Michael Starbird

The shrinkability of bing-whitehead decompositions

THIS PAPER is a study of a special class of toroidal decompositions of 3-manifolds called Bing-Whitehead decompositions. It is well known that a Bing-Whitehead decomposition of a 3-manifold is shrinkable if all successive stages are Bing nested; but it is not shrinkable if all successive stages are Whitehead nested. (See Figs 1 and 2.) Consider a Bing-Whitehead decomposition of a 3-manifold which is defined by h, successive Bing nested stages. followed by 1 Whitehead nested stage, followed by h, successive Bing nested stages, followed by 1 Whitehead nested stage, . . The principal result of this paper is that this decomposition is shrinkable if and only if

A complex which cannot be pushed around in

This paper contains an example of a finite complex C with triangulation T which admits two linear embeddings f and g into E3 so that although there is an isotopy of E3 taking the embedding f to g there is no continuous family of linear embeddings of C starting at f and ending at g. No such example can exist in E2.

Products with a compact factor

Abstract In this paper we consider spaces X X Y , where Y is a compact Hausdorff space. Most of this paper is devoted to giving new, simplified proofs to some recent results concerning normality and map extension properties in some products. The theorems are of two types. First, we assume that the product X X Y is normal and deduce separation and covering properties for X , for example, that X must be v ( Y )-collectionwise normal. Second, we assume that X has some special separation properties (namely, w ( Y )-collectionwise normality) and deduce some map extension properties for X X Y . For example, if A and B are closed subsets of X and Y , respectively, then maps from A X B into the real line R can be extended to all of X X Y regardless of whether X X Y is normal or not. The proofs of all the theorems take advantage of the natural one-to-one correspondence between maps ⨍: X X Y → R and maps ⨍ : X → C ( Y ).

Extending Maps from Products

If C be a compact, Hausdorff space and M be a metric space and if C × M × Y is normal and Z is the image of Y under a closed map, then C × M × Z is normal. This chapter presents an answer to whether X × Z is normal if X is a closed subset of C × M where C is a compact, Hausdorff space and M is a metric space and if X × Y is normal and Z is the image of Y under a closed map. The question is interesting because the class of spaces that can be embedded as a closed subset of a product of a compact, Hausdorff space with a metric space is the class of paracompact p -spaces, also called paracompact M -spaces, which have been studied extensively.

DECOMPOSITIONS OF E3 WITH COUNTABLY MANY NON-DEGENERATE ELEMENTS

This chapter presents the decompositions of E 3 with countably many nondegenerate elements. It presents an assumption where G is an upper semi-continuous decomposition of E 3 into points and countably many tame cellular polyhedral. It is shown in the chapter that E 3 / G is a homeomorphic to E 3 . This result extends Bings theorem, which states that E 3 / G is homeomorphic to E 3 if G is an upper semi-continuous decomposition of E 3 into points and countably many tame arcs. A subset X of E 3 is a tame polyhedron if and only if there is a homeomorphism h of E 3 to itself so that h ( X) is a rectilinear simplical complex.

A DECOMPOSITION OF S3 WITH A NULL SEQUENCE OF CELLULAR ARCS

Publisher Summary This chapter presents the decomposition of S 3 with a null sequence of cellular arcs. It presents a theorem that states that there exists a null sequence of disjoint cellular arcs { A i } in S 3 such that the decomposition spaces S 3 /{ A i } is topologically different from S 3 . To show that a cellular upper semi-continuous decomposition G of S 3 is not shrinkable, it is frequently convenient to construct a defining sequence for G each stage of which is a collection with the two disk property that is defined as: a finite collection of M of disjoint closed subsets in the interior of a solid torus T has the two disk property if and only if for every pair of disjoint meridonial disks D 0 , D 1 of T , there is an element of M which intersects both D 0 and D 1 . A disk D in T is meridonial if DBd T=Bd D and Bd D does not bound a disk in Bd T .

R.H. Bing’s Human and Mathematical Vitality

Two of Bing’s mathematical colleagues, Armentrout and Burgess, independently told us versions of this memorable evening. Those of us who knew Bing well avoided raising mathematical questions when he was driving.

A diagram oriented proof of Dehn's Lemma

Abstract We prove Dehns Lemma by a procedure which refers directly to the Dehn-diagram of singularities and makes no direct reference to covering spaces. The proof is an algorithmic translation of the usual proof.

APPROXIMATIONS AND DECOMPOSITIONS IN S3

Abstract In this paper we prove that given a closed 2-manifold S, e > 0, and positive integer k, there are k tame e-approximations {hi(S): i = 1, k} of S and a null sequence of disjoint e-disks {Dm: m = 1, ∞} on S such that each Dm intersects at most one hi(S) and that intersection is in hi(Dm). Furthermore, for each i, S ∩ hi(S) ⊊ (∪ Int Dm ∪ (a compact null sequence of simple closed curves on which hi is the identity)). The proof relies heavily on Eatons Two Sided Approximation Theorem [6]. Given a monotone upper semicontinuous decomposition of S3 whose non-degenerate elements miss S, we use the above result to obtain k disjoint approximations of S such that no non-degenerate element intersects two of them. This corollary, together with a Disjoint Disk criterion [8], gives an alternative proof of E. Woodruffs 2-Sphere Decomposition Theorem [10].