On handle decompositions and diffeomorphisms
Max K. Agoston · Transactions of the American Mathematical Society · 1969
In this paper we are concerned with two questions which arise naturally in the study of manifolds.One has to do with whether certain types of handle decompositions of a manifold are unique ; and the other, under what circumstances they are preserved by diffeomorphisms.The corollaries to Theorem 1 and 2 are partial answers to these questions.Corollary 2 may prove useful for the general study of diffeomorphisms.We shall begin with some notation.Rk shall denote real Ä>dimensional vector space with unit disc Dk and unit sphere S"'1.Let /=[0, 1].All manifolds are assumed to be compact and C°°.Homology groups are taken with integer coefficients.« means either diffeomorphic or isotopic.By a nice Morse function on a manifold Mn (see [5, p. 44]) we shall mean a function rj : Mn ->■ R which has only nondegenerate critical points, and if p is a critical point, then r¡(p) = index p.It is well known that in this case M^rD^rj'^O, k +1/2] is a n-submanifold of M for OSkSn, and