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Compact and finite-type support in the homology of big mapping class groups (2405.03512v2)
Published 6 May 2024 in math.GT, math.AT, and math.GR
Abstract: For any infinite-type surface $S$, a natural question is whether the homology of its mapping class group contains any non-trivial classes that are supported on (i) a compact subsurface or (ii) a finite-type subsurface. Our purpose here is to study this question, in particular giving an almost-complete answer when the genus of $S$ is positive (including infinite) and a partial answer when the genus of $S$ is zero. Our methods involve the notion of shiftable subsurfaces as well as homological stability for mapping class groups of finite-type surfaces.
- The first integral cohomology of pure mapping class groups. Int. Math. Res. Not. IMRN, 2020(22):8973–8996, 2020.
- Big mapping class groups: an overview. In Ken’ichi Ohshika and Athanase Papadopoulos, editors, In the Tradition of Thurston: Geometry and Topology, pages 459–496. Springer, 2020.
- Signature cocycles on the mapping class group and symplectic groups. J. Pure Appl. Algebra, 224(11):106400, 49, 2020.
- A. Jon Berrick. An approach to algebraic K𝐾Kitalic_K-theory, volume 56 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.
- A. J. Berrick. A topologist’s view of perfect and acyclic groups. In Invitations to geometry and topology, volume 7 of Oxf. Grad. Texts Math., pages 1–28. Oxford Univ. Press, Oxford, 2002.
- The classification of two-dimensional manifolds. Trans. Amer. Math. Soc., 255:377–402, 1979.
- Søren K. Boldsen. Improved homological stability for the mapping class group with integral or twisted coefficients. Math. Z., 270(1-2):297–329, 2012.
- Stripping and splitting decorated mapping class groups. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 47–57. Birkhäuser, Basel, 2001.
- George Domat. Big pure mapping class groups are never perfect. Math. Res. Lett., 29(3):691–726, 2022. Appendix with Ryan Dickmann.
- A fibre bundle description of Teichmüller theory. J. Differential Geometry, 3:19–43, 1969.
- C. J. Earle and A. Schatz. Teichmüller theory for surfaces with boundary. J. Differential Geometry, 4:169–185, 1970.
- A primer on mapping class groups. Princeton, NJ: Princeton University Press, 2011.
- Mary-Elizabeth Hamstrom. Homotopy groups of the space of homeomorphisms on a 2222-manifold. Illinois J. Math., 10:563–573, 1966.
- John L. Harer. Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2), 121(2):215–249, 1985.
- Morris W. Hirsch. Differential topology, volume No. 33 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.
- Stabilization for mapping class groups of 3-manifolds. Duke Math. J., 155(2):205–269, 2010.
- Nikolai V. Ivanov. On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients. In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), volume 150 of Contemp. Math., pages 149–194. Amer. Math. Soc., Providence, RI, 1993.
- Thomas Jech. Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, millennium edition, 2003.
- Mustafa Korkmaz. Low-dimensional homology groups of mapping class groups: a survey. Turkish J. Math., 26(1):101–114, 2002.
- Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah.
- John N. Mather. The vanishing of the homology of certain groups of homeomorphisms. Topology, 10:297–298, 1971.
- Contribution à la topologie des ensembles dénombrables. Fundam. Math., 1:17–27, 1920.
- Self-similar surfaces: involutions and perfection. ArXiv:2106.03681.
- David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, volume 36 of Progr. Math., pages 271–328. Birkhäuser Boston, Boston, MA, 1983.
- Ib Madsen and Michael Weiss. The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2), 165(3):843–941, 2007.
- On the homology of big mapping class groups. ArXiv:2211.07470.
- Big mapping class groups with uncountable integral homology. Doc. Math., 29(1):159–189, 2024.
- Ian Richards. On the classification of noncompact surfaces. Trans. Amer. Math. Soc., 106:259–269, 1963.
- Oscar Randal-Williams. Resolutions of moduli spaces and homological stability. J. Eur. Math. Soc. (JEMS), 18(1):1–81, 2016.
- Takuya Sakasai. Lagrangian mapping class groups from a group homological point of view. Algebr. Geom. Topol., 12(1):267–291, 2012.
- Wacław Sierpiński. Cardinal and ordinal numbers, volume Tom 34 of Polska Akademia Nauk. Monografie Matematyczne. Państwowe Wydawnictwo Naukowe, Warsaw, 1958.
- B. von Kerékjártó. Vorlesungen über Topologie. I.: Flächentopologie., volume 8 of Grundlehren Math. Wiss. Springer, Cham, 1923.
- Nathalie Wahl. Homological stability for mapping class groups of surfaces. In Handbook of moduli. Vol. III, volume 26 of Adv. Lect. Math. (ALM), pages 547–583. Int. Press, Somerville, MA, 2013.
- Tatsuhiko Yagasaki. Homotopy types of homeomorphism groups of noncompact 2222-manifolds. Topology Appl., 108(2):123–136, 2000.
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