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Semifree Isovariant Poincaré Spaces and the Gap Condition

Dominik Kirstein, Christian Kremer

TL;DR

The paper develops a homotopical framework for semifree isovariant G-Poincaré spaces, introducing Isov_G(X) as the moduli of isovariant structures and proving a gap-dependent connectivity result when G is periodic. The approach destabilises the stable normal data via join stabilisation and reassembles a nonequivariant embedding with a complement using Klein’s embedding theorem and obstruction theory, yielding a highly connected space of isovariant structures under codimension gaps. This yields a practical tool to construct and classify equivariant manifold structures on Poincaré spaces, with applications including a Browder–Straus-type lifting for semifree smooth G-manifolds and a path toward Nielsen realisation in semifree settings. Overall, the work provides a robust link between equivariant Poincaré duality, isovariant embeddings, and obstruction-theoretic constructions for semifree group actions, facilitating manifold constructions in equivariant topology with finite periodic groups.

Abstract

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

Semifree Isovariant Poincaré Spaces and the Gap Condition

TL;DR

The paper develops a homotopical framework for semifree isovariant G-Poincaré spaces, introducing Isov_G(X) as the moduli of isovariant structures and proving a gap-dependent connectivity result when G is periodic. The approach destabilises the stable normal data via join stabilisation and reassembles a nonequivariant embedding with a complement using Klein’s embedding theorem and obstruction theory, yielding a highly connected space of isovariant structures under codimension gaps. This yields a practical tool to construct and classify equivariant manifold structures on Poincaré spaces, with applications including a Browder–Straus-type lifting for semifree smooth G-manifolds and a path toward Nielsen realisation in semifree settings. Overall, the work provides a robust link between equivariant Poincaré duality, isovariant embeddings, and obstruction-theoretic constructions for semifree group actions, facilitating manifold constructions in equivariant topology with finite periodic groups.

Abstract

We introduce the notion of a semifree isovariant -Poincaré space, a homotopical notion interpolating between semifree closed smooth -manifolds and the equivariant Poincaré spaces of [HKK24b]. It carries the additional structure of an equivariant Poincaré embedding of the fixed points of a semifree -Poincaré space. Under suitable gap conditions on the codimension, we show that the space of isovariant structures on a semifree -Poincaré space for a periodic finite group is highly connected, giving a useful construction tool for manifold structures on equivariant Poincaré spaces.
Paper Structure (15 sections, 31 theorems, 54 equations)

This paper contains 15 sections, 31 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. Poincaré pairs and embeddings
  4. Equivariant Poincaré duality
  5. Semifree isovariant $G$-Poincaré spaces
  6. Destabilisations
  7. Generalised homotopy representations and their stabilisations
  8. Groups with periodic cohomology and free generalised homotopy representations
  9. Semifree $G$-spectra
  10. Join stabilisation
  11. Automorphisms versus stable automorphisms of representation spheres
  12. Constructing complements
  13. Complement problems
  14. Proof of the main theorem
  15. Equivariant spectra with specified isotropy

Key Result

Theorem A

Let $X$ be a semifree $G$-Poincaré space and $G$ a periodic finite group. Consider $k \ge -1$ such that for each component of $X^G$ and the corresponding component of $X^e$ containing it we have Then the space $\mathrm{Isov}_G(X) = \mathrm{PD}^{\mathrm{sf}}_{{G},\mathrm{isov}} \times_{\mathrm{PD}^{\mathrm{sf}}_{G}} \{X\}$ of isovariant structures on $X$ is $k$-connected.

Theorems & Definitions (74)

  • Theorem A: \ref{['thm:main_thm']}
  • Definition 1.1
  • Corollary 1.3
  • Corollary 1.6
  • Lemma 2.1.1
  • Definition 2.1.2
  • Theorem 2.1.3: Klein, Klein2002b
  • Theorem 2.2.1
  • proof
  • Theorem 2.2.2
  • ...and 64 more