Results 1  10
of
59
HOLOMORPHIC DISKS AND THREEMANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS
, 2001
"... ... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a ..."
Abstract

Cited by 204 (30 self)
 Add to MetaCart
(Show Context)
... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology.
FLOER MINIMAX THEORY, THE CERF DIAGRAM, AND THE SPECTRAL INVARIANTS
, 2007
"... ... function H as the minimax value of the action functional AH over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H; a) states that the minimax value is a critical value of the action functional AH. The main p ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
... function H as the minimax value of the action functional AH over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H; a) states that the minimax value is a critical value of the action functional AH. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, ω). We also prove that the spectral invariant function ρa: H ↦ → ρ(H; a) can be pushed down to a continuous function defined on the universal (étale) covering space ˜Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, ω). The proof relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic oneparameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer minimax theory.
EVERY CONTACT MANIFOLD CAN BE GIVEN A NONFILLABLE CONTACT STRUCTURE
, 2007
"... Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are nonfillable. Using contact surgery on his examples we create on every sphere S 2n−1, n ≥ 2, an exotic contact structure ξ − that also cont ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are nonfillable. Using contact surgery on his examples we create on every sphere S 2n−1, n ≥ 2, an exotic contact structure ξ − that also contains a plastikstufe. As a consequence, every closed contact manifold (M, ξ) (except S 1) can be converted into a contact manifold that is not (semipositively) fillable by taking the connected sum (M, ξ)#(S 2n−1, ξ−).
Morse Theory And Evasiveness
 Combinatorica
, 2000
"... Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices v 0 , v 1 ; : : : ; v n , and M a subcomplex of S, known to both the hider and the seeker. Let be a face of S, known only to the hider. The seeker is permitted ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices v 0 , v 1 ; : : : ; v n , and M a subcomplex of S, known to both the hider and the seeker. Let be a face of S, known only to the hider. The seeker is permitted to ask questions of the sort \Is vertex v i in ?" The seeker's goal is to determine whether is in M , using as few questions as possible. The seeker is permitted to use the answers to the earlier questions when choosing which vertex to ask about next. We assume that the seeker chooses each question, given the answers to the previous questions, according to a deterministic algorithm, which we call a decision tree algorithm. For any decision tree algor
The MorseWitten complex via dynamical systems
 Expo. Math
"... to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the GrobmanHartman theorem and the λlemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.
Heegaard splittings of graph manifolds
 Geometry & Topology
, 2004
"... Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V ∪S W be a Heegaard splitting. We prove that S is standard. In particular, S can be isotoped so that for each vertex manifold N of M, S ∩ N is either horizontal, pseudohorizontal, vertical or pseudovertical. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V ∪S W be a Heegaard splitting. We prove that S is standard. In particular, S can be isotoped so that for each vertex manifold N of M, S ∩ N is either horizontal, pseudohorizontal, vertical or pseudovertical. 1
RandalWilliams Stable moduli spaces of highdimensional mani
 folds, Acta Math
, 2014
"... ar ..."
(Show Context)
ON THE LUSTERNIKSCHNIRELMAN THEORY OF A REAL COHOMOLOGY CLASS
"... Farber developed a LusternikSchnirelman theory for finite CWcomplexes X and cohomology classes ξ ∈ H1 (X; R). This theory has similar properties as the classical LusternikSchnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X, ξ) as a generalization of the LusternikSc ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Farber developed a LusternikSchnirelman theory for finite CWcomplexes X and cohomology classes ξ ∈ H1 (X; R). This theory has similar properties as the classical LusternikSchnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X, ξ) as a generalization of the LusternikSchnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1form ω representing ξ. Namely, a closed 1form ω representing ξ which admits a gradientlike vector field with no homoclinic cycles has at least cat(X, ξ) zeros. In this paper we define an invariant F (X, ξ) for closed smooth manifolds X which gives the least number of zeros a closed 1form representing ξ can have such that it admits a gradientlike vector field without homoclinic cycles and give estimations for this number.