Robert J. Daverman

Decompositions into submanifolds that yield generalized manifolds

Abstract The paper sets forth several examples of decompositions of ( n + k )-manifolds ( k >2) such that the associated space B is not a generalized manifold. On the other hand, a fundamental technical result, derived from a spectral sequence argument, reveals that B always has cohomological dimension k ; the main result attests that if each g in G has trivial Cech homology in dimensions 1,2,…, k −1, where n ⩾ k ⩾2, and if B is finite dimensional, then it is a generalized k -manifold. By way of application, when each g ϵ G has the shape of S n , it follows that p : M → B is an approximate fibration.

Manifolds that induce approximate fibrations in the PL category

This paper provides quick recognition of approximate fibrations among certain PL maps by identifying fibrators. A closed connected orientable n-manifold N is called a codimension k PL fibrator if, for all PL maps p: M → B on a PL (n + k)-manifold M such that each p−1b collapses to a copy of N, p is an approximate fibration. As codimension 2 fibrators are fairly well understood, the paper emphasizes codimensions k > 2. Here are some of its key results, all stated for a codimension 2 PL fibrator N. If aspherical, N is a codimension 3 PL fibrator. If its universal cover is closed and (k − 1)-connected, k < 6, then N is a codimension k PL fibrator. If N is 3-dimensional and π1(N) ≠ Z2 ∗ Z2, then N is a codimension 3 fibrator. Again for 3-dimensional N, if π1(N) is normally cohopfian and either N is aspherical or H1(N) is infinite, then N is a codimension 4 PL fibrator. If N is aspherical, π1(N) is normally cohopfian, and π1(N) has no proper normal subgroup isomorphic to π1(N)A where A itself is an Abelian normal subgroup of π1(N), then N is a codimension k PL fibrator for all k. In particular, the final statement holds for all hyperbolic 3-manifolds as well as for those with two other geometric structures.

Hereditarily aspherical compacta and cell-like maps

Abstract This paper introduces a notion of strongly hereditarily aspherical compacta and gives a sufficient condition for an inverse limit to have this property. The main result shows that cell-like maps defined on strongly hereditarily aspherical compact metric spaces cannot raise dimension. It suggests why 2-dimensional examples of this sort are plentiful and then sets forth 3-dimensional and 4-dimensional examples.

Products of cell-like decompositions

Abstract This paper represents a survey concerning cell-like decompositions of manifolds. Primarily it summarizes the status of results and problems describing when the product of E 1 with such a decomposition space is again a manifold, and more generally it discusses conditions under which the product of two such decomposition spaces is also a manifold.

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 .

Decompositions into codimension two submanifolds: The nonorientable case

Abstract If G is an upper semicontinuous decomposition of an (n+2)-manifold into continua having the shape of closed n-manifolds, then the decomposition space is a 2-manifold with boundary.

Connected Sums of 4-Manifolds as Codimension-k Fibrators

The main result provides mild conditions under which a closed, orientable, PL 4-manifold

Decompositions into codimension two spheres and approximate fibrations

N = N_1\,\#\,N_2

Path concordances as detectors of codimension-one manifold factors

with

Weak flatness criteria for codimension 2 spheres in codimension 1 manifolds

\pi_1(N_e)