Homología de Morse en variedades compactas

Abstract

RESUMEN: Dada una variedad Riemanniana compacta (M, g) y una funci ́on de Morse f : M → R cuyo campo gradiente satisface la condici ́on de Morse–Smale, (i.e. las variedades estable e inestable de f tienen intersecci ́on transversal) construimos un complejo de cadena llamado el complejo de Morse–Witten. Nuestro objetivo en este art ́ıculo es mostrar que la homolog ́ıa del complejo de Morse–Witten es isomorfa a la homolog ́ıa singular de M.

ABSTRACT: Given a compact Riemannian manifold (M, g) and a Morse function f : M → R whose gradient flow satisfies the Morse–Smale condition, (i.e. the stable and unstable manifolds of f intersect transversely) we construct a chain complex called the Morse–Witten Complex. Our goal on this paper is to show that the homology of the Morse–Witten complex is isomorphic to the singular homology of M.