Geodesic
On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e82

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Nariman, Sam. On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e82. doi: 10.1017/fms.2025.38
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[BBS24] Boyd, R., Bregman, C. and Steinebrunner, J., ‘Moduli spaces of 3-manifolds with boundary are finitc’, Preprint, 2024, . Google Scholar | arXiv

[BK23] Bamler, R. H. and Kleiner, B., ‘Ricci flow and contractibility of spaces of metrics’, J. Amer. Math. Soc. 36(2) (2023), 563–589. Google Scholar | DOI

[BK24] Bamler, R. H. and Kleiner, B., ‘Diffeomorphism groups of prime 3-manifolds’, J. Reine Angew. Math. 806 (2024), 23–35. Google Scholar

[BL74] Burghelea, D. and Lashof, R., ‘The homotopy type of the space of diffeomorphisms. I, II’, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. (1974), 37–50. Google Scholar | DOI

[Bon83] Bonahon, F., ‘Cobordism of automorphisms of surfaces’, Ann. Sci. École Norm. Sup. (4) 16(2) (1983), 237–270. Google Scholar | DOI

[CdSR79] De Sá, E. C. and Rourke, C., ‘The homotopy type of homeomorphisms of -manifolds’, Bull. Amer. Math. Soc. (N.S.) 1(1) (1979), 251–254. Google Scholar | DOI

[Cer61] Cerf, J., ‘Topologie de certains espaces de plongements’, Bull. Soc. Math. France 89 (1961), 227–380. Google Scholar | DOI

[EK71] Edwards, R. D. and Kirby, R. C, ‘Deformations of spaces of imbeddings’, Ann. of Math. (1971), 63–88. Google Scholar | DOI

[ERW19] Ebert, J. and Randal-Williams, O., ‘Semisimplicial spaces’, Algebr. Geom. Topol. 19(4) (2019), 2099–2150. Google Scholar | DOI

[FOT08] Félix, Y., Oprea, J. and Tanré, D., Algebraic Models in Geometry (OUP, Oxford, 2008). Google Scholar | DOI

[Ham74] Hamstrom, M.-E., ‘Homotopy in homeomorphism spaces’, and . Bulletin of the American Mathematical Society 80(2) (1974), 207–230. Google Scholar | DOI

[Hat76] Hatcher, A., ‘Homeomorphisms of sufficiently large -irreducible -manifolds’, Topology 15(4) (1976), 343–347. Google Scholar | DOI

[Hat81] Hatcher, A., ‘On the diffeomorphism group of ’, Proc. Amer. Math. Soc. 83(2) (1981), 427–430. Google Scholar

[Hat83] Hatcher, A. E., ‘A proof of the Smale conjecture, ’, Ann. of Math. (1983), 553–607. Google Scholar | DOI

[Hem76] Hempel, J., -Manifolds (Annals of Mathematics Studies) no. 86 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1976). Google Scholar

[HKMR12] Hong, S., Kalliongis, J., Mccullough, D. and Rubinstein, J. H., Diffeomorphisms of Elliptic 3-Manifolds (Lecture Notes in Mathematics) vol. 2055 (Springer, Heidelberg, 2012). Google Scholar | DOI

[HL84] Hendriks, H. and Laudenbach, F., ‘Difféomorphismes des sommes connexes en dimension trois’, Topology 23(4) (1984), 423–443. Google Scholar | DOI

[HM87] Hendriks, H. and Mccullough, D., ‘On the diffeomorphism group of a reducible -manifold’, Topology Appl. 26(1) (1987), 25–31. Google Scholar | DOI

[HM90] Hatcher, A. and Mccullough, D., ‘Finite presentation of 3-manifold mapping class groups’, in Groups of Self-Equivalences and Related Topics Lecture Notes in Math., 1425 (Springer-Verlag, Berlin, 1990), 48–57. Google Scholar | DOI

[HM97] Hatcher, A. and Mccullough, D., ‘Finiteness of classifying spaces of relative diffeomorphism groups of -manifolds’, Geom. Topol. 1 (1997), 91–109. Google Scholar | DOI

[Iva76] Ivanov, N. V., ‘Groups of diffeomorphisms of Waldhausen manifolds’, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 172–176, 209. Studies in topology, II. Google Scholar

[Kir95] Kirby, R.. ‘Problems in low-dimensional topology’, in Proceedings of Georgia Topology Conference, Part 2 (Citeseer, 1995). Google Scholar

[KS77] Kirby, R. C. and Siebenmann, L. C., Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (Annals of Mathematics Studies) no. 88 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977). With notes by John Milnor and Michael Atiyah. Google Scholar

[Kup15] Kupers, A., ‘Proving homological stability for homeomorphisms of manifolds’, Preprint, 2015, . Google Scholar | arXiv

[Kup19a] Kupers, A., ‘Lectures on diffeomorphism groups of manifolds’, in University Lecture Notes (2019), 59. Google Scholar

[Kup19b] Kupers, A., ‘Some finiteness results for groups of automorphisms of manifolds’, Geom. Topol. 23(5) (2019), 2277–2333. Google Scholar | DOI

[Las76] Lashof, R., ‘Embedding spaces’, Illinois J. Math. 20(1) (1976), 144–154. Google Scholar | DOI

[McD80] Mcduff, D., ‘The homology of some groups of diffeomorphisms’, Comment. Math. Helv. 55(1) (1980), 97–129. Google Scholar | DOI

[Mil57] Milnor, J.. ‘The geometric realization of a semi-simplicial complex’, Ann. of Math. (2) 65 (1957), 357–362. Google Scholar | DOI

[MN20] Mann, K. and Nariman, S., ‘Dynamical and cohomological obstructions to extending group actions’, Math. Ann. 377(3) (2020), 1313–1338. Google Scholar | DOI

[Nar20] Nariman, S., ‘A local to global argument on low dimensional manifolds’, Trans. Amer. Math. Soc. 373(2) (2020), 1307–1342. Google Scholar | DOI

[Neu96] Neumann, W. D., Notes on Geometry and 3-Manifolds (Topology Atlas, 1996). Google Scholar

[Pal60] Palais, R. S., ‘Local triviality of the restriction map for embeddings’, Comment. Math. Helv. 34 (1960), 305–312. Google Scholar | DOI

[Sma59] Smale, S., ‘Diffeomorphisms of the -sphere’, Proc. Amer. Math. Soc. 10 (1959), 621–626,. Google Scholar | DOI

[Thu74] Thurston, W., ‘Foliations and groups of diffeomorphisms’, Bull. Amer. Math. Soc. 80(2) (1974), 304–307. Google Scholar | DOI

[Wal68] Waldhausen, F., ‘On irreducible -manifolds which are sufficiently large’, Ann. of Math. (2) 87 (1968), 56–88. Google Scholar | DOI

[Whi12] Whitehead, G. W., Elements of Homotopy Theory vol. 61 (Springer Science & Business Media, 2012). Google Scholar

[Wil66] Williamson, R. E. Jr., ‘Cobordism of combinatorial manifolds’, Ann. of Math. (2) 83 (1966), 1–33. Google Scholar | DOI

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