Related papers: Homological stability for spaces of commuting elem…

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…

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…

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 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…

We prove a representation stability result for the sequence of spaces $\overline M_{g, n}^A$ of pointed admissible $A$-covers of stable $n$-pointed genus-$g$ curves, for an abelian group $A$. For fixed genus $g$ and homology degree $i$, we…

We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of…

Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and…

In this paper we study stability patterns in the homology of the ordered configuration space of the once-punctured torus. In the last decade Church and Church-Ellenberg-Farb proved that the homology groups of the ordered configuration space…

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher…

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…

In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results.

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, \gamma). This is the space consisting of triples, (F_{g,n}, \phi, f), where F_{g,n} is a Riemann surface of genus g and…

In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are…

Homological stability has shown itself to be a powerful tool for the computation of homology of families of groups such as general linear groups, mapping class groups or automorphisms of free groups. We survey here tools and techniques for…

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for…

We will study homological stability of the diffeomorphism groups of the manifolds $W_{g,1}:=D^{2n} \# (S^n \times S^n)^{\#g }$ using $E_k$-algebras. This will lead to new improvements in the stability results, especially when working with…

In this paper, we develop a modified proof strategy for homological stability of linear groups, with the general linear groups serving as a primary example. Our arguments are more direct than those in the classical works of Quillen and…

We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras,…

Let $\{A_m\}$ be a pro system of associative commutative, not necessarily unital, rings. Assume that the pro systems $\{\mathrm{Tor}^{\mathbb{Z}\ltimes A_m}_i(\mathbb{Z},\mathbb{Z})\}_m$ vanish for all $i>0$. Then we prove that the sequence…

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with…