Related papers: Stability phenomena in the topology of moduli spac…
Some recent progress towards understanding the cohomology of moduli spaces of curves is described. Madsen and Weiss have given a proof of a generalisation of Mumford's conjecture on the stable cohomology of these moduli spaces M_g, and…
We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and…
A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…
We prove a homological stability theorem for moduli spaces of simply-connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer's stability theorem for the homology of…
We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology,…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration…
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable…
We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises Harer's stability theorem for the homology of mapping class groups. Combined with previous work of…
Twenty years ago, Mumford initiated the systematic study of the cohomology ring of moduli spaces of Riemann surfaces. Around the same time, Harer proved that the homology of the mapping class groups of oriented surfaces is independent of…
We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to $\#^{g}(S^{n+1}\times S^{n})$, provided $n \geq 4$. This is an odd dimensional analogue of a recent homological stability result of S. Galatius and…
The purpose of these notes is to show that the methods introduced by Bauer and Furuta in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from Riemann…
We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and…
We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with…
We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the…
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules…
We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only…
We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is…
Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and…
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…
We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was…