Aspherical 4-manifolds with elementary amenable fundamental group
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arXiv
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| Format: | Preprint |
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2025
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| author | Davis, James F. Hillman, J. A. |
| author_facet | Davis, James F. Hillman, J. A. |
| contents | We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Aspherical 4-manifolds with elementary amenable fundamental group Davis, James F. Hillman, J. A. Geometric Topology Algebraic Topology 57M05, 57K41, 20J05, 57R67, 57K10 We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold. |
| title | Aspherical 4-manifolds with elementary amenable fundamental group |
| topic | Geometric Topology Algebraic Topology 57M05, 57K41, 20J05, 57R67, 57K10 |
| url | https://arxiv.org/abs/2501.12512 |