Homotopy and uniform homotopy

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.

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. If the elementary matrices t\ and e' are of the same type, i.e.…”

Numerical invariants of homotopies into spheres

“…Of course in general the answer to the parametric problem may be infinity: If X = R and M -S\ the complex numbers of unit norm, then the exponential map exp(ί) = e 2niΐ is homotopic to the constant map, but not by a homotopy of finite width. However it was proved in [2] …”

“…THEOREM [4]. If(n,q) is a pair of integers such that n > 2, g >: 0, and(n, q) is not (2,2), (4,6), #r (8,14), then…”