Browsing by Author "RandalWilliams, Oscar"
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Abelian quotients of mapping class groups of highly connected manifolds
Galatius, Søren; RandalWilliams, Oscar (SpringerMathematische Annalen, 20151012)We compute the abelianisations of the mapping class groups of the manifolds W²ⁿ_g = g(Sⁿ × Sⁿ) for n ≥ 3 and g ≥ 5. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group ...

An upper bound for the pseudoisotopy stable range
RandalWilliams, Oscar (SpringerMathematische Annalen, 20170801)We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than (2n  2). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, ...

Cohomology of automorphism groups of free groups with twisted coefficients
RandalWilliams, OscarWe compute the groups H*(Aut(F$_{n}$);M) and H*(Out(F$_{n}$);M) in a stable range, where M is obtained by applying a Schur functor to H$_{Q}$ or H$_{Q}$, respectively the first rational homology and cohomology of F$_{n}$. ...

Homological stability for automorphism groups
RandalWilliams, Oscar; Wahl, N (Academic PressAdvances in Mathematics, 20171001)Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, ...

Homological stability for moduli spaces of high dimensional manifolds. I
Galatius, S; RandalWilliams, OscarWe prove a homological stability theorem for moduli spaces of simply connected manifolds of dimension 2n > 4, with respect to forming connected sum with S$^{n}$ x S$^{n}$ . This is analogous to Harer's stability theorem ...

Homological stability for moduli spaces of high dimensional manifolds. II
Galatius, S; RandalWilliams, OscarWe prove a homological stability theorem for moduli spaces of manifolds of dimension 2$\textit{n}$, for attaching handles of index at least $\textit{n}$, after these manifolds have been stabilised by countably many copies ...

Homological stability for spaces of embedded surfaces
Morán, FC; RandalWilliams, Oscar (Mathematical Sciences PublisherGeometry and Topology, 20170510)© 2017, Mathematical Sciences Publishers. All rights reserved.We study the space of oriented genusg subsurfaces of a fixed manifold M and, in particular, its homological properties. We construct a “scanning map” which ...

Homological Stability for Spaces of Embedded Surfaces
Moran, Federico Cantero; RandalWilliams, Oscar (Mathematical Sciences PublisherGeometry & Topology, 2016)We study the space of oriented genus $\textit{g}$ subsurfaces of a fixed manifold $\textit{M}$, and in particular its homological properties. We construct a “scanning map” which compares this space to the space of sections ...

Infinite loop spaces and positive scalar curvature
Botvinnik, B; Ebert, J; RandalWilliams, Oscar (SpringerInventiones Mathematicae, 20170901)We study the homotopy type of the space of metrics of positive scalar curvature on highdimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature ...

Tautological rings for highdimensional manifolds
Galatius, S; Grigoriev, I; RandalWilliams, Oscar (Cambridge University PressCompositio Mathematica, 201704)We study tautological rings for highdimensional manifolds, that is, for each smooth manifold $M$ the ring $R^*$($M$) of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised ...