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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

References

1. Huebsch-Morse, , Conditioned differentiable isotopies, Diff. Analysis, Bombay Coll. (1964), 1–25.Google Scholar
2. Huebsch-Morse, , A model nondegenerate function, Rev. Roumaine Math. Pures Appl. 10 (1965), 691–722.Google Scholar
3. Kalmbach, G., Ùber niederdimensionale CW-Komplexe in nichikompakten Mannigfaltigkeiten, Ph.D. Dissertation Gôttingen, 1966.Google Scholar
4. Kalmbach, G., Deformation retracts and weak deformation retracts of noncompact manifolds, Proc. Amer. Math. Soc. 20 (1969), 539–544.Google Scholar
5. Kalmbach, G., On sm∞th bounded manifolds, Proc. Amer. Math. Soc. 22 (1969), 466–469.Google Scholar
6. Kérekjarto, B. V., Vorlesungen iiber Topologie, I, Berlin, Grundlehren d. Math. Wiss. in Einzeldarst. VIII.Google Scholar
7. Milnor, J., Morse theory, Ann. of Math. Studies 51, 1963.Google Scholar
8. Milnor, J., Lectures on the h-cobordism theorem, Princeton, N.J., 1965.Google Scholar
9. Morse, M., The existence of polar nondegenerate functions on differentiate manifolds, Ann. of Math. 71 (1960), 352–383.Google Scholar
10. Morse, M., Quadratic forms 6 and 6-fibre bundles, Ann. of Math. 81 (1965), 303–340.Google Scholar
11. Morse, M., Bowls, f-fibre bundles and the alteration of critical values, An. Acad. Brasil. Ci. 36 (1964), 245–259.Google Scholar
12. Morse, M., The elimination of critical points of a nondegenerate function on a diff. manifold, J. Analyse Math. 13 (1964), 257–316.Google Scholar
13. Morse, M., Bowls of a nondegenerate function on a compact diff. manifold, Diff. and Combin. TopoL, Proc. of a symposium in honour of Morse, M., Princeton, N.J., 1964, 81–103.Google Scholar
14. Morse, M. and Cairns, S. S., Critical point theory in global analysis and differential topology (Academic Press, N.Y., 1969).Google Scholar
15. Munkres, J., Obstructions to the sm∞thing of piecewise differentiable homeomorphisms, Ann. of Math. 72 (1960), 521–554.Google Scholar
16. Munkres, J., Elementary differential topology, Ann. of Math. Studies 54, 1966.Google Scholar
17. Richards, J., On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259–269.Google Scholar
18. Seifert-Threlfall, , Lehrbuch d. Topologie, N.Y., 1934.Google Scholar
19. Smale, S., On gradient dynamical systems, Ann. of Math. 74 (1961), 199–206.Google Scholar
20. Smale, S., The generalized Poincaré conjecture in dimension greater than 4, Ann. of Math. 74 (1961), 391–406.Google Scholar
21. Spanier, E. H., Algebraic topology (McGraw Hill, New York, 1966).Google Scholar
22. Whitehead, J. H. C., Combinatorial homotopy, I, Math, works of J. H. C. Whitehead, Vol. III, 85–117.Google Scholar
23. Whitehead, J. H. C., Combinatorial homotopy, II, Math, works of J. H. C. Whitehead, Vol. III, 119–162.Google Scholar