Floer Homology and the Heat Flow

By Dietmar Salamon, Joa Weber. 2003

162 citations recorded by the source corpus.

Abstract.We study the heat flow in the loop space of a closed Riemannian manifold M as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2af222da4a4be11dbc8c6378bce6644bd2d407ba

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

An exact sequence for contact- and symplectic homology (2007, methodology): “1 In [28] the symbol Λa L stands for a sublevel set of the energy functional, and a ∈ R.”
Floer homology of cotangent bundles and the loop product (2008, unknown): “A different proof is due to D. Salamon and J. Weber, see [SW06].”
A Beginner’s Introduction to Fukaya Categories (2013, result): “This and other related results can be viewed as the culmination of over a decade of investigations of the deep connections between the symplectic topology of T ∗N and the algebraic topology of the loop space of N , as previously studied by Viterbo [48], Salamon-Weber [39], Abbondandolo-Schwarz [1, 2”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “This is developed, for example, in [1] or [71].”
Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, background): “It follows from Salamon and Weber [3] that if the manifold X is orientable and the degrees of noncontractible orbits are determined by the vertical Lagrangian subbundle of T .”
Symplectic Cohomology and q-Intersection Numbers (2010, methodology): “When M is a cotangent bundle, the relation with free loop space homology [42, 31, 1] can be used to study this issue.”
Contact geometry of the restricted three‐body problem (2010, result): “Combining this with the work of Viterbo [19], Salamon-Weber [18], Abbondandolo-Schwarz [1] SH ∗ ( T ∗≤ 1 S 2 ) ∼ = H ∗ (Λ S 2 ) where Λ S 2 denotes the free loop space of S 2 we see that in both cases, c H ( L 1 ) and c H ( L 1 )+ (cid:15) , Rabinowitz Floer homology of the energy hypersurface…”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, background): “Indeed, this was established in [21], [1], and [19].”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, background): “for a closed orientable spin manifold N [59, 53, 1, 2].”
Topological quantum field theory structure on symplectic cohomology (2010, result): “For closed n-manifolds N , Viterbo [36] proved SH(T N) ∼= Hn−∗(LN), and there are now other approaches by Abbondandolo-Schwarz [1] and Salamon-Weber [30].”
On the Floer homology of cotangent bundles (2004, background): “Thus it makes sense to talk about the Floer homology of T ∗M , which we denote by HF∗(T ∗M).”
The Conley Conjecture and Beyond (2014, background): “It is plausible that a C1-generic, or even C∞-generic, Hamiltonian diffeomorphism has an orbit in a non-trivial homology class when the fundamental group (or the first homology group) of M is large enough; see Tal (2013) for some possibly relevant results for surfaces.”