Morita classes in the homology of automorphism groups of free groups

Conant, James¹ ; Vogtmann, Karen²

¹ Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996, USA
² Department of Mathematics, Cornell Univeristy, Ithaca, New York 14853-4201, USA

Geometry & topology, Tome 8 (2004) no. 3, pp. 1471-1499.

Résumé

Using Kontsevich’s identification of the homology of the Lie algebra ${\ell }_{\infty }$ with the cohomology of $Out\left(F_r\right)$, Morita defined a sequence of $4k$–dimensional classes ${\mu }_k$ in the unstable rational homology of $Out\left(F_{2k+2}\right)$. He showed by a computer calculation that the first of these is non-trivial, so coincides with the unique non-trivial rational homology class for $Out\left(F_4\right)$. Using the “forested graph complex" introduced in an earlier paper, we reinterpret and generalize Morita’s cycles, obtaining an unstable cycle for every connected odd-valent graph. (Morita has independently found similar generalizations of these cycles.) The description of Morita’s original cycles becomes quite simple in this interpretation, and we are able to show that the second Morita cycle also gives a nontrivial homology class. Finally, we view things from the point of view of a different chain complex, one which is associated to Bestvina and Feighn’s bordification of outer space. We construct cycles which appear to be the same as the Morita cycles constructed in the first part of the paper. In this setting, a further generalization becomes apparent, giving cycles for objects more general than odd-valent graphs. Some of these cycles lie in the stable range. We also observe that these cycles lift to cycles for $Aut\left(F_r\right)$.

Détail

DOI : 10.2140/gt.2004.8.1471

Keywords: automorphism groups of free groups, graph homology

Bibliographie

[1] M Bestvina, M Feighn, The topology at infinity of $\mathrm{Out}(F_n)$, Invent. Math. 140 (2000) 651

[2] J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167

[3] J Conant, F Gerlits, K Vogtmann, Cut vertices in commutative graphs, Q. J. Math. 56 (2005) 321

[4] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91

[5] F Gerlits, PhD thesis, Cornell University (2002)

[6] A Hatcher, K Vogtmann, Cerf theory for graphs, J. London Math. Soc. $(2)$ 58 (1998) 633

[7] A Hatcher, K Vogtmann, Rational homology of $\mathrm{Aut}(F_n)$, Math. Res. Lett. 5 (1998) 759

[8] A Hatcher, K Vogtmann, Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol. 4 (2004) 1253

[9] M Kontsevich, Formal (non)commutative symplectic geometry, from: "The Gel'fand Mathematical Seminars, 1990–1992", Birkhäuser (1993) 173

[10] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97

[11] J Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group”, Algebr. Geom. Topol. 2 (2002) 1197

[12] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699

[13] S Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)", Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 349

[14] K Vogtmann, Automorphisms of free groups and outer space, from: "Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000)" (2002) 1