The homology of moduli spaces of 4-manifolds may be infinitely generated
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909404603875328 |
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| author | Konno, Hokuto |
| author_facet | Konno, Hokuto |
| contents | For a simply-connected closed manifold $X$ of $\dim X \neq 4$, the mapping class group $π_0(\mathrm{Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$, we prove that there are simply-connected closed smooth 4-manifolds $X$ for which $H_k(B\mathrm{Diff}(X);\mathbb{Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm{Diff}(X);\mathbb{Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$. The proof uses characteristic classes obtained from Seiberg-Witten theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_18695 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The homology of moduli spaces of 4-manifolds may be infinitely generated Konno, Hokuto Geometric Topology Algebraic Topology Differential Geometry For a simply-connected closed manifold $X$ of $\dim X \neq 4$, the mapping class group $π_0(\mathrm{Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$, we prove that there are simply-connected closed smooth 4-manifolds $X$ for which $H_k(B\mathrm{Diff}(X);\mathbb{Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm{Diff}(X);\mathbb{Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$. The proof uses characteristic classes obtained from Seiberg-Witten theory. |
| title | The homology of moduli spaces of 4-manifolds may be infinitely generated |
| topic | Geometric Topology Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2310.18695 |