Homotopy and Uniform Homotopy

Abstract

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.

Reduction of a Matrix Depending on Parameters to a Diagonal Form by Addition Operations

“…As a consequence of Theorem 2 (which is extracted here from results of [1,2]) we obtain that SLnA/EnA is a homotopy type invariant of X for finite dimensional spaces X if n > 3. This was proved in [6] for X = R and in [4] for X = R3 by different methods.…”

“…the corresponding map X -* SLnR is homotopic to the trivial map X -► 1". Now we invoke results of [1,2] to conclude that the map X -> SL"R is uniformly homotopic to the trivial map.…”

“…Note that Cx is endowed with the topology of the uniform convergence, and that the constant z" depends only on n and the dimension of X. We do not give any explicit bounds in this paper, although the proofs in [1,2] seem to be constructive enough to yield some explicit bounds.…”

“…Since n > 3, the fundamental group ^SOnR = Z/2Z is finite. Since X is finite dimensional and normal, Theorem 1 of [1] (see [2] for a shorter and a great deal more transparent proof) gives the desired conclusion.…”

“…We have 7t(q) = 0. By [1,2], a belongs to the connected component of 1", where SLnA is endowed with the topology induced by the uniform convergence topology on A (here we used that 7Tj(SL"R) is trivial).…”