An equivariant Laudenbach-Poénaru theorem
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910789158305792 |
|---|---|
| author | Meier, Jeffrey Scott, Evan |
| author_facet | Meier, Jeffrey Scott, Evan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An equivariant Laudenbach-Poénaru theorem Meier, Jeffrey Scott, Evan Geometric Topology 57M60, 57K40, 57R65 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. |
| title | An equivariant Laudenbach-Poénaru theorem |
| topic | Geometric Topology 57M60, 57K40, 57R65 |
| url | https://arxiv.org/abs/2501.10524 |