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Tags
- #todo/untagged
Refs:
- Comparisons of Floer homologies: https://web.ma.utexas.edu/users/vandyke/notes/242_notes/lecture27.pdf#page=2
- https://www.emis.de/journals/AM/20-5/Golovko_2020.pdf#page=7
Links:
- Novikov ring
- associahedra
- exact triangle
- A_infty categories
- Liouville manifold
- Fredholm operator
- Types of Floer homology:
  - Morse homology
  - Floer homology of (Hamiltonian) symplectomorphisms
  - Lagrangian Floer homology
  - Fukaya category
  - Heegaard Floer homology
  - Seiberg-Witten Floer homology
  - Embedded contact homology
  - Cylindrical contact homology
  - monopole Floer homology

Floer homology

Summary

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Notes:

Heegard-Floer homology (Osvath-Szabo) is Lagrangian-Floer theory for Lagrangian torii in the symmetric product of a Riemann surface.
- Expected to be equivalent to Seiberg-Witten theory.
- Major applications in knot theory and 3-manifolds, gives an algorithm for computing knot genus.
- Defining HF is very difficult – see 1000+ page Fukaya-Oh-Ohta-Ono monograph. The major issue is transversality for the moduli spaces, which requires perturbation theory on the almost complex structure, the PDE operator, Hamiltonian isotopies of the Lagrangians, etc. Dealt with via Kuranishi spaces and polyfolds.
- Nice case: exact symplectic manifolds which contain no holomorphic spheres and the Lagrangians bounds no holomorphic discs.
- Useful theory precisely because it’s computable, see Floer’s surgery exact triangle, and there are some functorially defined maps coming from cobordisms.
Floer cohomology of the diagonal recovers quantum cohomology with its quantum product, ie \(\operatorname{HF}(\Delta, \Delta) \cong {\mathrm{QH}}^*(M)\). The product counts holomorphic triangles.
Appears in homological mirror symmetry relating algebraic and symplectic geometry – the symplectic geometry side involves the geometry of Lagrangian submanifolds and their Floer homology groups. See the Fukaya category.
There is no general prediction for \(\operatorname{rank}\operatorname{HF}(L_0, L_1)\) when \(L_0\neq L_1\); its definition involves an elliptic operator corresponding to a PDE one has no expectation of being able to write down explicitly!
For closed varieties \(X\), there are techniques due to Seidel for studying it via the theory on \(Y = X\setminus\Sigma_0\) and deformation theory, which has been carried out for the quartic surface and genus 2 curves.
A Fredholm operator \(D\) on a Banach space is one such that the index \(\dim\ker D - \dim\operatorname{coker}D\) makes sense.