Floer Homology Research Articles

Contact invariants of [formula omitted]-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to•the Ehrhart (quasi-)polynomial of its toric diagram;•the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity;•the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve the Kapovich–Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also find, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.

Topologie

The lectures in the workshop covered various topics in modern topology, including algebraic and geometric topology, homotopy theory, geometric group theory, and manifold topology, as well as connections to neighboring areas, most prominently symplectic topology/geometry. The following current research topics received more attention during the workshop: manifolds and K-theory, symplectic topology and Floer homology, generalizations of hyperbolic techniques in geometric group theory, and equivariant and motivic homotopy theory. The aim of the various topics was to foster communication and provide chances for participants to see and experience driving questions and important methods in nearby fields within the realm of topology.

A calculus for bordered Floer homology

A calculus for bordered Floer homology

Constrained knots in lens spaces

In this paper, we study a special family of $(1,1)$ knots called constrained knots, which includes 2-bridge knots in the 3-sphere $S^3$ and simple knots in lens spaces. Constrained knots are parameterized by five integers and characterized by the distribution of spin$^c$ structures in the corresponding $(1,1)$ diagrams. The knot Floer homology $\widehat{HFK}$ of a constrained knot is thin. We obtain a complete classification of constrained knots based on the calculation of $\widehat{HFK}$ and presentations of knot groups. We provide many examples of constrained knots constructed from surgeries on links in $S^3$, which are related to 2-bridge knots and 1-bridge braids. We also show many examples of constrained knots whose knot complements are orientable hyperbolic 1-cusped manifolds with simple ideal triangulations.

Projective naturality in Heegaard Floer homology

Projective naturality in Heegaard Floer homology

Compatibility in Ozsváth–Szabó’s bordered HFK via higher representations

We equip the basic local crossing bimodules in Ozsv\'ath-Szab\'o's theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the $U_q(\mathfrak{gl}(1|1)^+)$-intertwining property of the corresponding maps between ordinary representations. Besides yielding a new connection between bordered knot Floer homology and higher representation theory in line with work of Rouquier and the second author, this structure gives an algebraic reformulation of a ``compatibility between summands'' property for Ozsv\'ath-Szab\'o's bimodules that is important when building their theory up from local crossings to more global tangles and knots.

On torsion in linearized Legendrian contact homology

In this short note, we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group [Formula: see text] and a positive integer [Formula: see text], [Formula: see text], we construct examples of Legendrian submanifolds of the standard contact vector space [Formula: see text], whose [Formula: see text]th linearized Legendrian contact (co)homology over [Formula: see text] computed with respect to a certain augmentation is isomorphic to [Formula: see text].

Split link detection for sl(P)$\mathfrak {sl}(P)$ link homology in characteristic P$P$

AbstractWe provide a sufficient condition for splitness of a link in terms of its reduced link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If is prime and the coefficient field is of characteristic , then the sufficient condition for splitness is also necessary. When , we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with coefficients.

Loop coproduct in Morse and Floer homology

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper, we extend the scope of this isomorphism in several directions. First, we give a direct definition of Rabinowitz loop homology in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to positive Floer homology, resp. loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce reduced loop homology, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo–Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism using linear Hamiltonians and the square root of the energy functional.

The Conley–Zehnder indices of the spatial Hill three-body problem

We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill’s system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems.

Quasipositive surfaces and decomposable Lagrangians

We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.

An Enhanced Euler Characteristic of Sutured Instanton Homology

Abstract For a balanced sutured manifold $(M,\gamma )$, we construct a decomposition of $SHI(M,\gamma )$ with respect to torsions in $H=H_{1}(M;\mathbb {Z})$, which generalizes the decomposition of $I^{\sharp }(Y)$ in a previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^{c}$ decompositions in $SFH(M,\gamma )$. Based on this decomposition, we define an enhanced Euler characteristic $\chi _{\textrm {en}}(SHI(M,\gamma ))\in \mathbb {Z}[H]/\pm H$ and prove that $\chi _{\textrm {en}}(SHI(M,\gamma ))=\chi (SFH(M,\gamma ))$. This provides a better lower bound on $\dim _{\mathbb {C}}SHI(M,\gamma )$ than the graded Euler characteristic $\chi _{\textrm {gr}}(SHI(M,\gamma ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat {HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.

On the Frøyshov invariant and monopole Lefschetz number

Given an involution on a rational homology $3$-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of $4$-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and $4$-dimensional topology.

Cosmetic surgery conjecture for three strand pretzel knots

The problem whether the purely cosmetic surgery conjecture holds for any knots is open. Using recent work of Hanselman, the conjecture was proved for any pretzel knots [A. I. Stipsicz and Z. Szabo, Purely cosmetic surgeries and pretzel knots (2020), arXiv:2006.06765]. In this paper, we prove the conjecture for any three strand pretzel knots in a way that does not depend on Heegaard Floer homology.

Augmentations and immersed Lagrangian fillings

AbstractFor a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings of . When is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635–722], for each augmentation of the LCH algebra of , there is an induced augmentation . With fixed, the set of homotopy classes of all such induced augmentations, , is a Legendrian isotopy invariant of . We establish methods to compute based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary , we give examples of Legendrian torus knots with distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when and , every ‐graded augmentation of can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ‐graded augmented Legendrian cobordism.

Lattice cohomology and q-series invariants of 3-manifolds

Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ⁢ ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant.

Mapping classes are almost determined by their finite quotient actions

Given any connected compact orientable surface, a pair of mapping classes is said to be procongruently conjugate if it induces a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.

On the kernel of the zero-surgery homomorphism from knot concordance

Kawauchi defined a group structure on the set of homology S1×S2's under an equivalence relation called H˜-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. We apply knot concordance invariants derived from knot Floer homology to study the kernel of the zero-surgery homomorphism. As a consequence, we show that the kernel contains a Z∞-subgroup generated by topologically slice knots in the smooth category.

Some computations on instanton knot homology

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard diagrams of [Formula: see text]-manifolds. In this paper, for a knot inside [Formula: see text], we further introduce an algorithm that computes an upper bound of the dimension of instanton knot homology from knot diagrams. We test the algorithm with all knots up to seven crossings as well as a more complicated knot [Formula: see text]. In the second half of the paper, we show that if the instanton knot Floer homology of a knot has a specified form, then the knot must be an instanton L-space knot.