Torelli spaces of high-dimensional manifolds

Abstract

The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology) of the cohomology of the classifying space of the Torelli group of certain high‐dimensional, highly connected manifolds, with rational coefficients and in a certain range of degrees. This is based on Galatius and Randal‐Williams’ work on the diffeomorphism groups of these manifolds, Borel's classical results on arithmetic groups, and methods from surgery theory and pseudoisotopy theory. As a corollary, we find that all Miller–Morita–Mumford characteristic classes are non‐trivial in the cohomology of the classifying space of the Torelli group, except for those associated with the Hirzebruch class, whose vanishing is forced by the family index theorem.

Generalised Miller–Morita–Mumford classes for block bundles and topological bundles

“…The same argument, using Section 3 instead, proves the surjectivity for the comparison map between diffeomorphisms and block diffeomorphisms. In [ERW,Theorem 5.1], we proved (or rather derived from results by Waldhausen, Igusa, Farrell-Hsiang and others) that D 2n ) induces an isomorphism in rational cohomology, in degrees * ≤ min 2n−7 2 , 2n−4…”

Mapping class groups of highly connected $$(4k+2)$$-manifolds

“…The description of n g up to these two extension problems has found a variety of applications [2,[5][6][7]18,23,29,33,38,39], especially in relation to the study of moduli spaces of manifolds [22]. The remaining extensions (1) and (2) have been studied more closely for particular values of g and n [15,19,21,36,37,48] but are generally not well-understood (see e.g.…”

Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

“…4.3 Applying the index formula. The degree-1 terms of the index formulas (1) and (2) give a system of linear equations: (16) q m =1 Im(q)≥0 c 1 (E q ) = σ/4, and for 1 ≤ r ≤ m − 1,…”

“…Since the diagram commutes and ψ * (y 1 ) = x 1 , we want to compute f * (x 1 ). By the index formulas (16) and (17), we have f * (x 1 + x −1 ) = σ/4 and f * (x 1 − x −1 ) = (e 1 + e 2 )/4, so…”