Conjugation spaces and edges of compatible torus actions

TL;DR

This paper introduces conjugation spaces, a topological concept inspired by symplectic geometry, providing a criterion to identify such spaces and exploring their relation to torus actions.

Contribution

It defines conjugation spaces and establishes a simple criterion for recognizing them, extending symplectic techniques to a broader topological context.

Findings

Conjugation spaces can be characterized by a specific topological criterion.

The concept generalizes real loci of Hamiltonian manifolds beyond symplectic structures.

The criterion simplifies the identification of conjugation spaces in topology.

Abstract

Duistermaat introduced the concept of ``real locus'' of a Hamiltonian manifold. In that and in others' subsequent works, it has been shown that many of the techniques developed in the symplectic category can be used to study real loci, so long as the coefficient ring is restricted to the integers modulo 2. It turns out that these results seem not necessarily to depend on the ambient symplectic structure, but rather to be topological in nature. This observation prompts the definition of ``conjugation space'' in a paper of the two authors with V. Puppe. Our main theorem in this paper gives a simple criterion for recognizing when a topological space is a conjugation space.