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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 ..."
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... 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.
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 ..."
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Cited by 9 (0 self)
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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
EVERY CONTACT MANIFOLD CAN BE GIVEN A NONFILLABLE CONTACT STRUCTURE
, 2007
"... Abstract. 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 ..."
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Cited by 7 (3 self)
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Abstract. 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−). Most of the natural examples of contact manifolds can be realized as convex boundaries of symplectic manifolds. These manifolds are called symplectically fillable. An important class of contact manifolds that do not fall into this category are socalled overtwisted manifolds ([Eli88], [Gro85]). Unfortunately, the notion of overtwistedness is only defined for 3–manifolds. A manifold is overtwisted if one finds an embedded disk D 2 such that T D 2 ∣ ∣ ∂D 2 ⊂ ξ, an overtwisted disk DOT. This topological definition gives an effective way to find many examples of contact 3–manifolds that are nonfillable.
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 ..."
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Cited by 6 (1 self)
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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. ..."
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Cited by 5 (1 self)
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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
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 ..."
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Cited by 3 (3 self)
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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.
An end theorem for stratified spaces
"... We show that a tame ended stratified space X is the interior of a compact stratified space if and only if a Ktheoretic obstruction γ∗(X) vanishes. The obstruction γ ∗ (X) is a localization of Quinn’s mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7 below. ..."
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We show that a tame ended stratified space X is the interior of a compact stratified space if and only if a Ktheoretic obstruction γ∗(X) vanishes. The obstruction γ ∗ (X) is a localization of Quinn’s mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7 below. In particular, this explains when a Gmanifold is the interior of a compact Gmanifold with boundary. Our methods include a new transversality theorem, Corollary 1.17. 1 Main Results, Background and Definitions. 1.1. A beautiful theorem of geometric topology characterizes those topological manifolds which can be the interiors of compact manifolds with boundary. The breakthrough in this direction, [1], due to Browder, Levine and Livesay, states that a smooth manifold M n is the interior of a compact smooth manifold with simply connected boundary provided only that n ≥ 6, H∗(M, Z) is finitely generated, and M is simply connected at infinity. This was promptly