Homology with multiple-valued functions applied to fixed points

Author:
Richard Jerrard

Journal:
Trans. Amer. Math. Soc. **213** (1975), 407-427

MSC:
Primary 55C20

DOI:
https://doi.org/10.1090/S0002-9947-1975-0380778-6

Erratum:
Trans. Amer. Math. Soc. **218** (1976), 406.

MathSciNet review:
0380778

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Abstract: Certain multiple-valued functions (*m*-functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an *m*-function from a standard simplex to a space and an *m*-function from one space to another induces a homomorphism of homology modules.

In a family of functions indexed by the fixed points of are taken to be the images at *x* of a multiple-valued function . In certain circumstances is an *m*-function, giving information about the behavior of the fixed points of as *x* varies over *X*. These facts are applied to self-maps of products of compact polyhedra and the question of whether such a product has the fixed point property for continuous functions is essentially reduced to the question of whether one of its factors has the fixed point property for *m*-functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for *m*-functions.

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0380778-6

Article copyright:
© Copyright 1975
American Mathematical Society