An extension of Brouwer's fixed-point theorem to nonacyclic, set valued functions

Author:
Robert Connelly

Journal:
Proc. Amer. Math. Soc. **43** (1974), 214-218

MSC:
Primary 55C20

DOI:
https://doi.org/10.1090/S0002-9939-1974-0339144-6

MathSciNet review:
0339144

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Abstract | References | Similar Articles | Additional Information

Abstract: If is a set valued function defined on an -ball such that each is a subset of the -ball, and the graph of is closed, then all that is needed to insure that there is a fixed point is that the singularity sets not be too high dimensional. I.e., the dimension of is . Examples are given to show that the dimension requirements are the best possible. The proof involves defining an analogue of the retraction in the ``no retraction'' proofs of the Brouwer theorem, and then applying the Leray spectral sequence to the projection of the graph of this retraction onto the -ball.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0339144-6

Keywords:
Brouwer fixed-point theorem,
sheaf cohomology,
sheaf,
Leray spectral sequence,
upper semicontinuous,
set valued function,
dimension,
multivalued function

Article copyright:
© Copyright 1974
American Mathematical Society