Constructing Morse Homology

Abstract

In this thesis we discuss the construction of the Morse homology groups for compact orientable manifolds. Morse homology is a way of describing information of manifolds in the form of algebraic groups, called the Morse homology groups. These groups encode information of manifolds that is invariant under diffeomorphisms in particular. Suppose that M is a n-dimensional manifold. The idea behind Morse homology is that we define a function f : M → R, that intuitively measures the height of each point in M. We are interested in such functions when they behave nicely around their critical points. For each such critical point, the behaviour of f around it can then be described by a number λ = 0, 1, ..., n. We call this number the index of that critical point. For such well-behaved functions, we define gradient-like objects X. We call these objects pseudo-gradients. The flow of a pseudo-gradient of a particular function descends through the level sets of this function. The flow-lines of this flow connect the critical points of the function. These connections form spaces themselves. If a pseudo-gradient is well-behaved, then also the spaces of connections between critical points are well-behaved. For any manifold M, one can find well-behaved functions and well-behaved pseudo-gradients. Suppose that the indices of two critical points, as mentioned above, differ by one. In this case, we can count the connections between these two critical points. By looking at critical points and counting connections between them, we can construct a sequence of modules that is a chain complex. We call this complex, the Morse complex. The Morse complex is dependent on the function and pseudo-gradient that are used to construct it. However, we can derive algebraic groups from it that are not dependent on the choice of Morse function and pseudo-gradient. These groups are the Morse homology groups.