Allan Calder

Homotopy and uniform homotopy

An elementary proof of the Bounded Lifting Lemma is given, together with a proof that homotopy and uniform homotopy do not agree for maps into compact spaces with infinite fundamental groups even though they can agree for maps into a noncompact space with infinite fundamental group.

ON THE WIDTH OF HOMOTOPIES

The purpose of this paper is to initiate an examination of this phenomenon and to try and determine to what extent Theorem 0.1 is a special case. In the process we show the existence of a sequence of invariants of the structure of a Riemannian manifold that does not appear to have been noted before. The ideas we will be concerned with are as follows: If (Y,

Bounded linear mappings of finite rank

Abstract This paper comprises a constructive investigation of the relationship between compactness, finite rank and located kernel for a bounded linear mapping into a finite-dimensional normed space. The main result is that a bounded linear mapping of a normed space into a finite-dimensional normed space is constructively compact if and only if its kernel is located. Several examples are given which highlight the constructive distinction between a mapping into a finite-dimensional space and a mapping with finite-dimensional range.

The width of homotopies into spheres

If IHI < ~0, H is said to have bounded width. Natural questions are: (1) If two maps f, g: X+ M are homotopic, are they connected by a homotopy of bounded width? (2) If so, what can be said about the bound? Clearly some restrictions on X and M are necessary in order to give a positive answer to (1). For example, letting X = R’ and M = S’, the constant map and the exponential map t +exp (it) are homotopic but not by a homotopy of bounded width. The fundamental group of M turns out to be the key. It is shown in [3] that if M is a closed Riemannian manifold with finite fundamental group, and X is a finite dimensional normal space, the answer to question (1) is yes. In fact, there is a finite bound b such that any two homotopic maps are connected by a homotopy of width less than b. Finally, if b(X, M) denotes the infimum of all such bounds, the number

Picard’s theorem

This paper deals with the numerical content of Picards Thsorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.

Homotopies of bounded width are almost Lipschitz

Abstract Let Y be a compact Riemannian manifold. Let X be locally compact and paracompact. It is shown that every uniform homotopy and every homotopy of bounded width from X to Y may be approximated by a Lipschitz homotopy. Applications to the study of uniform homotopy and homotopies of bounded width are given.