|
Spaces of graphs and surfaces: on the work of Søren Galatius
Author:
Ulrike Tillmann
Journal:
Bull. Amer. Math. Soc. 49 (2012), 73-90
MSC (2010):
Primary 20J06
Posted:
October 19, 2011
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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
- 1.
Tatsuji
Kudo and Shôrô
Araki, Topology of 𝐻_{𝑛}-spaces and
𝐻-squaring operations, Mem. Fac. Sci. Kyūsyū
Univ. Ser. A. 10 (1956), 85–120. MR 0087948
(19,442b)
- 2.
Michael
Barratt and Stewart
Priddy, On the homology of non-connected monoids and their
associated groups, Comment. Math. Helv. 47 (1972),
1–14. MR
0314940 (47 #3489)
- 3.
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
(95j:57014), http://dx.doi.org/10.1007/BF02937522
- 4.
S. K. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, Mat. Zeit., to appear arXiv:0904.3269
- 5.
Kenneth
S. Brown, Cohomology of groups, Graduate Texts in Mathematics,
vol. 87, Springer-Verlag, New York, 1982. MR 672956
(83k:20002)
- 6.
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
(88m:55024), http://dx.doi.org/10.1016/0022-4049(87)90020-X
- 7.
Marc
Culler and Karen
Vogtmann, Moduli of graphs and automorphisms of free groups,
Invent. Math. 84 (1986), no. 1, 91–119. MR 830040
(87f:20048), http://dx.doi.org/10.1007/BF01388734
- 8.
Clifford
J. Earle and James
Eells, A fibre bundle description of Teichmüller theory,
J. Differential Geometry 3 (1969), 19–43. MR 0276999
(43 #2737a)
- 9.
Eldon
Dyer and R.
K. Lashof, Homology of iterated loop spaces, Amer. J. Math.
84 (1962), 35–88. MR 0141112
(25 #4523)
- 10.
J. Ebert, A vanishing theorem for characteristic classes of odd-dimensional manifold bundles, arXiv:0902.4719
- 11.
Johannes
Ebert, Algebraic independence of generalized MMM-classes,
Algebr. Geom. Topol. 11 (2011), no. 1, 69–105.
MR
2764037, http://dx.doi.org/10.2140/agt.2011.11.69
- 12.
B. Farb, D. Margalit, A primer on mapping class groups, Princeton University Press, to appear.
- 13.
Søren
Galatius, Mod 𝑝 homology of the stable mapping class
group, Topology 43 (2004), no. 5,
1105–1132. MR 2079997
(2006a:57020), http://dx.doi.org/10.1016/j.top.2004.01.011
- 14.
Søren
Galatius, Stable homology of automorphism groups of free
groups, Ann. of Math. (2) 173 (2011), no. 2,
705–768. MR 2784914
(2012c:20149), http://dx.doi.org/10.4007/annals.2011.173.2.3
- 15.
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
(2006m:57039), http://dx.doi.org/10.1090/S0894-0347-06-00523-6
- 16.
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
(2011c:55022), http://dx.doi.org/10.1007/s11511-009-0036-9
- 17.
Søren
Galatius and Oscar
Randal-Williams, Monoids of moduli spaces of manifolds, Geom.
Topol. 14 (2010), no. 3, 1243–1302. MR 2653727
(2011j:57047), http://dx.doi.org/10.2140/gt.2010.14.1243
- 18.
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
(86f:20041), http://dx.doi.org/10.1016/0022-4049(84)90062-8
- 19.
J. Giansiracusa, U. Tillmann, Vanishing of universal charcteristic classes for handlebody groups and boundary bundles, arXiv:0910.5367
- 20.
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
(84g:57006), http://dx.doi.org/10.1007/BF01389321
- 21.
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
(87f:57009), http://dx.doi.org/10.2307/1971172
- 22.
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
(87c:32030), http://dx.doi.org/10.1007/BF01388737
- 23.
Allen
E. Hatcher, A proof of the Smale conjecture,
𝐷𝑖𝑓𝑓(𝑆³)≃𝑂(4),
Ann. of Math. (2) 117 (1983), no. 3, 553–607.
MR 701256
(85c:57008), http://dx.doi.org/10.2307/2007035
- 24.
