Inequalities of Critical Point Theory

A purpose of critical point theory is the counting of critical points of functions. Principal theorems in the subject state in precise terms that topological complexity of the underlying space is reflected in the existence and nature of critical points of any smooth real-valued function defined on the space. The initial development of critical point theory is peculiarly the work of one man, Marston Morse. Readers of the fundamental papers of Morse, particularly his Calculus of Variations in the Large [MM1 ], have found them difficult not only because of the intrinsic difficulties but for another reason. The work was done at a time when the requisite algebraic topology was not so adequately or systematically developed as at present. As a consequence, a substantial part of his exposition is concerned with proof of purely topological results in the special context of critical point theory. Thus part of his exposition deals with various aspects of the exactness of the homology sequence of a pair of spaces and part with the relation between deformations and the homomorphisms of homology theory. I t will be supposed that the reader knows a modest amount of homology theory, which may be summarized in the statement that the axioms of Eilenberg and Steenrod [E-Sl ] are theorems in singular homology theory. In this paper an account is given of a specific problem in critical point theory, namely the problem of a smooth function on a Riemannian manifold. The form of statements is chosen in such fashion that they may be extended reasonably to a wider class of problems. Thus this is intended simultaneously as an exposition of a particular useful case and a model. Critical points are defined locally and are classified locally in the neighborhood of separated sets of such points. Theorem 7.3 states that the local classification is possible, Corollary 7.4 permits direct sum decomposition, and Theorem 10.2 details the computation for nondegenerate critical points. The end result of this