Equivariant Cerf theory and perturbative $SU(n)$ Casson invariants

TL;DR

This paper develops an equivariant Cerf theory for Morse functions with group actions and applies it to define perturbative $SU(n)$ Casson invariants on integer homology spheres, extending previous work.

Contribution

It introduces an equivariant Cerf theory framework for both finite and infinite-dimensional manifolds and establishes the existence of perturbative $SU(n)$ Casson invariants for all $n\,\geq 3$.

Findings

Proves existence of perturbative $SU(n)$ Casson invariants for all $n\ge 3$

Provides explicit formula for $n=4$ case

Generalizes previous results of Boden and Herald

Abstract

We develop an equivariant Cerf theory for Morse functions on finite-dimensional manifolds with group actions, and adapt the technique to the infinite-dimensional setting to study the moduli space of perturbed flat $SU(n)$-connections. As a consequence, we prove the existence of perturbative $SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$, and write down an explicit formula when $n=4$. This generalizes the previous works of Boden and Herald.