Ivancevic_Applied-Diff-Geom

412 Applied Differential Geometry: A Modern IntroductionThis operator satisfies δ 2 = 0 and therefore defines a cohomology theory.Using Conley’s continuation principle, Floer [Floer (1988)] showed thatthe resulting cohomology theories resulting from different choices of f arecanonically isomorphic.In his QFT–based rewriting the Morse topology, Ed Witten [Witten(1982)] considered also the operators:d t = e −tf de tf , their adjoints : d ∗ t = e tf de −tf ,as well as their Laplacian: ∆ t = d t d ∗ t + d ∗ t d t .For t = 0, ∆ 0 is the standard Hodge–de Rham Laplacian, whereas fort → ∞, one has the following expansion∆ t = dd ∗ + d ∗ d + t 2 ‖df‖ 2 + t ∑ k,j∂ 2 h∂x k ∂x j [i ∂ x k, dxj ],where (∂ x k) k=1,...,n is an orthonormal frame at the point under consideration.This becomes very large for t → ∞, except at the critical points of f,i.e., where df = 0. Therefore, the eigenvalues of ∆ t will concentrate nearthe critical points of f for t → ∞, and we get an interpolation between deRham cohomology and Morse cohomology.Morse Homology on M. Now, following [Milinkovic (1999); Ivancevicand Pearce (2006)], for any Morse function f on the configuration manifoldM we denote by Crit p (f) the set of its critical points of index p and defineC p (f) as a free Abelian group generated by Crit p (f). Consider the gradientflow generated by (3.228). Denote by M f,g (M) the set of all γ : R → Msatisfying (3.228) such thatThe spaces∫ +∞−∞2dγ∣ dt ∣ dt < ∞.M f,g (x − , x + ) = {γ ∈ M f,g (M) | γ(t) → x ± as t → ±∞}are smooth manifolds of dimension m(x + ) − m(x − ), where m(x) denotesthe Morse index of a critical point x. Note thatM f,g (x, y) ∼ = W u g (x) ∩ W s g (y),