Gerard A. Venema

Embeddings of compacta with shape dimension in the trivial range

In this paper a loop condition is defined which generalizes the cellularity criterion and applies to compacta with nontrivial shape. It is shown that if X, Y c El, n > 5, are compacta which satisfy this loop condition and whose shape classes include a space having dimension in the trivial range with respect to n, then Sh(X) = Sh(Y) is equivalent to El X El Y. An application is given to compacta with the shape of a compact connected abelian topological group.

CE equivalence and shape equivalence of 1-dimensional compacta

Abstract In this paper the relationship between CE equivalence and shape equivalence for locally connected, 1-dimensional compacta is investigated. Two theorems are proved. The first asserts that every path connected planar continuum is CE equivalent either to a bouquet of circles or to the Hawaiian earring. The second asserts that for every locally connected, 1-dimensional continuum X there is a cell-like map of X onto a planar continuum. It follows that CE equivalence and shape equivalence are the same for the class of all locally connected, 1-dimensional compacta. In addition, an example of Ferry is generalized to show that for every n ⩾1 there exists an n -dimensional, LC n −2 continuum Y such that Sh( Y )=Sh( S 1 ) but Y is not CE equivalent to S 1 .

CLASSIFYING POLYGONAL CHAINS OF SIX SEGMENTS

A polygonal chain is the union of a finite number of straight line segments in ℝ3 that are connected end-to-end. Two chains are considered to be equivalent if there is an isotopy of ℝ3 that moves one chain to the other while keeping the segments rigid. Each segment must remain straight during the isotopy and the lengths of the segments may not change, but bending and twisting are allowed at the joints between the segments. Chains may be knotted and stuck in this category even though all chains are topologically trivial. Cantarella and Johnston have classified polygonal chains with five or fewer segments. In this paper we classify polygonal chains of six segments.

On the asphericity of knot complements

Two examples of topological embeddings of S 2 in S 4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is non-trivial. The second example is a non-locally flat embedding whose complement exhibits this property locally. Two theorems are proved. The first answers the question of just when good π 1 implies the vanishing of the higher homotopy groups for knot complements in 54. The second theorem characterizes local flatness for 2-spheres in S 4 in terms of a local π 1 condition

A WEAK FLATTENING CRITERION FOR COMPACTA IN 4-SPACE

This chapter presents a weak flattening criterion for compacta in 4-space. It presents a theorem that states that if X, Y ⊂ E 4 are compacts such that Sd (X) ≤ 1, Sd (Y) ≤ 1 and each has the disk pushing property. Then Sh (X)=Sh(Y) if and only if E 4 –X is homeomorphic with E 4 –Y. If X ⊂ E 4 is compact, then the following are equivalent: X has arbitrarily small PL neighborhoods with 1-spines; and Sd (X) ≤ 1 and X has the disk pushing property.

A manifold that does not contain a compact core

Abstract A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.

A HOMOTOPY EQUIVALENCE THAT IS NOT HOMOTOPIC TO A TOPOLOGICAL EMBEDDING

A b st r a c t . An open subset W of S n , n ‚ 6, and a homotopy equivalence f : S 2 ◊S ni4 ! W are constructed having the property that f is not homotopic to any topological embedding.

Characterization of knot complements in the 4-sphere☆☆☆

Abstract Knot complements in S4 are characterized as follows: A connected open set W ⊂ S4 is homeomorphic to the complement of some locally flat 2-sphere in S4 if and only if H1(W) is infinite cyclic, W has one end, and the fundamental group of that end is infinite cyclic. Applications include a characterization of weakly flat 2-spheres in S4 and a complement theorem for 2-spheres in S4.

Cell-like images and UVm groups

Abstract Let X and A be compact metric spaces. The main problem studied in this paper is that of finding conditions under which a map ƒ:A→X can be lifted to a cell-like space; i.e., conditions are sought under which there exist a cell-like continuum Z and continuous maps g:Z→X and ƒ′:A→Z such that g∘ƒ′=ƒ . A theorem is proved which spells out technical conditions on the embedding of A into X and on the homotopy pro-groups of X under which such a lifting exists. The main corollary asserts the following: If X is UVk+1, dim A⩽k, and ƒ′:A→X is continuous, then there exist a cell-like continuum Z and maps g:Z→X and ƒ′:A→Z such that g∘ƒ′= ƒ . An example is constructed which shows that the hypothesis cannot be weakened to UVk. The theorem and example are both applied to the problem of calculating UVm groups.

Neighborhoods of compacta in 4-manifolds

Abstract In this paper we investigate conditions under which a compact set in a piecewise linear 4-manifold has close neighborhoods with 1-dimensional spines. We prove that such neighborhoods exist in case the compactum satisfies the inessential loops condition and either has fundamental dimension 0 or has the shape of S 1 . Corollaries include two complement theorems and a weak flatness theorem for compacta in S 4 .