On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds
Kontsevich ([Kir95, Problem 3.48]) conjectured that $\mathrm {BDiff}(M, \text {rel }\partial )$ has the homotopy type of a finite CW complex for all compact $3$ -manifolds with nonempty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We prove a homolo...
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000386/type/journal_article |
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| Summary: | Kontsevich ([Kir95, Problem 3.48]) conjectured that
$\mathrm {BDiff}(M, \text {rel }\partial )$
has the homotopy type of a finite CW complex for all compact
$3$
-manifolds with nonempty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We prove a homological version of Kontsevich’s conjecture. More precisely, we show that
$\mathrm {BDiff}(M, \text {rel }\partial )$
has finitely many nonzero homology groups each finitely generated when M is a connected sum of irreducible
$3$
-manifolds that each have a nontrivial and non-spherical boundary. |
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| ISSN: | 2050-5094 |