arXiv research

Homological stability aids in computing group homology.

Homology of torus knots stabilizes to loop space homology.

Knot Floer homology stabilizes with twists.

Homological stability fails for Cremona groups, rational varieties, and function fields.

New stability results for homology groups of Torelli subgroups and congruence subgroups.

Homological stability proved for handlebody mapping class groups.

New proof of homological stability for surface mapping classes.

The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequ…

Study homological stabilization in Hurwitz spaces using plant complexes.

The abstract discusses homological stability in topological moduli spaces.

New stabilization method in graph braid homology yields polynomial growth.

We give another proof of a theorem of Hatcher and Vogtmann stating that the sequence $Aut(F_n)$ satisfies integral homological stability. The paper is for the most part expository, and we also explain Quillen's method for proving homological stability.

New stability theorem for nonorientable surfaces mapping class groups.

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 give explicit stability ranges in many new cases. In each of these cases the stable…

In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $S_g (K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $S_{g,n} (K)$ was studied by the authors in \cite{…

We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological …

Homological stability for mapping class groups on fixed subsurfaces.

The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.

Enhances stability ranges for Torelli and congruence subgroup homologies.

Studied knot Floer homology stabilization and prime mutant knots.

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 mapping class groups. Combined with previous work of the authors, it gives a cal…

New method proves representation stability for linear groups, resolving homological questions.

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 O. Randal Williams for the moduli space of manifolds diffeomorphic to $\#^{g}(S…

We prove that the quotient map from Aut(F_n) to Out(F_n) induces an isomorphism on homology in dimension i for n at least 2i+4. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which a…

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…

Abstract: Necessary conditions for stabilizing subsets in systems are found.

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.

We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.

The paper defines and calculates stabilization distances between surfaces in 4-manifolds.

The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.

Homological stability fails for 4-manifold moduli spaces, detected by new class.

We correct the proof of Theorem 5 of the paper Homology stability for outer automorphism groups of free groups, by the first two authors.

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups.…

This paper extends homological stability results for configuration spaces of manifolds.

Stability in homology of moduli spaces of admissible covers.

The cohomology of complex irreducible polynomials stabilizes with degree or variables.

The study examines how twisting a knot affects its homology and stability properties.

Study rational homology of moduli space via Morse functions, proving stability phenomena.

New stability result for Milnor fibers of pure braid groups.

Paper stabilizes persistent homology rank functions for statistical inference.

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the un…

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 analogous to recent results of S. Galatius and O. Randal-Williams regarding the homo…

Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there…

We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an …

Decouples homotopy quotients of generalised configuration spaces on surfaces.

We give a complete and detailed proof of Harer's stability theorem for the homology of mapping class groups of surfaces, with the best stability range presently known. This theorem and its proof have seen several improvements since Harer's original proof in the mid-80's, and our purpose here is to assemble these many a…

The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.