arXiv research

Investigates local indicability of groups with circle homology presentations.

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

Circle graph complexes reveal link properties via Khovanov homology.

Proves Khovanov homology has no torsion for bipartite circle graphs.

Classifies torus bundles bounding 4-manifolds with rational homology.

An introduction to circle valued Morse theory and Novikov homology, from an algebraic point of view.

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is al…

The paper identifies manifolds with free circle actions.

Solving polynomial equations finds circle packings on surfaces.

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…

The paper calculates a specific invariant for a manifold with a circle action.

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

A new invariant captures geometric features of circle embeddings.

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss th…

Persistent homology reveals geometric features of metric spaces, especially geodesic circles.

Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurst…

Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian …

We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …

Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…

Floer homology applied to inscribing rectangles into curves.

Generalizes Seifert algorithm to integral homology spheres.

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Sigma_g of genus g with S^1. We determine HF^+ for this 3-manifold completely for the spin^c structure having trivial first Chern class, which for g>2 was previously unknown. We show that in this case HF^\infty is closely related…

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …

A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic La…

In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.

Study maps surface configurations to Heisenberg homologies for mapping class groups.

Study symplectic and Hamiltonian actions on irrational ruled surfaces, proving existence and non-existence of extensions.

Let $K$ be a null-homologous knot in a three-manifold $Y$. We give a description of the Heegaard Floer homology of integer surgeries on $Y$ along $K$ in terms of the filtered homotopy type of the knot invariant for $K$. As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles ov…

We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-man…

We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number by using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó using the Seiberg-Witten invariants. To prove these relations, we give a fairly exp…

Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle …

We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible ci…

Given a torus bundle $Y$ over the circle and a cohomology class $[ω]\in H^2(Y;\mathbb{Z})$ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of $Y$ with twisted coefficients in the universal Novikov ring.

This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbo…

Computes homotopy groups of embedding spaces of arcs or circles in 4-manifolds.

Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author.

An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.

The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions. We then show that for simply connected 5--manifolds these restrictions are neces…

Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links

In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together wi…

Novikov homology refined for 2D integer group coverings.

We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for certain extremal spin^c structures. As applications, we give new calculations of He…

Improved bounds linking entropy and volume in hyperbolic 3-manifolds.

We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.

Open 2D TFTs extend to closed theories with circle value as Hochschild homology.

This article offers an introductory look at Khovanov homology and its historical context.

Given an irreducible closed 3--manifold $Y$, we show that its twisted Heegaard Floer homology determines whether $Y$ is a torus bundle over the circle. Another result we will prove is, if $K$ is a genus 1 null-homologous knot in an $L$--space, and the 0--surgery on $K$ is fibered, then $K$ itself is fibered. These two …