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Proceedings of the American Mathematical Society

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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
MathSciNet review: 0339144
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Abstract: If $ f$ is a set valued function defined on an $ n$-ball such that each $ f(x)$ is a subset of the $ n$-ball, and the graph of $ f$ is closed, then all that is needed to insure that there is a fixed point $ (x \in f(x))$ is that the singularity sets not be too high dimensional. I.e., the dimension of $ \{ x \in {B^n}\vert{\tilde H^q}(f(x)) \ne 0\} $ is $ \leqq n - q - 2$. 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 $ n$-ball.

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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