Floer Cohomology Research Articles

Affine varieties, singularities and the growth rate of wrapped Floer cohomology

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if [Formula: see text] is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

Framed cobordism and flow category moves

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse-Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus. We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.

Khovanov homology from Floer cohomology

This paper realises the Khovanov homology of a link in S 3 S^3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k \mathbf {k} of characteristic zero. Here we prove the symplectic cup and cap bimodules, which relate different symplectic arc algebras, are themselves formal over k \mathbf {k} , and we construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over Z \mathbb {Z} in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.

Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S3×S1 under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to Fintushel and Stern. We make use of the Z-fold covering space of homology S3×S1 and the instantons on it.

Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

In the 2016 preprint “Kauffman states, bordered algebras, and a bigraded knot invariant,” Ozsváth and Szabó introduced a set of algebraic constructions in the spirit of bordered Heegaard Floer homology. Their constructions can be used to compute knot Floer homology algebraically for knots in the 3-sphere. In this paper we investigate a relationship between Ozsváth–Szabó's bordered theory and the algebras and bimodules constructed by Khovanov and Seidel in “Quivers, Floer cohomology, and braid group actions” (2002). Specifically, we show that the Khovanov–Seidel quiver algebras are isomorphic to quotients of idempotent truncations of some of Ozsváth–Szabó's algebras. Furthermore, we show that the dg bimodule associated to a braid generator by Khovanov–Seidel, with the right action restricted using the quotient map, is homotopy equivalent to Ozsváth–Szabó's DA bimodule with the left action induced using the quotient map.

The Künneth theorem for the Fukaya algebra of a product of Lagrangians

Given a compact Lagrangian submanifold [Formula: see text] of a symplectic manifold [Formula: see text], Fukaya, Oh, Ohta and Ono construct a filtered [Formula: see text]-algebra [Formula: see text], on the cohomology of [Formula: see text], which we call the Fukaya algebra of [Formula: see text]. In this paper, we describe the Fukaya algebra of a product of two Lagrangians submanifolds [Formula: see text]. Namely, we show that [Formula: see text] is quasi-isomorphic to [Formula: see text], where [Formula: see text] is the tensor product of filtered [Formula: see text]-algebras defined in [L. Amorim, Tensor product of filtered [Formula: see text]-algebras, J. Pure Appl. Algebra 220(12) (2016) 3984–4016]. As a corollary of this quasi-isomorphism, we obtain a description of the bounding cochains on [Formula: see text] and of the Floer cohomology of [Formula: see text].

Symplectic topology of Lagrangian submanifolds of ℂPn with intermediate minimal Maslov numbers

Abstract We examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in ℂPn . First we give cohomological constraints on a Lagrangian submanifold in ℂPn whose first integral homology is p-torsion. In the case where (n, p) = (5,3), (8, 3), we prove that the cohomologies with coefficients in ℤ2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)ℤ3) and SU(3)/ℤ3, respectively. Then we calculate the Floer cohomology with coefficients in ℤ2 of a monotone Lagrangian submanifold SU(p)/ℤ p in C P p 2 − 1 . ${\mathbb C}P^{p^2-1}.$

Antisymplectic involution and Floer cohomology

The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions $\tau$ on a symplectic manifold. We introduce the notion of $\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and study how the orientations on the moduli space behave under the involution $\tau$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $\tau$-fixed point set of symplectic manifolds and in particular prove its unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit calculation of Floer cohomology of $\R P^{2n+1}$ over $\Lambda_{0,nov}^{\Z}$ which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of symplectic manifold in complete generality.

Floer trajectories and stabilizing divisors

We incorporate pearly Floer trajectories into the tranversality scheme for pseudoholomorphic maps introduced by Cieliebak–Mohnke (J Symplectic Geom 5(3): 281–356, 2007). By choosing generic domain-dependent almost complex structures, we obtain zero- and one-dimensional moduli spaces with the structure of cell complexes with rational fundamental classes. Integrating over these moduli spaces gives a definition of Floer cohomology over Novikov rings via stabilizing divisors for rational Lagrangians that are fixed point sets of anti-symplectic involutions satisfying certain Maslov index conditions as well as Hamiltonian Floer cohomology of compact rational symplectic manifolds.

Morse moves in flow categories

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction associates a stable homotopy type to a framed flow category whose cohomology is designed to recover the corresponding Morse or Floer cohomology. We obtain analogues of the Whitney trick and of handle cancellation for framed flow categories: in this new setting these are moves that can be performed to simplify a framed flow category without changing the associated stable homotopy type. These moves often enable one to compute by hand the stable homotopy type associated to a framed flow category. We apply this in the setting of the Lipshitz-Sarkar stable homotopy type (corresponding to Khovanov cohomology) and the stable homotopy type of a matched diagram due to the authors (corresponding to sln Khovanov-Rozansky cohomology).

A flexible construction of equivariant Floer homology and applications

Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsv\'ath-Szab\'o, Lipshitz-Treumann, Szab\'o, Sarkar-Seed-Szab\'o, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith-type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar-Seed-Szab\'o's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

A Spectral Sequence of the Floer Cohomology of Symplectomorphisms of Trivial Polarization Class

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $\phi$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hat{\phi}$ such that neither $\hat{\phi}$ nor $\hat{\phi}^2$ has fixed points. We give conditions under which Seidel and Smith's localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $\mathit{HF}(\phi^2)\otimes \mathbb Z_2((\theta))$ to $\mathit{HF}(\phi)\otimes \mathbb Z_2((\theta))$.

Floer cohomologies of non-torus fibers of the Gelfand–Cetlin system

The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundary strata of the moment polytope. We compute Floer cohomologies of such non-torus Lagrangian fibers in the cases of the 3-dimensional full flag manifold and the Grassmannian of 2-planes in a 4-space.

Commuting symplectomorphisms and Dehn twists in divisors

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of $HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

Floer cohomology of -equivariant Lagrangian branes

Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal.22(2012), 443–477], we define the notion of a$\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold$M$, where$\mathfrak{g}\subset SH^{1}(M)$is a sub-Lie algebra of the symplectic cohomology of$M$. When$M$is a (symplectic) mirror to an (algebraic) homogeneous space$G/P$, homological mirror symmetry predicts that there is an embedding of$\mathfrak{g}$in$SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of$\mathfrak{sl}_{2}$as representations on the Floer cohomology of an$\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in Cm

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $\Pi_1\cup\Pi_2$ of two transverse special Lagrangian planes $\Pi_1,\Pi_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $\Pi_1\cup\Pi_2$ of two transverse Lagrangian planes $\Pi_1,\Pi_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=\Pi_1\cup\Pi_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.

The equivariant pair-of-pants product in fixed point Floer cohomology

We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.

Floer cohomology of the Chiang Lagrangian

We study holomorphic discs with boundary on a Lagrangian submanifold \(L\) in a symplectic manifold admitting a Hamiltonian action of a group \(K\) which has \(L\) as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in \(\mathbf {C}\mathbf {P}^3\) first noticed by Chiang (Int Math Res Not 45:2437–2441, 2004). We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case, it generates the split-closed derived Fukaya category as a triangulated category.

Floer cohomology in the mirror of the projective plane and a binodal cubic curve

We construct a family of Lagrangian submanifolds in the Landau--Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf O(1), and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. The algebra structure on the Floer cohomology is computed by counting sections of Lefschetz fibrations. Our results agree with the tropical analog proposed by Abouzaid--Gross--Siebert. An extension to mirrors of the complements of components of the anticanonical divisor is discussed.