An equivariant Laudenbach-Po\'enaru theorem

A foundational theorem of Laudenbach and Po\'enaru 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.