Semifree Isovariant Poincar\'e Spaces and the Gap Condition

TL;DR

This paper introduces semifree isovariant Poincaré spaces, a new homotopical framework bridging smooth G-manifolds and equivariant Poincaré spaces, with results on their structure and connectivity under certain conditions.

Contribution

It defines semifree isovariant G-Poincaré spaces and proves their isovariant structure space is highly connected under gap codimension conditions.

Findings

Highly connected space of isovariant structures

Provides a construction tool for manifold structures

Bridges smooth G-manifolds and equivariant Poincaré spaces

Abstract

We introduce the notion of a semifree isovariant $G$-Poincar\'e space, a homotopical notion interpolating between semifree closed smooth $G$-manifolds and the equivariant Poincar\'e spaces of [HKK24b]. It carries the additional structure of an equivariant Poincar\'e embedding of the fixed points of a semifree $G$-Poincar\'e space. Under suitable gap conditions on the codimension, we show that the space of isovariant structures on a semifree $G$-Poincar\'e space for a periodic finite group $G$ is highly connected, giving a useful construction tool for manifold structures on equivariant Poincar\'e spaces.