Mistake in Hirsch's proof

of the Brouwer Fixed Point Theorem

Author:
Kapil D. Joshi

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1523-1525

MSC (1991):
Primary 55M20, 54H25

Published electronically:
August 3, 1999

MathSciNet review:
1646193

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Abstract | References | Similar Articles | Additional Information

Abstract: A mistake in Hirsch's proof of the Brouwer Fixed Point Theorem, based on the Simplicial Approximation Theorem, is pointed out along with some elaboration of its nature and possible ways to salvage Hirsch's proof.

**[1]**R. H. Bing,*The elusive fixed point property*, Amer. Math. Monthly**76**(1969), 119–132. MR**0236908****[2]**J. Dugundji*in*N. Steenrod,*Reviews of Papers in Algebraic and Differential Topology, Topological Groups and Homological Algebra,*p. 565, American Mathematical Society, Providence, R. I., 1968.**[3]**Samuel Eilenberg and Norman Steenrod,*Foundations of algebraic topology*, Princeton University Press, Princeton, New Jersey, 1952. MR**0050886****[4]**Victor Guillemin and Alan Pollack,*Differential topology*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR**0348781****[5]**M. W. Hirsch,*A proof of the non-retractability of a cell onto its boundary*, Proc. Amer. Math. Soc., 14(1963), 364-365.**[6]**K. D. Joshi,*Introduction to general topology*, A Halsted Press Book, John Wiley & Sons, Inc., New York, 1983. MR**709260****[7]**K. D. Joshi,*Applied Discrete Structures*, New Age International, New Delhi, 1997**[8]**S. Lefschetz,*Topology*, American Mathematical Society Colloquium Publication, vol. XII, New York, 1930**[9]**Colin Patrick Rourke and Brian Joseph Sanderson,*Introduction to piecewise-linear topology*, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR**665919****[10]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[11]**E. C. Zeeman,*Relative simplicial approximation*, Proc. Cambridge Philos. Soc.**60**(1964), 39–43. MR**0158403**

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Additional Information

**Kapil D. Joshi**

Affiliation:
Department of Mathematics, Indian Institute of Technology, Powai, Mumbai, 400 076 India

Email:
kdjoshi@math.iitb.ernet.in

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05205-3

Received by editor(s):
November 4, 1997

Received by editor(s) in revised form:
March 24, 1998, and June 19, 1998

Published electronically:
August 3, 1999

Communicated by:
Ralph Cohen

Article copyright:
© Copyright 2000
American Mathematical Society