Geodesic
The homology of moduli spaces of 4-manifolds may be infinitely generated
Forum of Mathematics, Pi, Tome 12 (2024)

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For a simply-connected closed manifold X of $\dim X \neq 4$, the mapping class group $\pi _0(\mathrm {Diff}(X))$ 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 $S^2 \times S^2$. These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory. 
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Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum of Mathematics, Pi, Tome 12 (2024). doi: 10.1017/fmp.2024.26

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