Homology with multiple-valued functions applied to fixed points

Abstract

Certain multiple-valued functions ( <italic>m</italic> -functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an <italic>m</italic> -function from a standard simplex to a space and an <italic>m</italic> -function from one space to another induces a homomorphism of homology modules. In a family of functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript x Baseline colon upper Y right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{f_x}:Y \to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> indexed by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the fixed points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript x"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{f_x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are taken to be the images at <italic>x</italic> of a multiple-valued function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :X \to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In certain circumstances <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <italic>m</italic> -function, giving information about the behavior of the fixed points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript x"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{f_x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <italic>x</italic> varies over <italic>X</italic> . 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 <italic>m</italic> -functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for <italic>m</italic> -functions.