arXiv research

This paper computes fixed point Floer cohomology for Dehn twists on surfaces.

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality f…

Researchers compare two homological invariants for mapping classes of surfaces.

Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poinca…

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $φ$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hatφ$ such that neither $\hatφ$ nor $\hatφ^2$…

Introduces linear K-systems for Hamiltonian Floer theory.

Knot Floer homology matches fixed point Floer for fibred knots.

Knot Floer homology reveals fixed points of monodromy.

Paper connects Khovanov homology to Floer cohomology.

Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Kh_{symp,inv}(K) for a knot K inside S^{3}, as well as a spectral sequence converging to the Heegaard Floer homology-hat group for the connected sum of the double branched cover with a copy of S^{2}xS^{1}. The E^{1}-page of this spe…

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we…

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…

Formula estimates pseudo-Anosov maps' fixed points, linking to surface properties.

Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltot…

Study shows hyperbolic knots' monodromy without fixed points.

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…

Compact manifolds without odd cohomology have almost fixed points.

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …

Develops Floer cohomology for 4-manifolds with involutions and links.

Survey on equivariant cohomology of Lie group actions.

We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…

Study torus orbifolds with two fixed points and their cohomology.

Develops Lefschetz theory for noncompact manifolds.

We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…

We show that there is a well-defined cap-product structure on the Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological poin…

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

Study computes SL(2,C) Floer cohomology for surgeries on knots.

New proof for 6D symplectic manifold with 4 fixed points.

Geometrically generates Fukaya categories of Weinstein manifolds.

We compute the Floer homology of mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The formula for the Floer homology is obtained from a topological separation of fixed points and a separation mechanism for Floer connecting orbits. As examples, …

Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

Develops Floer theory for 3-manifold covers using equivariant structures.

Study calculates non-trivial link cohomologies and applies to branched double covers.

Study new conjectures linking knot volume and knot cohomology.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

New knot invariant from equivariant Heegaard Floer cohomology.

We give a presentation for the Floer cohomology ring $HF^*(Σ\times S^1)$, where $Σ$ is a Riemann surface of genus bigger than one, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of flat SO(3)-connections of odd degree over $Σ$. We study the spectrum of the action o…

Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and o…

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

Paper constructs new equivariant Floer cohomology and proves invariance properties.

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Constructs a cyclic, filtered, strictly unital curved $A_{\infty}$ category for Lagrangian submanifolds and develops Floer theory.

Real Heegaard Floer homology gets a new grading for certain 3-manifolds.