Monopole Floer Homology and Real Structures

TL;DR

This paper introduces a real version of monopole Floer homology for 3-manifolds with involutions, leading to new link invariants via double branched covers, based on the concept of real spin-c structures.

Contribution

It develops a novel 'real' monopole Floer homology framework incorporating anti-linear involutions on spinor bundles, extending existing invariants to new settings.

Findings

Defined real monopole Floer homology for involutive 3-manifolds

Constructed link invariants via double branched covers

Introduced the concept of real spin-c structures

Abstract

We define a "real" version of Kronheimer-Mrowka's monopole Floer homology for a 3-manifold equipped with an involution. As a special case, we obtain invariants for links via their double branched covers. The new input is the notion of a real spin-c structure, which consists of a spin-c structure along with a compatible anti-linear involution on the spinor bundle.