The Borel Conjecture for manifolds with boundary
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915114279501824 |
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| author | Davis, James F. Hillman, J. A. |
| author_facet | Davis, James F. Hillman, J. A. |
| contents | We undertake a systematic investigation of compact aspherical manifolds with boundary; motivated by the plethora of examples in the bounded case and by the beauty of the theory in the closed case. Our main theorems give a homological criterion for when a closed manifold, together with maps from the fundamental groups of its components to a fixed group, can be realized as the boundary of a compact aspherical manifold. This is done in two steps: we first produce a Poincaré pair and then apply surgery theory to obtain a manifold. We illustrate this in the case of abelian fundamental group. The results of this paper will be applied in a sequel where we classify compact aspherical 4-manifolds with elementary amenable fundamental group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12509 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Borel Conjecture for manifolds with boundary Davis, James F. Hillman, J. A. Geometric Topology Algebraic Topology Group Theory 57R67, 57P10, 20J05, 57K10 We undertake a systematic investigation of compact aspherical manifolds with boundary; motivated by the plethora of examples in the bounded case and by the beauty of the theory in the closed case. Our main theorems give a homological criterion for when a closed manifold, together with maps from the fundamental groups of its components to a fixed group, can be realized as the boundary of a compact aspherical manifold. This is done in two steps: we first produce a Poincaré pair and then apply surgery theory to obtain a manifold. We illustrate this in the case of abelian fundamental group. The results of this paper will be applied in a sequel where we classify compact aspherical 4-manifolds with elementary amenable fundamental group. |
| title | The Borel Conjecture for manifolds with boundary |
| topic | Geometric Topology Algebraic Topology Group Theory 57R67, 57P10, 20J05, 57K10 |
| url | https://arxiv.org/abs/2501.12509 |