Rigidity and gluing for Morse and Novikov complexes

Abstract

Abstract. We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closedThe rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C 0 close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.

Lagrangian intersections and the Serre spectral sequence

“…The results summarized in §1.3.2 as well as the inequalitities (5) and (6) are first proved under the assumption at (1). However, we then show that our spectral sequence may also be constructed (with minor modifications) when L and L are Hamiltonian isotopic under the single additional assumption ω| π2(M,L) = 0 and as a consequence these three results also remain true in this setting.…”

“…It has been proven by the second author together with Andrew Ranicki in §2.1 of [6] that under the assumptions of the theorem and for the case of periodic orbits, the Floer comparison morphism admits a retract. More precisely, there exist monotone homotopies H 01 and G 01 so that V G 01 • V H 01 is an isomorphism whose matrix is upper triangular with 1's on the diagonal.…”

On the Floer homology of cotangent bundles

“…The above result is due to Cornea and Ranicki, [6] (in the case of Floer homology for a class of compact symplectic manifolds). See [20], Lemma 6.3, for an earlier application of the same argument.…”

“…Such a solution is a regular one (in the sense of transversality theory) because of (6). We conclude that, if we order the generators of CM k (E) and CF k (H) by increasing action, the homomorphism Θ k is given by a square matrix which is lower triangular and has ±1 on each diagonal entry.…”

“…The compactness expressed by Proposition 2.5 and the transversality expressed by Proposition 2.6 imply that W u (x) ∩ W s (y) consists of finitely many flow lines. The flow line through p -denote it by [p] -has the orientation defined above, and we define ǫ([p]) to be +1 if the tangent vector −∇ g f (p) is positively oriented, to 6 The fact that we get transversal intersections only for index difference not exceeding h − 1 is related to the fact that we are assuming f to be only of class C h . The possibility of keeping the regularity requirements low is important in nonlinear analysis, because functionals arising from smooth problems have often low regularity, and because in infinite dimensions C h+1 functionals are not dense in the space of C h ones, when h ≥ 1.…”