Mutation and Gauge Theory I: Yang-Mills Invariants

TL;DR

This paper demonstrates that mutation operations on 3- and 4-manifolds preserve key invariants like Floer homology and Donaldson invariants, advancing understanding of manifold symmetries and invariants.

Contribution

It proves that mutation preserves instanton Floer homology for homology 3-spheres and Donaldson invariants for 4-manifolds, revealing new invariance properties.

Findings

Mutation preserves Floer homology of homology 3-spheres.

Mutation preserves Donaldson invariants of 4-manifolds.

Supports invariance of manifold invariants under specific topological operations.

Abstract

Mutation is an operation on 3-manifolds containing an embedded surface of genus 2. It is defined by cutting along the surface and regluing using the `hyperelliptic' involution, and is known to preserve many 3-manifold invariants. I show that mutation of a homology 3-sphere preserves its (instanton) Floer homology, and that a related operation on 4-manifolds preserves the Donaldson invariants. A companion article (in preparation) will treat invariants based on the Seiberg-Witten equations.