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…
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48 results for “trivial homology”
Homology handles with trivial Alexander polynomial bound a 3D sphere.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for -homology 3-spheres.
result A distinguished homology handle with trivial Alexander polynomial bounds a homology .
Study shows non-trivial knot Floer homology for specific knots.
problem Proving non-triviality of knot Floer homology for certain knots.
method Analyzing Alexander grading and proving non-triviality in specific knots.
result Non-trivial knot Floer homology for certain knots, answering a question posed by Baldwin and Vela-Vick.
Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.
problem Classifying spaces as almost homology n-manifolds.
method Providing a necessary and sufficient condition for locally compact homogeneous ANR-spaces or strongly locally homogeneous ANR-spaces to be almost homology n-manifolds.
result Spaces are classified as almost homology n-manifolds based on their homology groups.
We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.
New knots found with non-trivial Steenrod operations on Khovanov homology.
problem Identifying knots with non-trivial Steenrod operations on Khovanov homology.
method Examined prime, hyperbolic, and satellite knots using Steenrod operations.
result Found knots (prime, hyperbolic, satellite) with non-trivial Steenrod operations on Khovanov homology.
Non-trivial action on surface configuration homology.
problem Understanding the Johnson filtration's action on surface configuration homology.
method Action of mapping class group on configuration space homology, focusing on Johnson filtration stages.
result Non-trivial action on the stage of the Johnson filtration for all and .
Tangle contact homologymath.SG
The paper extends knot contact homology to tangles and proves a gluing formula.
problem Defining and proving invariants for tangles in .
method Extending knot contact homology to tangles and proving a gluing formula.
result Tangle contact homology detects triviality of tangles.
Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.
problem Understanding the homology cobordism group of 3-manifolds formed by knots and their mirrors.
method Using involutive Floer homology and gauge theory.
result Locally trivial involutive Floer homology on certain splices of knots and their mirrors.
We present some non-trivial calculations of Baldwin-Ozsváth-Szabó cohomology of links, and applications to Heegaard-Floer homology of branched double covers.
Study proves Kotschick's conjecture for certain compact Kähler manifolds.
problem Proving a conjecture about one-forms without zeros on compact Kähler manifolds.
method Using a conjecture about homologically trivial fibrations and properties of Albanese torus.
result Proves Kotschick's conjecture for specific compact Kähler manifolds.
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
problem Analyzing Khovanov homology of Turaev genus one links.
method Examined Khovanov homology in specific polynomial gradings.
result Found a trivial summand in Khovanov homology of Turaev genus one links.
New conditions prevent non-trivial relations in local equivalence group.
problem Understanding non-trivial relations in homology cobordism groups.
method Filtered instanton Floer homology and -invariants.
result Certain conditions prevent non-trivial relations in the local equivalence group.
We prove a conjecture of Rozansky's concerning his categorification of the tail of the colored Jones polynomial for an -adequate link. We show that the tail homology groups he constructs are trivial for non -adequate links.
It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.
This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot inside a homology sphere , the involution on the knot Floer homology of which corresponds to moving the basepoints by o…
In this paper we prove that a complete noncompact manifold with nonnegative Ricci curvature has a trivial codimension one homology unless it is a split or flat normal bundle over a compact totally geodesic submanifold. In particular, we prove the conjecture that a complete noncompact manifold with positive Ricci curvat…
Existence of non-trivial monopoles on 3-spheres proven.
problem Existence of non-trivial SU(2)-monopoles on 3-manifolds.
method Gluing construction to prove existence on rational homology 3-spheres.
result Existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities.
We construct a graph TQFT for the minus flavor of Heegaard Floer homology. Our graph TQFT extends Ozsváth and Szabó's TQFT for closed and connected 3-manifolds, and allows for cobordisms with disconnected ends. As an application, we give an explicit formula for the chain homotopy type of the -action on Heegaard Fl…
We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this coro…
The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial t…
Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all o…
In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.
New families of Brieskorn spheres bound rational homology balls.
problem Identifying new Brieskorn spheres that bound rational homology balls.
method Using techniques from Akbulut and Larson's work, we present new families of Brieskorn spheres.
result We discover new infinite families of Brieskorn spheres that non-trivially bound rational homology balls.
Flip symmetry on knot diagrams affects Khovanov homology.
problem Understanding the flip map on Khovanov homology.
method Analyzing the behavior of the flip map on unlinks and using it to determine the involution.
result The flip map is the identity map over \(\mathbb{F}_2\), confirming a conjecture.
We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or …
We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.
Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actionsmath.SG
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…
We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.
We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebrai…
We give a Khovanov homology proof that hyperbolic twist knots do not admit non-trivial Dehn surgeries with finite fundamental group.
Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.
problem Investigating whether topologically isotopic surfaces are smoothly isotopic in stabilised 4-manifolds.
method Analyzing the triviality of surfaces in the 2-homology of the ambient 4-manifold and constructing stabilisations.
result The result holds for surfaces trivial in the 2-homology of the 4-manifold and for certain fundamental groups.
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
Paper studies Lagrangian submanifolds and their homological monodromy.
problem Understanding the homological monodromy of Lagrangian submanifolds.
method Proves triviality of homological Lagrangian monodromy under specific conditions.
result Homological Lagrangian monodromy is trivial if Hofer energy is less than minimum energy of J-holomorphic spheres and discs.
New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.
problem Finding non-smoothable homeomorphisms of 4-manifolds with boundary.
method Constructing specific examples of homeomorphisms that fix the boundary and act trivially on homology.
result First examples of non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary.
We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.
Pochette surgery of 4-spheremath.GT
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.
problem Locally flat, compact, genus g surfaces with boundary a fixed knot in simply-connected 4-manifolds with boundary the 3-sphere.
method Analyzes homology and isotopy of surfaces with trivial fundamental group complements.
result Homologous surfaces are topologically ambiently isotopic rel. boundary.
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.
problem Finding groups with non-trivial fundamental groups in the complements of surfaces in 4-manifolds.
method Combining topological constructions with homological properties of simple groups.
result The complement of a surface in a simply connected 4-manifold can have a non-trivial fundamental group.
We show a spectral sequence for the rational Khovanov homology of an oriented link in terms of the rational Khovanov complexes and homologies of the link surgeries along an admissible cut. As a non trivial corollary, we give an explicit splitting formula for the Jones polynomial.
This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods of constructing non-trivial cohomology classes.
For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part.
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
Let be a null-homologous knot in a three-manifold . We give a description of the Heegaard Floer homology of integer surgeries on along in terms of the filtered homotopy type of the knot invariant for . As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles ov…