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
Full-text PDF Free Access
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.
- [1] Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221500
- [2] Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 16676, https://doi.org/10.2307/2371832
- [3] Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
- [4] B. O'Neill, Fixed points of multi-valued functions, Duke Math. J. 24 (1957), 61-62. MR 18, 752.
- [5] Barrett O’Neill, Induced homology homomorphisms for set-valued maps, Pacific J. Math. 7 (1957), 1179–1184. MR 104226
- [6] E. G. Skljarenko, Some applications of the theory of sheaves in general topology, Uspehi Mat. Nauk 19 (1964), no. 6 (120), 47–70 (Russian). MR 0171259
- [7] E. Skljarenko, A theorem on mappings which lower the dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 429–432 (Russian, with English summary). MR 149445
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55C20
Retrieve articles in all journals with MSC: 55C20
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