Abstract

We present a new, intrinsic approach to Morse Theory which has interesting applications in geometry. We show that a Morse function f on a manifold determines a submanifold T of the product X \Theta X, and that (in the sense that Stokes theorem is valid) T has boundary consisting of the diagonal \Delta ae X \Theta X and a sum P = X p2Cr(f) Up \Theta Sp where Sp and Up are the stable and unstable manifolds at the critical point p. In the language of currents, @T = \Delta \Gamma P:(Stokes Theorem) This current (or kernel) equation on X \Theta X is equivalent to an operator equation d ffi T+T ffi d = I \Gamma P; ((Chain Homotopy)) where P is a chain map onto the finite complex of currents S f spanned by (integration over) the stable manifolds of f . The operator P can be defd on an exterior form ff by P(ff) = lim t!1 '

Citations