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.
We prove that in dimension = 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of R n and various types of automorphisms of 2-connected manifolds.Theorem A. Let n = 4, 5, 7, then BDiff ∂ (D n ) is of finite type. Corollary B. Let n = 4, 5, 7, then BDiff(S n ) is of finite type Corollary C. Let n = 4, 5, 7. Suppose that M is a closed 2-connected oriented smooth manifold of dimension n, then BDiff + (M ) is of homologically finite type.It is convention to denote PL(R n ) by PL(n) and Top(R n ) by Top(n). Corollary D.Let n = 4, 5, 7, then BTop(n) and BPL(n) are of finite type.Corollary E. Let n = 4, 5, 7, then BTop(S n ) and BPL(S n ) are of finite type.
We define bounded generation for En-algebras in chain complexes and prove that for n ≥ 2 this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, which says that if an En-algebra A has homological stability (or equivalently the topological chiral homology R n A has homology stability), then so has the topological chiral homology M A of any connected non-compact manifold M . Using scanning, we reformulate the local-to-global homological stability principle so that it applies to compact manifolds. We also give several applications of our results.Here k M X denotes the charge k component of M X, B T M X is a bundle over M with fiber given by the n-fold delooping B n X of X, and Γ c k (−) denotes the space of compactly
The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$ . In this article we prove that for $2n \geq 6$ and $g \geq 2$ , 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.
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