Four-manifolds, two-complexes and the quadratic bias invariant
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918323248168960 |
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| author | Hambleton, Ian Nicholson, John |
| author_facet | Hambleton, Ian Nicholson, John |
| contents | Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k \ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Four-manifolds, two-complexes and the quadratic bias invariant Hambleton, Ian Nicholson, John Geometric Topology Algebraic Topology 57K41, 57N65, 20F05, 55U15, 57R67 Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k \ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent. |
| title | Four-manifolds, two-complexes and the quadratic bias invariant |
| topic | Geometric Topology Algebraic Topology 57K41, 57N65, 20F05, 55U15, 57R67 |
| url | https://arxiv.org/abs/2412.15089 |