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On Some Results in Morse Theory

Published online by Cambridge University Press:  20 November 2018

Gudrun Kalmbach
Gudrun Kalmbach*
Affiliation:
Pennsylvania State University, University Park, Pennsylvania
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The /z-cobordism theorem in [8], the generalized Poincaré conjecture in higher dimensions in [20] and several other results in differential topology are proved by using the following theorems of Morse theory:

(1) the elimination of critical points;

(2) the existence of nondegenerate functions for which the descending and ascending bowls have normal intersection;

(3) the alteration of function values at critical points. (For the details see below.)

We shall give short and elementary pr∞fs of these theorems together with some stronger statements than the ones given in [8-13] or [19].

The theorems are proved for noncompact manifolds rather than for compact manifolds since, by a trivial modification of the manifold (deleting the boundary or one point) the case of compact manifolds is included.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 88 - 105
Copyright
Copyright © Canadian Mathematical Society 1975

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