Spaces of graphs and surfaces: on the work of Søren Galatius
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Abstract:
We put Soren Galatius’s result on the homology of the automorphism group of free groups into context. In particular we explain its relation to the Mumford conjecture and the main ideas of the proofs.References
- Tatsuji Kudo and Shôrô Araki, Topology of $H_n$-spaces and $H$-squaring operations, Mem. Fac. Sci. Kyūsyū Univ. A 10 (1956), 85–120. MR 87948
- Michael Barratt and Stewart Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972), 1–14. MR 314940, DOI 10.1007/BF02566785
- Joan S. Birman and Bronislaw Wajnryb, Presentations of the mapping class group. Errata: “$3$-fold branched coverings and the mapping class group of a surface” [in Geometry and topology (College Park, MD, 1983/84), 24–46, Lecture Notes in Math., 1167, Springer, Berlin, 1985; MR0827260 (87g:57019)] and “A simple presentation of the mapping class group of an orientable surface” [Israel J. Math. 45 (1983), no. 2-3, 157–174; MR0719117 (85g:57007)] by Wajnryb, Israel J. Math. 88 (1994), no. 1-3, 425–427. MR 1303506, DOI 10.1007/BF02937522
- S. K. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, Mat. Zeit., to appear arXiv:0904.3269
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- Ruth Charney and Ronnie Lee, An application of homotopy theory to mapping class groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 127–135. MR 885100, DOI 10.1016/0022-4049(87)90020-X
- Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. MR 830040, DOI 10.1007/BF01388734
- Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
- Eldon Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 35–88. MR 141112, DOI 10.2307/2372804
- J. Ebert, A vanishing theorem for characteristic classes of odd-dimensional manifold bundles, arXiv:0902.4719
- Johannes Ebert, Algebraic independence of generalized MMM-classes, Algebr. Geom. Topol. 11 (2011), no. 1, 69–105. MR 2764037, DOI 10.2140/agt.2011.11.69
- B. Farb, D. Margalit, A primer on mapping class groups, Princeton University Press, to appear.
- Søren Galatius, Mod $p$ homology of the stable mapping class group, Topology 43 (2004), no. 5, 1105–1132. MR 2079997, DOI 10.1016/j.top.2004.01.011
- Søren Galatius, Stable homology of automorphism groups of free groups, Ann. of Math. (2) 173 (2011), no. 2, 705–768. MR 2784914, DOI 10.4007/annals.2011.173.2.3
- Soren Galatius, Ib Madsen, and Ulrike Tillmann, Divisibility of the stable Miller-Morita-Mumford classes, J. Amer. Math. Soc. 19 (2006), no. 4, 759–779. MR 2219303, DOI 10.1090/S0894-0347-06-00523-6
- Søren Galatius, Ulrike Tillmann, Ib Madsen, and Michael Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009), no. 2, 195–239. MR 2506750, DOI 10.1007/s11511-009-0036-9
- Søren Galatius and Oscar Randal-Williams, Monoids of moduli spaces of manifolds, Geom. Topol. 14 (2010), no. 3, 1243–1302. MR 2653727, DOI 10.2140/gt.2010.14.1243
- S. M. Gersten, A presentation for the special automorphism group of a free group, J. Pure Appl. Algebra 33 (1984), no. 3, 269–279. MR 761633, DOI 10.1016/0022-4049(84)90062-8
- J. Giansiracusa, U. Tillmann, Vanishing of universal charcteristic classes for handlebody groups and boundary bundles, arXiv:0910.5367
- John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221–239. MR 700769, DOI 10.1007/BF01389321
- John L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215–249. MR 786348, DOI 10.2307/1971172
- John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, DOI 10.1007/BF01388737
- Allen E. Hatcher, A proof of the Smale conjecture, $\textrm {Diff}(S^{3})\simeq \textrm {O}(4)$, Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR 701256, DOI 10.2307/2007035
- Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), no. 1, 39–62. MR 1314940, DOI 10.1007/BF02565999
- Allen Hatcher and Karen Vogtmann, Rational homology of $\textrm {Aut}(F_n)$, Math. Res. Lett. 5 (1998), no. 6, 759–780. MR 1671188, DOI 10.4310/MRL.1998.v5.n6.a6
- Allen Hatcher and Karen Vogtmann, Cerf theory for graphs, J. London Math. Soc. (2) 58 (1998), no. 3, 633–655. MR 1678155, DOI 10.1112/S0024610798006644
- Allen Hatcher and Karen Vogtmann, Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol. 4 (2004), 1253–1272. MR 2113904, DOI 10.2140/agt.2004.4.1253
- Allan Hatcher, Karen Vogtmann, and Natalie Wahl, Erratum to: “Homology stability for outer automorphism groups of free groups [Algebr. Geom. Topol. 4 (2004), 1253–1272 (electronic); MR2113904] by Hatcher and Vogtmann, Algebr. Geom. Topol. 6 (2006), 573–579. MR 2220689, DOI 10.2140/agt.2006.6.573
- Allen Hatcher and Nathalie Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J. 155 (2010), no. 2, 205–269. MR 2736166, DOI 10.1215/00127094-2010-055
- Kiyoshi Igusa, Higher Franz-Reidemeister torsion, AMS/IP Studies in Advanced Mathematics, vol. 31, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1945530, DOI 10.1090/amsip/031
- N. V. Ivanov, Stabilization of the homology of Teichmüller modular groups, Algebra i Analiz 1 (1989), no. 3, 110–126 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 675–691. MR 1015128
- Ib Madsen and Ulrike Tillmann, The stable mapping class group and $Q(\Bbb C \mathrm P^\infty _+)$, Invent. Math. 145 (2001), no. 3, 509–544. MR 1856399, DOI 10.1007/PL00005807
- Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941. MR 2335797, DOI 10.4007/annals.2007.165.843
- Edward Y. Miller, The homology of the mapping class group, J. Differential Geom. 24 (1986), no. 1, 1–14. MR 857372
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
- David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602
- David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR 717614
- Minoru Nakaoka, Decomposition theorem for homology groups of symmetric groups, Ann. of Math. (2) 71 (1960), 16–42. MR 112134, DOI 10.2307/1969878
- Eric M. Friedlander and Barry Mazur, Filtrations on the homology of algebraic varieties, Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110. With an appendix by Daniel Quillen. MR 1211371, DOI 10.1090/memo/0529
- O. Randal-Williams, Resolutions of moduli spaces and homological stability, arXiv:0909.4278
- Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
- Ulrike Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997), no. 2, 257–275. MR 1474157, DOI 10.1007/s002220050184
- Karen Vogtmann, The cohomology of automorphism groups of free groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1101–1117. MR 2275637
- Nathalie Wahl, Homological stability for the mapping class groups of non-orientable surfaces, Invent. Math. 171 (2008), no. 2, 389–424. MR 2367024, DOI 10.1007/s00222-007-0085-7
- Bronislaw Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), no. 2-3, 157–174. MR 719117, DOI 10.1007/BF02774014
Additional Information
- Ulrike Tillmann
- Affiliation: Mathematical Institute, Oxford University, 24-29 St Giles St., Oxford OX1 3LB, United Kingdom
- Email: tillmann@maths.ox.ac.uk
- Received by editor(s): July 18, 2011
- Received by editor(s) in revised form: August 7, 2011
- Published electronically: October 19, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 49 (2012), 73-90
- MSC (2010): Primary 20J06
- DOI: https://doi.org/10.1090/S0273-0979-2011-01360-X
- MathSciNet review: 2869008