Stable moduli spaces of high-dimensional manifolds

Galatius, Søren; Randal-Williams, Oscar

doi:10.1007/s11511-014-0112-7

Cited by 86 publications

(157 citation statements)

References 27 publications

Supporting

Mentioning

154

Contrasting

Order By: Relevance

“…In the case when d > 2, the fact is a consequence of two theorems of Galatius and Randal-Williams [10,9]. For d = 1, the range of degrees can be improved to * ≤ 2g/3.…”

Section: Manifolds With a Fixed Disk And Homological Stabilitymentioning

confidence: 99%

Relations among characteristic classes of manifold bundles

Grigoriev

2017

Geom. Topol.

View full text Add to dashboard Cite

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of g copies of a product of spheres S d × S d and odd d , we find numerous algebraic relations among so-called "generalized Miller-Morita-Mumford classes". For all g > 1, we show that these infinitely many classes are algebraically generated by a finite subset.Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with g, according to recent homological stability results. In the case of surface bundles (d = 1), our approach recovers some previously known results about the structure of the classical "tautological ring", as introduced by Mumford, using only the tools of algebraic topology. 55R40; 57R22, 55T10 IntroductionLet M be a 2d -dimensional closed oriented smooth manifold. We denote by Diff M the topological group of orientation-preserving diffeomorphisms of M . The bar construction can be used to construct the space BDiff(M) that classifies bundles with fiber M . For any characteristic class of vector bundles p ∈ H * +2d (BSO 2d ; Q), we will define a generalized Miller-Morita-Mumford class (or just kappa class) κ p ∈ H * (BDiff(M); Q). These are the simplest examples of characteristic classes of bundles 1 with fiber M and structure group Diff M .We are mainly interested in the case where the fiber is #g S d × S d , the connected sum of g copies of S d × S d . More generally, we let the fiber to be a highly-connected manifold (see Definition 2.5) of genus g and dimension 2d , denoted M 2d g or M g . Recall that H * (BSO 2d ; Q) = Q[p 1 , . . . , p d−1 , e], where p i is the Pontryagin class of degree 4i and e is the Euler class of degree 2d . Let S ⊂ H * (BSO 2d ; Q) consist of the monomials 1 A geometric example of such a bundle is a proper submersion f : E → B of smooth, oriented manifolds that has M as its fiber.arXiv:1310.6804v3 [math.AT]

show abstract

“…In the case when d > 2, the fact is a consequence of two theorems of Galatius and Randal-Williams [10,9]. For d = 1, the range of degrees can be improved to * ≤ 2g/3.…”

Section: Manifolds With a Fixed Disk And Homological Stabilitymentioning

confidence: 99%

Relations among characteristic classes of manifold bundles

Grigoriev

2017

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…Let MTθ n 2n denote the Thom spectrum associated to the virtual vector bundle −(θ n 2n ) * γ 2n over BO(2n) n . The main result from [4] proves the isomorphism H i (B Diff((S n × S n ) g , D 2n )) ∼ = H i (Ω ∞ 0 MTθ n 2n ), for n 3, when i g and where D 2n is an embedded disk in (S n × S n ) g . This result is proven by the techniques of cobordism categories and parameterized surgery first introduced by Madsen and Weiss in their proof of the Mumford Conjecture [10].…”

Section: Motivationmentioning

confidence: 87%

“…The main result of this paper is a direct analogue of Hatcher's result for higher-dimensional handlebodies (D n+1 × S n ) g . On the other hand, our technique is closely related to (and motivated by) the work of Galatius and Randal-Williams from [4]. Let θ n 2n : BO(2n) n −→ BO(2n) denote the n-connective cover.…”

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Stable moduli spaces of high‐dimensional handlebodies

Botvinnik

Perlmutter

2017

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Theorem 2.4 (Galatius and Randal-Williams [13,14]). The map (2.2) induces an isomorphism in integral homology in degrees * (g − 3)/2.…”

Section: Madsen-tillmann-weiss Theorymentioning

confidence: 99%

Torelli spaces of high-dimensional manifolds

Ebert

Randal-Williams

2014

Journal of Topology

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Stable moduli spaces of high-dimensional manifolds

Cited by 86 publications

References 27 publications

Relations among characteristic classes of manifold bundles

Relations among characteristic classes of manifold bundles

Stable moduli spaces of high‐dimensional handlebodies

Torelli spaces of high-dimensional manifolds

Contact Info

Product

Resources

About