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123
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
Holomorphic disks and threemanifold invariants: properties and applications
"... ̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The pr ..."
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Cited by 119 (29 self)
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̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. 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. 1.
SYMPLECTIC TOPOLOGY AS THE GEOMETRY OF ACTION FUNCTIONAL. I  RELATIVE FLOER THEORY ON THE COTANGENT BUNDLE
, 1997
"... ..."
Morse homology
 Progress in Mathematics
, 1993
"... Abstract. An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudocycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such ..."
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Cited by 36 (3 self)
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Abstract. An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudocycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudocycle and a welldefined integral homology class in singular homology. 1.
Asymptotic stability equals exponential stability, and ISS equals finite energy gain  if you twist your eyes
, 1999
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Asymptotic Expansion of the Witten deformation of the analytic torsion
, 1993
"... Given a compact Riemannianmanifold (M d ; g), a finite dimensional representation ae : 1 (M) ! GL(V ) of the fundamental group 1 (M) on a vectorspace V of dimension l and a Hermitian structure on the flat vector bundle E p !M associated to ae, RaySinger [RS] have introduced the analytic to ..."
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Cited by 20 (7 self)
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Given a compact Riemannianmanifold (M d ; g), a finite dimensional representation ae : 1 (M) ! GL(V ) of the fundamental group 1 (M) on a vectorspace V of dimension l and a Hermitian structure on the flat vector bundle E p !M associated to ae, RaySinger [RS] have introduced the analytic torsion T = T (M; ae; g; ) ? 0: Witten's deformation dq (t) of the exterior derivative dq ; dq (t) = e \Gammaht dq e ht ; with h : M ! R a smooth Morse function, can be used to define a deformation T (h; t) ? 0 of the analytic torsion T with T (h; 0) = T: The main results of this paper are to provide, assuming that grad g h is Morse Smale, an asymptotic expansion for log T (h; t) for t ! 1 of the form P d+1 j=0 a j t j + b log t + O( 1 p t ) and to present two different formulae for a 0 : As an application we obtain a shorter derivation of results due to RaySinger [RS], Cheeger [Ch], Muller [Mu1,2] and BismutZhang [BZ] which, in increasing generality, concern the equality for ...
Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group
 Duke Math. J
, 2005
"... Abstract. In this paper, we apply spectral invariants, constructed in [Oh5,8], to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds (M, ω). Using spectral invariants, we first construct an invariant norm called the spectral norm on the Hamiltonian diffeomorphism group and obtai ..."
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Cited by 18 (4 self)
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Abstract. In this paper, we apply spectral invariants, constructed in [Oh5,8], to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds (M, ω). Using spectral invariants, we first construct an invariant norm called the spectral norm on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the ǫregularity theorem and the symplectic area of certain pseudoholomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths among all paths. In addition to the construction of spectral invariants, these applications rely heavily on the chain level Floer theory and on some existence theorems with energy bounds of pseudoholomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this paper in turn relies on some careful geometric analysis involving adiabatic degeneration and thickthin decomposition of the Floer moduli spaces which has an independent interest of its own. We assume that (M, ω) is strongly semipositive throughout, which will be removed in a sequel.
Exact Lagrangian submanifolds in simplyconnected cotangent bundles,” math.SG/0701783
"... Abstract. We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second StiefelWhitney class of the Lagrangian submanifold) we prove such submanifolds are Floerc ..."
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Cited by 17 (1 self)
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Abstract. We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second StiefelWhitney class of the Lagrangian submanifold) we prove such submanifolds are Floercohomologically indistinguishable from the zerosection. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach. 1.
Closed orbits of gradient flows and logarithms of nonabelian Witt vectors
, 2000
"... We consider the flows generated by generic gradients of Morse maps f: M → S 1. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called the eta funct ..."
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Cited by 17 (1 self)
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We consider the flows generated by generic gradients of Morse maps f: M → S 1. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called the eta function, and lies in a suitable quotient of the Novikov completion of the group ring of the fundamental group of M. Its abelianization coincides with the logarithm of the twisted Lefschetz zeta function of the flow. For C 0generic gradients we obtain a formula expressing the eta function in terms of the torsion of a special homotopy equivalence between the Novikov complex of the gradient flow and the completed simplicial chain complex of the universal cover.