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Extending families of homeomorphisms over 4-dimensional handlebodies

Rachael Boyd, Corey Bregman, Jan Steinebrunner

TL;DR

The paper proves that for every genus $g\ge0$ the boundary-restriction map $B\mathop{\mathrm{Homeo}}(H_g)\to B\mathop{\mathrm{Homeo}}(U_g)$ admits a section, enabling fiberwise extension of $U_g$-bundles to $H_g$-bundles. It develops a robust simplicial and topological framework of separating systems, introducing $\mathrm{Sep}^{\rm top}(U_g)$ and $\operatorname{DSep}^{\rm top}(H_g)$ and showing their diagonals are contractible, with weak equivalences connecting discrete and topological models. The main theorem is established via an Alexander-trick–style argument and the Orbit–Stabiliser lemma to compare homotopy orbit spaces, supported by a disc-system dual-graph construction that matches boundary data with a disc system on $H_g$. Consequences include that any topological $U_g$-bundle can be fiberwise filled by an $H_g$-bundle and, for a 4-manifold $M$ with $U_g\subset\partial M$, the restriction map $B\mathop{\mathrm{Homeo}}(M\cup_{U_g} H_g, H_g)\to B\mathop{\mathrm{Homeo}}(M,U_g)$ admits a section; this framework also facilitates rational characteristic class constructions on $B\mathop{\mathrm{Homeo}}(U_g)$ via the section.

Abstract

Let $H_g$ denote the 4-dimensional handlebody of genus $g$ and $U_g$ its boundary. We show that for all $g \ge 0$ the map from $B Homeo(H_g)$ to $B Homeo(U_g)$ induced by restriction to the boundary admits a section.

Extending families of homeomorphisms over 4-dimensional handlebodies

TL;DR

The paper proves that for every genus the boundary-restriction map admits a section, enabling fiberwise extension of -bundles to -bundles. It develops a robust simplicial and topological framework of separating systems, introducing and and showing their diagonals are contractible, with weak equivalences connecting discrete and topological models. The main theorem is established via an Alexander-trick–style argument and the Orbit–Stabiliser lemma to compare homotopy orbit spaces, supported by a disc-system dual-graph construction that matches boundary data with a disc system on . Consequences include that any topological -bundle can be fiberwise filled by an -bundle and, for a 4-manifold with , the restriction map admits a section; this framework also facilitates rational characteristic class constructions on via the section.

Abstract

Let denote the 4-dimensional handlebody of genus and its boundary. We show that for all the map from to induced by restriction to the boundary admits a section.

Paper Structure

This paper contains 3 sections, 10 theorems, 32 equations.

Table of Contents

  1. Simplicial preliminaries
  2. Topological separating systems
  3. Proof of the main theorem

Key Result

Lemma 1

Let $\varphi\colon G \to H$ be a simplicial group homomorphism and $f\colon X \to Y$ a map of simplicial sets such that $G$ acts on $X$, $H$ acts on $Y$, and $f$ is $G$-equivariant. Assume that for each $y \in Y_0$ there is an $h \in H_0$ and $x \in X_0$ such that $h\cdot f(x)$ is in the same path c Then the induced map on homotopy orbits is a weak equivalence

Theorems & Definitions (22)

  • Lemma 1: Orbit stabiliser lemma
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 12 more