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From Path Lifting and Unique Arc Lifting to Unique Path Lifting

Published online by Cambridge University Press:  20 November 2018

Harold Bell  and
Gerald S. Ungar
Harold Bell
Affiliation:
University of Cincinnati, Cincinnati, Ohio
Gerald S. Ungar
Affiliation:
University of Cincinnati, Cincinnati, Ohio
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In [4] it was conjectured that a light map p : E → B for which paths can be lifted and lifting of arcs is unique is a Serre fibration. As is well-known this implies that paths have unique liftings. In this paper we shall prove several special cases of this conjecture.

The two main theorems are: (3.5) Let p be a light compact map of a metric space E onto a connected semi-locally contractible along arcs metric space B. If arcs can be lifted uniquely then p is locally trivial.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 204 - 212
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bing, R. H., A convex metric with unique segments, Proc. Amer. Math. Soc. 4 (1953), 167–174.Google Scholar
2. Floyd, E., Some characterizations of interior maps, Ann. of Math. 51 (1950), 571–575.Google Scholar
3. Rolfsen, D., Geometric methods in topological spaces, Topology Conference, Arizona State University, 1967 (Edited by Grace, E. E.).Google Scholar
4. Ungar, G. S., Light fiber maps, Fund. Math. 62 (1968), 31–45.Google Scholar