Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-10T19:51:26.459Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of the Mapping Degree

Published online by Cambridge University Press:  20 November 2018

D. G. Bourgin
D. G. Bourgin*
Affiliation:
University of Houston, Houston, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For the single-valued case the notion of degree has been given recent expression by papers of Dold [5] for the finite dimensional case, and by Leray-Schauder [8] for the locally convex linear topological space. Klee [7] has removed this restriction by use of shrinkable in place of convex neighborhoods with the central role filled by a form of (2.15) below. For set-valued maps a modern formulation is, for instance, to be found in Gorniewicz-Granas [6]. These contributions relate the degree to the Lefschetz number, and the set-valued maps are required to map points into acyclic sets; that is to say, into "swollen points".

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1109 - 1117
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bourgin, D. G., Set valued transformations, Bull. Amer. Math. Soc. 78 (1972), 597–599.Google Scholar
2. Bourgin, D. G., Fixed points and saddle points, Notices Amer. Math. Soc. 19 (1972), A–724.Google Scholar
3. Bourgin, D. G., Fixed point and min max theorems, Pacific J. Math. 45 (1973) 403–412.Google Scholar
4. Bourgin, D. G., Modern algebraic topology (Macmillan, New York, 1963).Google Scholar
5. Dold, A., Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965), 1–8.Google Scholar
6. Gornewicz, L. and Granas, A., Fixed point theorems for multi-valued mappings of the absolute neighborhood retracts, J. Math Pures Appl. 49 (1970), 381–395.Google Scholar
7. Klee, V., Leray-Schauder Theory without local convexity, Math. Ann. 141 (1962), 380–296.Google Scholar
8. Leray, J. and Schauder, J., Topologie et equations functionelles, Ann. Sci. Ecole Norm. Sup. 51 (1934), 45–78.Google Scholar
9. Sklyarenko, E. G., Some applications of the theory of sheaves in general topology, Uspehi, Mat. Nauk. 19 (1964), 47–70.Google Scholar