arXiv research

3-manifolds with specific homology are cobordant if homeomorphic.

This paper explores how many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space.

Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.

The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.

New homology $q_2$ from 3D cobordism to integers.

Study shows quotient group is infinitely generated.

Study exact surgery formula in involutive Heegaard Floer homology.

For each integral homology sphere $Y$, a function $Γ_Y$ on the set of integers is constructed. It is established that $Γ_Y$ depends only on the homology cobordism of $Y$ and it recovers the Frøyshov invariant. A relation between $Γ_Y$ and Fintushel-Stern's $R$-invariant is stated. It is shown that the value of $Γ_Y$ at…

The main result of this paper is that any $3$-dimensional manifold with a finite group action is equivariantly, invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivarian…

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…

Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…

Study knot invariants and surgeries, proving properties of instanton homologies.

The study counts critical points in knot cobordisms using abelian and metacyclic invariants.

Refines Khovanov homology using signed Burnside categories.

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…

Defines cobordism maps connecting Khovanov and instanton homologies.

This paper analyzes $Δ_a$ invariants in non-perturbative complex Chern-Simons theory.

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the…

Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.

Study on knot concordance and homology cobordism using Heegaard Floer homology.

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equiv…

Paper extends cobordism maps in Khovanov and instanton homologies.

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic tors…

New example shows figure eight knot not smoothly concordant but homology cobordant.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

We define four versions of equivariant instanton Floer homology ($I^+, I^-, I^\infty$ and $\widetilde I$) for a class of 3-manifolds and $SO(3)$-bundles over them including all rational homology spheres. These versions are analogous to the four flavors of monopole and Heegaard Floer homology theories. This construction…

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…

We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a …

New cobordism maps for instanton knot homology help compute surgeries on knots.

The paper studies hyperbolic three-manifolds and their geometric constraints.

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

We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that $β=-\barμ$ for Seifert integral homology three-spheres. We show that the Manolescu invari…

New instanton invariants for rational homology spheres defined and shown to be functorial.

We use Floer's exact triangle to study the u-map (cup product with the 4-dimensional class) in the Floer cohomology groups of admissible SO(3) bundles over closed, oriented 3-manifolds. In the case of non-trivial bundles we show that (u^2-64)^n = 0 for some positive integer n. For homology 3-spheres Y the same holds fo…

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

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …

Study ribbon homology concordances using link Floer homology.

It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …

New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.

Let S be a compact oriented surface. A homology cobordism of S is a cobordism C between two copies of S, such that both the "top" inclusion and the "bottom" inclusion of S in C induce isomorphisms in homology. Homology cobordisms of S form a monoid, into which the mapping class group of S embeds by the mapping cylinder…

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps in…

A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …

Trisections help compute Heegaard Floer homology invariants.

We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.

For any $s \in [-\infty, 0]$ and oriented homology 3-sphere $Y$, we introduce a homology cobordism invariant $r_s(Y)\in (0,\infty]$. The values $\{r_s(Y)\}$ are included in the critical values of the $SU(2)$-Chern-Simons functional of $Y$, and we show a negative definite cobordism inequality and a connected sum formul…

The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.

We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on …