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On the functoriality of the space of equivariant smooth $h$-cobordisms

Thomas Goodwillie, Kiyoshi Igusa, Cary Malkiewich, Mona Merling

TL;DR

The paper constructs an $(\\infty,1)$-functorial framework for equivariant smooth $h$-cobordisms, associating to each compact smooth $G$-manifold $M$ the space ${\\mathcal H}_{\\mathrm{Diff}}(M)$ of equivariant $h$-cobordisms and its stabilized variant ${\\mathcal H}^{\\mathcal{U}}_{\\mathrm{Diff}}(M)$ stabilized via representation discs. It introduces polar stabilization plus mirror structures and round bundles to achieve coherently compatible stabilization and to model functoriality at the level of simplicial spaces, via left fibrations of Segal spaces and a straightening-unstraightening approach. The work proves the unstable functor can be enhanced to an $(\\infty,1)$-functor on the category of smooth $G$-manifolds with corners, and extends the stabilized functor to all $G$-CW complexes, yielding a robust, coherent equivariant stable parametrized $h$-cobordism theory. These constructions underpin future results on splitting of equivariant stable $h$-cobordism spaces and interact with equivariant Reidemeister torsion and stable parametrized theorems. The framework generalizes the non-equivariant case and provides a flexible apparatus for compatibility across multiple stabilizations and embeddings.

Abstract

We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal H}_{\mathrm{Diff}}(M)$. We also give a stable analogue ${\mathcal H}^{\mathcal U}_{\mathrm{Diff}}(M)$ where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In the non-equivariant case $G=e$, our $(\infty,1)$-functor agrees with previous constructions of the smooth $h$-cobordism space as a functor to the homotopy category.

On the functoriality of the space of equivariant smooth $h$-cobordisms

TL;DR

The paper constructs an -functorial framework for equivariant smooth -cobordisms, associating to each compact smooth -manifold the space of equivariant -cobordisms and its stabilized variant stabilized via representation discs. It introduces polar stabilization plus mirror structures and round bundles to achieve coherently compatible stabilization and to model functoriality at the level of simplicial spaces, via left fibrations of Segal spaces and a straightening-unstraightening approach. The work proves the unstable functor can be enhanced to an -functor on the category of smooth -manifolds with corners, and extends the stabilized functor to all -CW complexes, yielding a robust, coherent equivariant stable parametrized -cobordism theory. These constructions underpin future results on splitting of equivariant stable -cobordism spaces and interact with equivariant Reidemeister torsion and stable parametrized theorems. The framework generalizes the non-equivariant case and provides a flexible apparatus for compatibility across multiple stabilizations and embeddings.

Abstract

We construct an -functor that takes each smooth -manifold with corners to the space of equivariant smooth -cobordisms . We also give a stable analogue where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In the non-equivariant case , our -functor agrees with previous constructions of the smooth -cobordism space as a functor to the homotopy category.
Paper Structure (27 sections, 82 theorems, 111 equations, 18 figures)

This paper contains 27 sections, 82 theorems, 111 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. Preliminaries on $G$-manifolds with corners
  5. $G$-manifolds with corners
  6. Existence of smooth extensions
  7. Embeddings, tubular neighborhoods, and submersions
  8. Families of maps and families of tubular neighborhoods
  9. Trimmings, faces, and collars
  10. Smooth simplices of diffeomorphisms
  11. Pseudoisotopies on manifolds with corners
  12. Pseudoisotopies and mirror pseudoisotopies
  13. Smoothness properties of even and odd functions
  14. Polar stabilization
  15. $h$-cobordisms on manifolds with corners
  16. ...and 12 more sections

Key Result

Theorem 1.1

There is a simplicially enriched functor sending each compact $G$-manifold $M$ to a space equivalent to $\mathcal{H}_{\textup{Diff}\,}(M)$, and each homotopy class of equivariant embeddings $M_0 \to M_1$ to the homotopy class of maps $\mathcal{H}_{\textup{Diff}\,}(M_0) \to \mathcal{H}_{\textup{Diff}\,}(M_1)$ given by the stabilization depi

Figures (18)

  • Figure 1: The U-shaped stabilization of an $h$-cobordism $W_0$ from $M_0$ to $N_0$ along an embedding $M_0 \to M_1$ with normal bundle $\nu$. Each strip in the $U$-shape given by the disk bundle $D\nu$ is a copy of $W_0$. In the top region given by the lower hemisphere disk bundle $D^-(\nu\times \mathbb R)$ we glue in copies of the top manifold $N_0$.
  • Figure 2: The polar stabilization of an $h$-cobordism $W_0$ from $M_0$ to $N_0$ along an embedding $M_0 \to M_1$ with normal bundle $\nu$. As in \ref{['fig:intro_stab']}, each strip along the U-shape given by the disk bundle $D\nu$ is a copy of $W_0$, but now the copies of the top manifold $N_0$ are identified at the cone point.
  • Figure 3: Polar stabilization defined using round bundles.
  • Figure 4: The smooth boundary of a square consists of four line segments.
  • Figure 5: A partial boundary $A \subseteq \partial M$ and an open neighborhood of $A$.
  • ...and 13 more figures

Theorems & Definitions (201)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Definition 2.5
  • ...and 191 more