$4$-manifolds with boundary and fundamental group $\mathbb{Z}$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913472452755456 |
|---|---|
| author | Conway, Anthony Piccirillo, Lisa Powell, Mark |
| author_facet | Conway, Anthony Piccirillo, Lisa Powell, Mark |
| contents | We classify topological $4$-manifolds with boundary and fundamental group $\mathbb{Z}$, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected $4$-manifolds with $S^3$ boundary, where the fundamental group of the surface complement is $\mathbb{Z}$. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over $\mathbb{Z}[t^{\pm 1}]$ arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group $\mathbb{Z}$. For surfaces we have a similar result, and in particular we show that every $2$-handlebody with $S^3$ boundary contains a pair of exotic discs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12774 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $4$-manifolds with boundary and fundamental group $\mathbb{Z}$ Conway, Anthony Piccirillo, Lisa Powell, Mark Geometric Topology 57N35, 57K40, 57R55, 57K10, 57K43, 57R40 We classify topological $4$-manifolds with boundary and fundamental group $\mathbb{Z}$, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected $4$-manifolds with $S^3$ boundary, where the fundamental group of the surface complement is $\mathbb{Z}$. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over $\mathbb{Z}[t^{\pm 1}]$ arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group $\mathbb{Z}$. For surfaces we have a similar result, and in particular we show that every $2$-handlebody with $S^3$ boundary contains a pair of exotic discs. |
| title | $4$-manifolds with boundary and fundamental group $\mathbb{Z}$ |
| topic | Geometric Topology 57N35, 57K40, 57R55, 57K10, 57K43, 57R40 |
| url | https://arxiv.org/abs/2205.12774 |