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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An extension of Brouwer’s fixed-point theorem to nonacyclic, set valued functions
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by Robert Connelly PDF
Proc. Amer. Math. Soc. 43 (1974), 214-218 Request permission

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 $\{ x \in {B^n}|{\tilde H^q}(f(x)) \ne 0\}$ 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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Additional Information
  • © Copyright 1974 American Mathematical Society
  • 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