Morse homology for generating functions of Lagrangian submanifolds

Abstract

Abstract. The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the "finite dimensional" symplectic invariants constructed via generating functions to the "infinite dimensional" ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional.

Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

“…can be understood as a Morse family, which generates the usual Lagrangian submanifold dL h (G) ⊂ T * G. This is closely related to the work of Milinković (1999), who studied a similar action principle on T * Q, in connection with the Morse homology of generating functions for Lagrangian submanifolds. In Milinković's formulation, the Lagrangian submanifold is determined by the time-1 Hamiltonian isotopy of H, which in this case, takes the zero section to dL h (G).…”

“…We begin, in section 2, by giving a brief review of Lagrangian mechanics, including the continuous Hamilton-Pontryagin principle, as well as summarizing the existing frameworks for discrete Lagrangian mechanics on Q × Q and on Lie groupoids G ⇒ Q. Next, in section 3, we introduce the discrete Hamilton-Pontryagin principle, which is defined with respect to paths in the cotangent groupoid T * G ⇒ A * G beginning at the zero section; the approach is related to that used by Milinković (1999) in studying the Morse homology of generating functions. This variational principle and its solutions, which imply those of the previous approaches to discrete Lagrangian mechanics, are derived first in cotangent bundle coordinates and then given intrinsically.…”

A brief survey of the spectral numbers in floer homology

“…depends only on . It turns out that these two constructions give the same result (see [56,57] for more details; see also [42] for similar relations in a different context and for the surfaces).…”

Equivariant homology for generating functions and orderability of lens spaces

“…The only case that does not follow immediately from the discussion above is when * = 2nl and a ≤ (l − 1)R. In this case we have G B(R) with lR < a ′ < (l + 1)R, but the exact sequences for a < a ′′ < ∞ and a ′′ < a ′ < ∞ with (l − 1)R < a ′′ < lR do not allow us to conclude, since both G (a ′′ ,a ′ ] Z k , * B(R) and G (a,a ′′ ] Z k , * −1 B(R) do not vanish. To get the result we will follow the approach of Morse homology for generating functions, as introduced by Milinković [Mil99,Mil97]. In order to turn the generating function of ρ κ into a (Z k -invariant) Morse function we will perturb it by a Z k -invariant Morse function f on S 2n−1 with k critical points { a 0 0,j , • • • , a 0 k−1,j } of index 2j and k critical points { a 1 0,j , • • • , a 1 k−1,j } of index 2j + 1 for each j = 0, • • • , n − 1 (see [Miln,p26]).…”