Correntes, teoria de Morse para variedades com bordo e dualidade de Lefschetz

Abstract

Some results obtained by Harvey-Lawson Jr in the Morse-Stokes and Stokes Theorem article will be adapted for onboard varieties when the gradient flow is tangent to the edge. More precisely, given a compact Riemannian manifold with edge, a Morse function with a weak transversality condition and a flat metric in a vicinity of the critical points, we will prove two isomorphisms inspired by the Lefschetz Duality and that rescue the homology of two complexes defined by Kronheimer and Mrowka (Monopoles and Three-Manifolds). The isomorphisms can be with real or integer coefficients. The above results are adapted to a hypersurface contained in a non-mapped variety and a Morse function with gradient field tangent to hypersurface.