An extension of Brouwer's fixedpoint theorem to nonacyclic, set valued functions
Author:
Robert Connelly
Journal:
Proc. Amer. Math. Soc. 43 (1974), 214218
MSC:
Primary 55C20
MathSciNet review:
0339144
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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:
http://dx.doi.org/10.1090/S00029939197403391446
PII:
S 00029939(1974)03391446
Keywords:
Brouwer fixedpoint theorem,
sheaf cohomology,
sheaf,
Leray spectral sequence,
upper semicontinuous,
set valued function,
dimension,
multivalued function
Article copyright:
© Copyright 1974
American Mathematical Society
