arXiv research

Study homology groups for non-orientable surfaces with boundary.

New invariant distinguishes non-orientable surfaces.

Introduces integer-valued Heegaard Floer theory with canonical orientations.

Study non-orientable link cobordisms using Floer homologies to prove inequalities.

Study the first homology group for a specific mapping class group.

Method calculates $\mathbb{Z}_2$-Thurston norm for Seifert 3-manifolds using pseudo-horizontal surfaces.

Grid homology properties for MOY graphs studied.

We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.

Finite group action on a surface yields a special homology subspace.

We determine the second homology group of the homological Goldman Lie algebra for an oriented surface.

We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…

Extends Khovanov homology to surfaces with singularities.

Rational homology ribbon cobordism defines a partial order on 3-manifolds.

By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov…

Clarifies sign ambiguity in Khovanov homology functoriality.

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that t…

Paper introduces Bar-Natan homology for special links in a modified space.

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 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 …

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expandi…

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

Study homology products on loop spaces with orientation reversal.

We give an alternative presentation of Khovanov homology of links with strict functoriality result over integers. The construction uses an oriented $sl(2)$ state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces.

New operations defined on moduli spaces for bundles with orientations.

Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket. This includes a categorification of brackets skein relation. Then we incorporat…

Study shows Seifert fibered spaces don't bound rational homology balls.

In this paper we prove that every Khovanov homology associated to a Frobenius algebra of rank $2$ can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented links to the homotopy category of complexes.

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó $d$-invariants or …

The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

In this paper we present our results on the homology cobordism group $\Th$ of the oriented integral homology 3-spheres. We specially emphasize the role played in the subject by the gauge theory including Floer homology and invariants by Donaldson and Seiberg -- Witten.

An excision theorem connects Heegaard Floer homology of 3-manifolds.

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…

We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

Generalizes Seifert algorithm to integral homology spheres.

Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with ext…

In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We construct such a representative explicitly. We describe the analogous construction…

This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these t…

We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.

We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…

Extends Kauffman's formula to 3-manifolds with markings.

Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group of automorphisms on the first homology group with $\mathbb Z_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\c…

The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.

New shifting chain map enhances quandle invariants for links.

Acyclicity proven for curve complex on surfaces.

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…