Homology with multiple-valued functions applied to fixed points

Author:
Richard Jerrard

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

MSC:
Primary 55C20

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.

**[1]**Robert F. Brown,*The Lefschetz fixed point theorem*, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR**0283793****[2]**Albrecht Dold,*Fixed point index and fixed point theorem for Euclidean neighborhood retracts*, Topology**4**(1965), 1–8. MR**0193634****[3]**E. Fadell,*Some examples in fixed point theory*, Pacific J. Math.**33**(1970), 89–100. MR**0261582****[4]**R. P. Jerrard,*Inscribed squares in plane curves*, Trans. Amer. Math. Soc.**98**(1961), 234–241. MR**0120604**, 10.1090/S0002-9947-1961-0120604-3**[5]**R. P. Jerrard,*On Knaster’s conjecture*, Trans. Amer. Math. Soc.**170**(1972), 385–402. MR**0309101**, 10.1090/S0002-9947-1972-0309101-7**[6]**Ronald J. Knill,*Cones, products and fixed points*, Fund. Math.**60**(1967), 35–46. MR**0211389****[7]**K. Kuratowski,*Problem*49, Fund. Math.**15**(1930), 356.**[8]**William Lopez,*An example in the fixed point theory of polyhedra*, Bull. Amer. Math. Soc.**73**(1967), 922–924. MR**0216490**, 10.1090/S0002-9904-1967-11848-2

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

Article copyright:
© Copyright 1975
American Mathematical Society