arXiv research

Authors compute stable homology of torus knots using a new deformation technique.

Refines quantum annular homology using stable homotopy methods.

New stable homotopy refinement of quantum annular Khovanov homology.

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k\to \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify t…

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

Refines Khovanov homology using signed Burnside categories.

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.

Study calculates stable norm of slit tori using Farey sequence.

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute t…

The family of negative torus links $T_{p,q}$ over a fixed number of strands $p$ admits a stable limit in reduced Khovanov homology as $q$ grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for $p=2,3,4$. As an application, w…

We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…

We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X…

The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl(3)-homology. Special attention is paid to torsion. In addition, explicit conjectura…

Computes homology of an obstruction chain complex in grid homology.

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 surface diffeomorphism groups made discrete. The stable homology is isomorphic to the…

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology o…

We review the construction and context of a stable homotopy refinement of Khovanov homology.

We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.

Study abelian cycles in Torelli group homology, proving new results in stable rational homology.

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, γ). This is the space consisting of triples, (F_{g,n}, φ, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, φis a parameterization of the boundary, and f : F_{g,n} \to X is a contin…

The real homology of a compact, n-dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm on H_{n-1}(M,R) is a homogenized version of the Riemannian (n-1)-volume. We study…

Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.

Study shows configuration spaces' homological dimension increases monotonically.

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

The paper develops methods for calculating equivariant homology from Morse functions.

Contradicts claims about Poincaré complexes and homology manifolds.

Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.

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

The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.

Researchers find a Steenrod square for link Floer homology.

Given a closed, oriented surface M, the algebraic intersection of closed curves induces a symplectic form Int(.,.) on the first homology group of M. If M is equipped with a Riemannian metric g, the first homology group of M inherits a norm, called the stable norm. We study the norm of the bilinear form Int(.,.), with r…

C. Armond, S. Garoufalidis and T.Le have shown that a unicolored Jones polynomial of a B-adequate link has a stable tail at large colors. We categorify this tail by showing that Khovanov homology of a unicolored link also has a stable tail, whose graded Euler characteristic coincides with the tail of the Jones polynomi…

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …

Torelli groups' homology is finitely generated in stable range.

Homological stability aids in computing group homology.

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.

The WRT invariant of a link L in S2xS1 at sufficiently high values of the level r can be expresses as an evaluation of a special polynomial invariant of L at 2r-th root of unity. We categorify this polynomial invariant by associating to L a bigraded homology whose graded Euler characteristic is equal to this polynomial…

We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater tha…

Explains Khovanov homology and its applications.

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

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the inf…

We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the "edge" in our title.) Applying our results to auto…

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Flo…

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

In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration usi…