Morse homology

Abstract

Morse homology were developed during the rst half of the twentieth century. The underlying idea and various in nite dimensional versions, such as Floer homology, continue to be of interest to researchers in mathematics and theoretical physics today. In the rst chapter of this thesis, we brie y describe the nite dimensional Morse theory and its cellular singular homology . Our main focuses are stable/ unstable manifolds, the associated CW-complex. Suppose M is a smooth compact nite dimensional Riemannian manifold and f is a real valued function de ned on M with all critical points nondegenerate. We consider the gradient ow line, that is, a smooth curve : R ! M satisfying the following di erential equation: d (t) dt +5f( (t)) = 0 Using these graident ow lines, we can decompose M into cells and construct a CW-complex associated to f. In chapter two, we introduce Morse homology, whose chain complex is generated by critical points of f and the boundary operators is de ned by counting certain gradient ow lines connecting two critical points of relative index one. In order to check that this boundary is well-de ned, We need to study the analysis of moduli spaces of gradient ow lines. These moduli spaces can be identi ed as zero sets of some Fredholm operators between in nite dimensional Banach manifolds. Since the moduli spaces may fail to be compact, we will discuss about the compacti cation of the moduli space. Then we obtain the fact that there are nitely many ow lines between critical points with relative index 1. After de ning an coherent orientation on moduli spaces, we can construct boundary operator of Morse complex by counting ow lines with sign. This approach is very useful in studying a generalization of Morse theory on certain in nite-dimensional manifolds. In chapter 3, we show that the two kinds of homology we construct in the previous two chapters are isomorphic. In fact, we show that two chain complexes are identical. In the nal chapter, we give an overview of Floer theory and very brie y introduce Lagragian-Floer theory and instanton Floer theory. Show more