B. J. Ball

Alternative Approaches to Proper Shape Theory

Publisher Summary The notion of proper shape, recently introduced by the author and R. B. Sher, is a modification of proper homotopy type analogous to the modification of homotopy type given by Borsuks shape theory for compacta. The presentation in R. B. Sher closely parallels Borsuks original development, using a notion of proper fundamental net in place of Borsuks fundamental sequence. For some purposes, it might be preferable to use an approach to proper shape theory modeled on the ANR-systerns of Mardesic-Segal, on the mutations of Fox, or on the shapings of Mardesic. This chapter discusses how this might be done.

Extending cell-like maps on manifolds

Let X be a closed subset of a manifold M and Go be a cell. like upper semicontinuous decomposition of X. We consider the problem of extending G to a cell-like upper semicontinuous decomposition G of M such that M/G t M. Under fairly weak restrictions (which vanish if M = En or Sn and n ; 4) we show that such a G exists if and only if the trivial extension of Go obtained by adjoining to Go the singletons of M X, has the desired property. In particular, the nondegenerate elements of Bings dogbone decomposition of E3 are not elements of any cell-like upper semicontinuous decomposition G of E3 such that E3/G = E3. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if X/G : X and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in E3 are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold (k, m f 4). Examples are given of an everywhere wild simply embedded simple closed curve in E3 and of a compact absolute retract which embeds in E3 yet has no simple embedding in E3.

Certain collections of arcs in

t. Introduction. In considering upper semicontinuous decompositions of E3, it is sometimes useful to know whether a given collection of continua can be transformed, by a homeomorphism of E3 onto itself, into another collection which is simpler in some respects; for example, a collection of straight line intervals might be transformed into a collection of vertical intervals, or a collection of arcs into a collectioin of straight line intervals. It might also be useful to know conditions under which such a transformation can be effected by means of a particular type of homeomorphism of E3 onto itself. In this paper, the following questions of this type will be considered. Suppose a and ,3 are horizontal planes and G is a continuous collection of mutually exclusive arcs, each of which is irreducible from a to / and no one of which contains two points of any horizontal plane, such that the sum of the elements of G is compact and intersects a in a totally disconnected set. Under what conditions is there a homeomorphism of E3 onto itself which takes each element of G onto a vertical interval and does not change the z-coordinate of any point? It is shown, with the aid of certain results due to Bing [1] and Fort [5 ], that such a transformation is not always possible, even when the elements of G are straight line intervals. The following condition is found to be necessary and sufficient for the existence of such a transformation (see ?3 for definitions of unfamiliar terms): For every positive number e there exists a finite set K1, K2, * .* Kn of topological cylinders with bases on a and : such that (1) the solid cylinders determined by K1, K2, , Kn. are mutually exclusive, (2) each arc of G is enclosed by some K? and (3) each Ki has horizontal diameter less than e.

Some Theorems Concerning Spirals in the Plane

Throughout this paper the space under consideration will be the Euclidean plane. The definitions given by R. L. Moore in [6] are essentially equivalent to the following, also due to Professor Moore. An arc t is said to spiral down on a point 0 provided there does not exist a finite sequence of rays 1 each starting from 0 such that the first has no point other than 0 in common with t, the last is a straight ray, and no two adjacent rays of the sequence have a point other than 0 in common. The arc t is said to go n steps toward spiraling down on 0 provided there does not exist a sequence of n rays satisfying the above conditions. An arc which spirals down on some point is called a spiral and a point set which contains no spiral is said to be spiral free. In this paper an example is given of a compact, totally disconnected, closed point set such that every arc containinig it spirals down on each of uncountably many points. In order that the compact, totally disconnected, closed point set ][[ should be a subset of a spiral free arc it is sufficient that the projection of M onto some straight line be totally disconnected and it is necessary and sufficient that M be a subset of the sum of the elements of a continuous and equicontinuous collection G of mutually exclusive spiral free arcs such that G, with respect to its elements as points, is totally disconnected. A number of related results are also obtained.