An equivariant Laudenbach-Poénaru theorem

TL;DR

The paper generalizes the Laudenbach–Poénaru extension theorem to finite group actions by introducing linearly parted $G$-actions on 4D $1$-handlebodies, proving that any finite group action on $Y=\#^n(S^1\times S^2)$ extends to a linearly parted $G$-action on $X=\natural^n(S^1\times B^3)$, with two extensions equivariantly diffeomorphic rel-boundary. It builds the theory from 3D equivariant topology (e.g., the Equivariant Sphere and Loop Theorems) and linear action characterizations to control attaching spheres and handle-cancellations, yielding a robust equivariant handlebody framework. The work also extends to pairs $(Y,L)$ with $G$-invariant unlink $L$ and $G$-equivariant boundary-parallel disk-tangles $\mathcal{D}$, proving existence and uniqueness of equivariant fillings $(X,\mathcal{D})$ up to $G$-diffeomorphism rel-boundary. These results lay groundwork for equivariant trisections and Kirby calculus in dimension four, and motivate open questions about four-dimensional linearization and fixed-point data determining smooth action classes.

Abstract

A foundational theorem of Laudenbach and Poénaru states that any diffeomorphism of $\#^n(S^1\times S^2)$ extends to a diffeomorphism of $\natural^n(S^1\times B^3)$. We prove a generalization of this theorem that accounts for the presence of a finite group action on $\#^n(S^1\times S^2)$. Our proof is independent of the classical theorem, so by considering the trivial group action, we give a new proof of the classical theorem. Specifically, we show that any finite group action on $\#^n(S^1\times S^2)$ extends to a $\textit{linearly parted}$ action on $\natural^n(S^1\times B^3)$ and that any two such extensions are equivariantly diffeomorphic. Roughly, a linearly parted action respects a decomposition into equivariant $0$-handles and $1$-handles, where, for each handle in the decomposition, its stabilizer acts linearly on that handle. The restriction to linearly parted actions is important, because there are infinitely many distinct nonlinear actions on $B^4$ with identical actions on $\partial B^4$; these nonlinear actions give extensions of the same action on $\partial B^4$ which are $\textit{not}$ equivariantly diffeomorphic. We also prove a more general theorem: Every finite group action on $\left(\#^n(S^1\times S^2),L\right)$, with $L$ an invariant unlink, extends across a pair $\left(\natural^n(S^1\times B^3),\mathcal{D}\right)$, with $\mathcal{D}$ an equivariantly boundary-parallel disk-tangle, and any two such extensions are equivariantly diffeomorphic.

Figures (3)

• Figure 1: An action on $B^n$ fixing a point $p$ in the boundary is the equivariant cone on an invariant $B^{n-1}$ in its boundary with cone point $p$.
• Figure 2: A solid torus, linearly parted into 3--balls with respect to the obvious, cyclic $\mathbb{Z}_6$--action
• Figure 3: Left: Removing the purple disk on the right from this punctured $3$--ball splits it into two punctured $3$--balls, and attaching the green disk on the left splits the puncture in two, so that the left-hand punctured $3$--ball turns into another punctured $3$--ball with one more puncture, as in Lemma \ref{['lem:innermost']}. Right: A schematic for the tube-and-cap approach in Lemma \ref{['lem:new_ball-system']}, with an initial choice of innermost sphere $S_I$ and capping 3--ball $B_I$ indicated.

Theorems & Definitions (49)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Corollary 3.6