spaces, and the RFC agrees with the virtual fundamental class (VFC) constructed by J. Pardon via implicit atlases in all cases where both are defined.