Abstract. It is shown that the sets, homotopy and uniform homotopy classes of maps from a finite dimensional normal space to a space of finite type with finite fundamental group, coincide. Applications of this result to the study of remainders of Stone-Cech compactifications, Kan extensions, and other areas are given.In this paper we consider the question of when the existence of homotopies implies existence of uniform homotopies for given maps of one space to another. Not surprisingly, the positive results we obtain require strong hypotheses on the spaces and maps in question. The domain must be finite dimensional normal and the codomain must have the homotopy type of a Clf-complex of finite type with finite fundamental group. However, as will be seen below, these hypotheses are sufficiently broad so as to admit interesting applications.A major stumbling block to carrying out an investigation of when homotopic maps must also be uniformly homotopic seems to have been the example of the real line and the circle. While all maps of the real line to the circle are homotopic, a simple computation shows that there are uncountably many uniform homotopy classes of such maps [7]. What we show below is that this example is the exceptional case, the culprit being the circle. We prove the following: Theorem 1. Let X be a finite dimensional normal space. Let Y be a compact space with the homotopy type of a CW-complex and with mX(Y) finite. Then two maps f g: X -* Y are homotopic iff they are uniformly homotopic.Indeed, in the paper itself we prove a somewhat stronger theorem, relaxing the hypothesis on Y in various ways.This result established, we proceed to a series of applications, somewhat unrelated in statement, but all depending heavily on a form of Theorem 1. We give two examples.
It is shown that the sets, homotopy and uniform homotopy classes of maps from a finite dimensional normal space to a space of finite type with finite fundamental group, coincide. Applications of this result to the study of remainders of Stone-Cech compactifications, Kan extensions, and other areas are given.
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