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Transactions of the American Mathematical Society

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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 $ {f_x}:Y \to Y$ indexed by $ x \in X$ the fixed points of $ {f_x}$ are taken to be the images at x of a multiple-valued function $ \phi :X \to Y$. In certain circumstances $ \phi $ is an m-function, giving information about the behavior of the fixed points of $ {f_x}$ 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0380778-6
Article copyright: © Copyright 1975 American Mathematical Society

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