Aspherical 4-manifolds with positive Euler characteristic and their geography
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917092912005120 |
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| author | Capovilla, Pietro |
| author_facet | Capovilla, Pietro |
| contents | We present an explicit construction of closed oriented aspherical smooth 4-manifolds with $χ= σ= n$ for every positive integer $n$. This proves a conjecture of Edmonds by providing a closed oriented aspherical 4-manifold with Euler characteristic 1, and it shows that the real analogue of the Bogomolov-Miyaoka-Yau inequality fails for aspherical 4-manifolds. By the Hitchin-Thorpe inequality, these manifolds do not admit Einstein metrics. As a further consequence of our construction, we show that every closed aspherical 3-manifold with amenable fundamental group is virtually the $π_1$-injective boundary of an aspherical 4-manifold with vanishing Euler characteristic and vanishing simplicial volume, thereby answering questions of Edmonds and of Löh-Moraschini-Raptis up to finite covers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Aspherical 4-manifolds with positive Euler characteristic and their geography Capovilla, Pietro Geometric Topology We present an explicit construction of closed oriented aspherical smooth 4-manifolds with $χ= σ= n$ for every positive integer $n$. This proves a conjecture of Edmonds by providing a closed oriented aspherical 4-manifold with Euler characteristic 1, and it shows that the real analogue of the Bogomolov-Miyaoka-Yau inequality fails for aspherical 4-manifolds. By the Hitchin-Thorpe inequality, these manifolds do not admit Einstein metrics. As a further consequence of our construction, we show that every closed aspherical 3-manifold with amenable fundamental group is virtually the $π_1$-injective boundary of an aspherical 4-manifold with vanishing Euler characteristic and vanishing simplicial volume, thereby answering questions of Edmonds and of Löh-Moraschini-Raptis up to finite covers. |
| title | Aspherical 4-manifolds with positive Euler characteristic and their geography |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2511.15577 |