Morse Inequalities
Introduction
The Morse Inequalities are a set of fundamental results in the field of differential topology, named after American mathematician Marston Morse. These inequalities provide a relationship between the topology of a manifold and the critical points of a Morse function defined on that manifold.
Definition of Morse Function
A Morse function is a smooth function on a manifold that has only non-degenerate critical points. In other words, at each critical point, the Hessian matrix of second derivatives of the function is non-singular. This property allows for a local analysis around each critical point, leading to the concept of the Morse complex.
Morse Complex
The Morse complex is a chain complex that is constructed from the critical points of a Morse function. Each critical point contributes a generator to a certain degree in the complex, corresponding to the index of the critical point. The differential in the Morse complex counts the number of gradient flow lines between critical points of consecutive indices.
Morse Inequalities
The Morse Inequalities provide a relationship between the algebraic topology of the manifold and the critical points of a Morse function on the manifold. Specifically, they relate the Betti numbers of the manifold, which are the ranks of the homology groups, to the number of critical points of each index.
The inequalities state that for each integer k, the k-th Betti number of the manifold is less than or equal to the number of critical points of index k in the Morse function. Furthermore, alternating sums of these quantities are equal.
Proof of Morse Inequalities
The proof of the Morse Inequalities involves a detailed analysis of the Morse complex and its homology. The key tool is the Morse lemma, which provides a local normal form for a Morse function near a non-degenerate critical point.
Applications of Morse Inequalities
The Morse Inequalities have wide-ranging applications in differential topology and related fields. They provide a method to estimate the topological complexity of a manifold based on a suitable Morse function. This has been used in the study of minimal surfaces, symplectic geometry, and string theory, among others.