Equivariant Cerf theory and perturbative SU(n) Casson invariants

Bai, Shaoyun; Zhang, Boyu

doi:10.2140/gt.2026.30.1203

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Equivariant Cerf theory and perturbative $\mathrm{SU}(n)$ Casson invariants

Shaoyun Bai and Boyu Zhang

Geometry & Topology 30 (2026) 1203–1276

DOI: 10.2140/gt.2026.30.1203

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 $n \geq 3$ , and write down an explicit formula when $n = 4$ . This generalizes the previous works of Boden and Herald (1998) and Herald (2006).

Keywords

Casson invariant, gauge theory, equivariant transversality

Mathematical Subject Classification

Primary: 57K31

References

Publication

Received: 11 May 2021

Revised: 15 February 2024

Accepted: 10 November 2025

Published: 9 July 2026

Proposed: András I Stipsicz

Seconded: Ciprian Manolescu, Simon Donaldson

Authors

Shaoyun Bai
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Boyu Zhang
Department of Mathematics University of Maryland at College Park College Park, MD United States

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Volume 30, issue 4 (2026)