On the Cohomology of Torelli Groups

Abstract

The Torelli group of Wg = # g S n × S n is the subgroup of the diffeomorphisms of Wg fixing a disk which act trivially on Hn(Wg; Z). The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of Sp 2g (Z) or Og,g(Z). In this paper we prove that for 2n ≥ 6 and g ≥ 2, they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.Date: October 31, 2019. Corollary B. Let 2n ≥ 6, g ≥ 2, V be an algebraic G ′ g -representation, and if n is even then assume that G ′′ g is not entirely contained in SO g,g (Z). Then the natural mapwhich is a split injection by transfer, is an isomorphism in degrees * < g − e, with e = 0 if n is odd and e = 1 if n is even.Our techniques can also be used to prove a second property of Torelli groups:Theorem C. For 2n ≥ 6, the spaces BTor (W g , D 2n ) are nilpotent.The spaces BDiff(W g , D 2n ) classify smooth fibre bundles with fibre W g containing a trivialised disc bundle, and can be considered as moduli spaces of such manifolds. In Section 8 we prove the natural generalisations of Theorem A, Corollary B, and Theorem C to moduli spaces of manifolds with certain tangential structures (such as framings).Acknowledgements. The authors are grateful to Manuel Krannich for his comments.

Stable cohomology of graph complexes

“…We shall also note that results partially overlapping ours have independently been obtained by Kupers and Randal-Williams [18].…”

The cohomology of Torelli groups is algebraic

“…This only uses that ′′ is an arithmetic subgroup of G(Q) and so also holds with ′′ replaced with ′ . Now by Theorem A the representation ( Tor ( ,1 ); Q) ⊗ is algebraic, and by our assumptions both ′ and ′′ are Zariski-dense in G(Q) (see Subsection 2.1.1 of [34]). Thus, the ′ -and ′′invariants coincide, so the map of total spaces induces an isomorphism on homology in total degrees * < − .…”

“…We usually denote a representation ( , ) by , leaving the action of on implicit. Properties (a), (b) and (c) of algebraic representations listed below are obtained in Subsection 2.1 of [34] by combining several results in the literature, and properties (d) and (e) are direct consequences of the definition.…”

“…Theorem A may be used in conjunction with the main result of [34]: for 2 ≥ 6 that paper computes the maximal algebraic subrepresentation of * ( Tor( , 2 ); Q) in a stable range; by Theorem A this is in fact the entire cohomology.…”

“…To compare the left-and right-hand sides, we shall use work of Borel. By Corollary 5.5 of [33], each ( Tor ( ,1 ); Q) ⊗ is a finite-dimensional ′ -representation, so by theorems of Borel [5,6] (see Theorem 2.3 of [34] for a description of Borel's results adapted to this situation, using the bounds from [46]) the map…”