Homotopy classification of $4$-manifolds with $3$-manifold fundamental group
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918121154019328 |
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| author | Hillman, Jonathan Kasprowski, Daniel Powell, Mark Ray, Arunima |
| author_facet | Hillman, Jonathan Kasprowski, Daniel Powell, Mark Ray, Arunima |
| contents | We give a criterion on a group $π$ and a homomorphism $w \colon π\to C_2$ under which closed $4$-manifolds with fundamental group $π$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic $2$-types. We verify the criterion for a large class of $3$-manifold groups and orientation characters, in particular for the fundamental group $π$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $π$ of finite order. We deduce a homeomorphism classification of closed, orientable $4$-manifolds with infinite dihedral fundamental group $\mathbb{Z}/2 * \mathbb{Z}/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy classification of $4$-manifolds with $3$-manifold fundamental group Hillman, Jonathan Kasprowski, Daniel Powell, Mark Ray, Arunima Geometric Topology 57K40 We give a criterion on a group $π$ and a homomorphism $w \colon π\to C_2$ under which closed $4$-manifolds with fundamental group $π$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic $2$-types. We verify the criterion for a large class of $3$-manifold groups and orientation characters, in particular for the fundamental group $π$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $π$ of finite order. We deduce a homeomorphism classification of closed, orientable $4$-manifolds with infinite dihedral fundamental group $\mathbb{Z}/2 * \mathbb{Z}/2$. |
| title | Homotopy classification of $4$-manifolds with $3$-manifold fundamental group |
| topic | Geometric Topology 57K40 |
| url | https://arxiv.org/abs/2508.07504 |