Allen
Hatcher, Homological stability for automorphism groups of free
groups, Comment. Math. Helv. 70 (1995), no. 1,
39–62. MR
1314940 (95k:20030), http://dx.doi.org/10.1007/BF02565999
- 25.
Allen
Hatcher and Karen
Vogtmann, Rational homology of
𝐴𝑢𝑡(𝐹_{𝑛}), Math. Res. Lett.
5 (1998), no. 6, 759–780. MR 1671188
(99m:20127)
- 26.
Allen
Hatcher and Karen
Vogtmann, Cerf theory for graphs, J. London Math. Soc. (2)
58 (1998), no. 3, 633–655. MR 1678155
(2000e:20041), http://dx.doi.org/10.1112/S0024610798006644
- 27.
Allen
Hatcher and Karen
Vogtmann, Homology stability for outer automorphism groups of free
groups, Algebr. Geom. Topol. 4 (2004),
1253–1272. MR 2113904
(2005j:20038), http://dx.doi.org/10.2140/agt.2004.4.1253
- 28.
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); MR 2113904] by Hatcher and Vogtmann, Algebr. Geom.
Topol. 6 (2006), 573–579 (electronic). MR 2220689
(2006k:20069), http://dx.doi.org/10.2140/agt.2006.6.573
- 29.
Allen
Hatcher and Nathalie
Wahl, Stabilization for mapping class groups of 3-manifolds,
Duke Math. J. 155 (2010), no. 2, 205–269. MR 2736166
(2012c:57001), http://dx.doi.org/10.1215/00127094-2010-055
- 30.
Kiyoshi
Igusa, Higher Franz-Reidemeister torsion, AMS/IP Studies in
Advanced Mathematics, vol. 31, American Mathematical Society,
Providence, RI, 2002. MR 1945530
(2004f:19003)
- 31.
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
(91g:57010)
- 32.
Ib
Madsen and Ulrike
Tillmann, The stable mapping class group and
𝑄(ℂ\𝕣𝕠𝕞𝕒𝕟ℙ^{∞}₊),
Invent. Math. 145 (2001), no. 3, 509–544. MR 1856399
(2002h:55011), http://dx.doi.org/10.1007/PL00005807
- 33.
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
(2009b:14051), http://dx.doi.org/10.4007/annals.2007.165.843
- 34.
Edward
Y. Miller, The homology of the mapping class group, J.
Differential Geom. 24 (1986), no. 1, 1–14. MR 857372
(88b:32051)
- 35.
John
W. Milnor and James
D. Stasheff, Characteristic classes, Princeton University
Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76. MR 0440554
(55 #13428)
- 36.
David
Mumford, Geometric invariant theory, Ergebnisse der Mathematik
und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin, 1965.
MR
0214602 (35 #5451)
- 37.
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
(85j:14046)
- 38.
Minoru
Nakaoka, Decomposition theorem for homology groups of symmetric
groups, Ann. of Math. (2) 71 (1960), 16–42. MR 0112134
(22 #2989)
- 39.
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
(95a:14023)
- 40.
O. Randal-Williams, Resolutions of moduli spaces and homological stability, arXiv:0909.4278
- 41.
Graeme
Segal, Configuration-spaces and iterated loop-spaces, Invent.
Math. 21 (1973), 213–221. MR 0331377
(48 #9710)
- 42.
Ulrike
Tillmann, On the homotopy of the stable mapping class group,
Invent. Math. 130 (1997), no. 2, 257–275. MR 1474157
(99k:57036), http://dx.doi.org/10.1007/s002220050184
- 43.
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
(2007k:20090)
- 44.
Nathalie
Wahl, Homological stability for the mapping class groups of
non-orientable surfaces, Invent. Math. 171 (2008),
no. 2, 389–424. MR 2367024
(2008m:57047), http://dx.doi.org/10.1007/s00222-007-0085-7
- 45.
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
(85g:57007), http://dx.doi.org/10.1007/BF02774014
Similar Articles
Retrieve articles in Bulletin of the American Mathematical Society
with MSC (2010):
20J06
Retrieve articles in all journals
with MSC (2010):
20J06
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
DOI:
http://dx.doi.org/10.1090/S0273-0979-2011-01360-X
PII:
S 0273-0979(2011)01360-X
Received by editor(s):
July, 18, 2011,
Received by editor(s) in revised form:
August 7, 2011
Posted:
October 19, 2011
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|