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We give an alternative proof of Madsen-Weiss' generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss' original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem about folded maps. Comment: 67...
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We prove that group homology of the diffeomorphism group of $\#^g S^n \times
S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$.
This answers the high dimensional version of a question posed by Morita about
surface diffeomorphism groups made discrete. The stable homology is isomorphic
to the homology of a certain infinit...
Citations
... There are now multiple different proofs of the Madsen-Weiss theorem, including [MW07,GMTW09,EGM11,GRW10]. Interestingly, the proof of [GRW10] (see also [Hat11]) interprets the Madsen-Weiss isomorphism as an instance of a scanning map, making the parallel between this story and the Madsen-Weiss theorem even closer. ...
We compute the stable homology of the braid group with coefficients in any Schur functor applied to the integral reduced Burau representation. This may be considered as a hyperelliptic analogue of the Mumford conjecture (Madsen-Weiss theorem) with twisted coefficients. We relate the result to the function field case of conjectures of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of families of quadratic L-functions. In particular, we formulate a purely topological homological stability conjecture, which when combined with our calculations would imply a precise asymptotic formula for all moments in the rational function field case.
... Remark 1. 4 The construction does not use sophisticated results on groups of diffeomorphisms. For every integer r 1, Theorem A holds as well in the differentiability class C r , with the same proof. ...
... In the same way, let us say that the c-principle holds when every formal object is cobordant to a genuine object through the formal objects. Examples of results pertaining to the c-principle are the Mather-Thurston theorem for foliated products or bundles, the Madsen-Weiss theorem for fibrations whose fibres are surfaces (see [4]), and the realization of taut compactly generated pseudogroups by foliations of dimension 2 and codimension 1; see Meigniez [26]. In the same spirit, see Fuchs' early paper [9] and Kupers' recent one [18]. ...
... Let us resume the proof of Lemma 4.12. By (4) and (5) of Claim 4.13, the vector field z r is horizontal on Op˛ D q .˛ @D q / and on Op˛ D q .ˇ ...
This third version is more detailed and has more figures
... The "Hurewicz theorem for bordism groups" (see e.g. [4] appendix B) allowing one to translate homology into cobordism, one thus gets the following paraphrase. ...
The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1\le r\le\infty$. The corollaries deal with the existence of foliations, with the homology and the perfectness of the diffeomorphism groups, with the existence of foliated products, and of foliated bundles.
... The standard Mumford conjecture asserts that for d = 2 and for a closed oriented surface F g of genus g, the map α| BDiff F g induces an isomorphism of rational cohomology rings in stable range of dimensions * g. The Mumford conjecture was proved in the positive by Madsen and Weiss in [11], and later several other proofs of the Madsen-Weiss theorem were given in [7,4,8,10]. or, equivalently, the map α| BDiff F g is a rational homology equivalence in a stable range of dimensions. ...
... Then f is bordant to a fiber bundle by bordism which is a broken submersion itself. Theorem 1.3 relies heavily on the Harer stability theorem, and its proof is very much in spirit of a singularity theoretic argument by Eliashberg, Galatius and Mishachev in [4]. ...
... In particular, the moduli space of colored broken submersions of dimension d is an appropriate homotopy colimit of classifying spaces BDiff M of diffeomorphism groups of manifolds of dimension d with certain boundary components, compare with the original paper [11]. Colored broken submersions should be compared with enriched fold maps from [4] of Galatius-Eliashberg-Michachev who used them to give a topological proof of the Madsen-Weiss theorem. Note, however, that in contrast to enriched fold maps, colored broken submersions behave well with respect to taking pullbacks and possess a moduli space ( §8). ...
In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.
... • It is a natural setting for geometric objects such as spaces of smooth structures or foliations. For example, it was used in [20,40]. ...
... It is a well-known fact that a map X → Y of simplicial sets or topological spaces induces an isomorphism on homology if and only if it induces an isomorphism on oriented bordism. A reference for this is Appendix B of [40], but it also follows from the Atiyah-Hirzebruch spectral sequence. Note that by Lemma 91 we may replace Δ by Δ e . ...
In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contractibility of the space of framed functions, and a version of Mather-Thurston theory.
... In their papers [10][11][12][15][16][17], Eliashberg and Mishachev exploit this wrinkling strategy to prove a number of results in flexible geometric topology. Together with Galatius, they give a further application in [18]. The theorem on wrinkled embeddings from [15], which is particularly relevant for our purposes, has gained greater significance after it was used by Murphy in [34] to establish the existence of loose Legendrians in high-dimensional contact man- The difference between wiggling and wrinkling ifolds. ...
We establish a full h-principle (C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document}-close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy.
... Theorem 1.3 amounts to say that c induces an isomorphism in oriented bordism. Equivalently, by the "Hurewicz theorem for bordism groups" ([1], see also [5], appendix B), c induces an isomorphism in integral homology. That last wording is the classical one. ...
... In the same way, let us say that the c-principle holds when every formal solution is cobordant to a genuine solution through the formal solutions. As far as the author knows, three existence results pertain to the c-principle: the Mather-Thurston theorem for foliated products, the Madsen-Weiss theorem for fibrations whose fibres are surfaces (see [5]), and the realization of taut compactly generated pseudogroups by foliations of dimension 2 and codimension 1 [18]. It could be interesting to understand some general conditions under which the c-principle applies. ...
We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.
... Figure 1. A summary of what is known about the cohomology of Mod(S g ); filled boxes denote nontrivial cohomology groups, the shaded region denotes the unstable region, and in this region X's denote trivial cohomology groups Mumford's conjecture was proved in 2006 by Madsen-Weiss [105]; see also [48,76,152]. This gives a complete picture of the cohomology in the stable range: it is the polynomial algebra generated by the κ i (for fixed g these are necessarily trivial for 2i > 4g − 5). ...
We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.
... Our main reference for this part is Chapter 5 of [3]. For more details about stable cohomology and Mumford's conjecture see [25,122,26]. is denoted by RH * (M g,n ). In genus zero, the equality ...
This note is about invariants of moduli spaces of curves. It includes their intersection theory and cohomology. Our main focus in on the distinguished piece containing the so called tautological classes. These are the most natural classes on the moduli space. We give a review of known results and discuss their conjectural descriptions.
... More recently, Eliashberg & Mishachev with S. Galatius [5] have shown that wrinkles apply in an area usually reserved to homotopy theorists. The question is to compute the stable 2 homology of the mapping class group of a Riemann surface. ...
