J. L. Bryant

Topology of Homology Manifolds

ANR homology n-manifolds are nite-dimensional absolute neighborhood retracts X with the property that for every x 2 X, Hi(X;X fxg) is 0 for i 6= n and Z for i = n. Topological manifolds are natural examples of such spaces. To obtain nonmanifold examples, we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components. The resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group. It is a consequence of the double suspension theorem of Cannon and Edwards that, as in the examples above, the singularities of polyhedral ANR homology manifolds are isolated. There are, however, many examples of ANR homology manifolds which have no manifold points whatever. See [12] for a good exposition of the relevant theory. The purpose of this paper is to begin a surgical classi cation of ANR homology manifolds, sometimes referred to in the sequel, simply as homology manifolds. One way to approach this circle of ideas is via the problem of characterizing topological manifolds among ANR homology manifolds. In Cannons work on the double suspension problem [6], it became clear that in dimensions greater than 4, the right transversality hypothesis is the following (weak) form of general position.

Topology of homology manifolds

We construct examples of nonresolvable generalized

A model for the asymptotic behavior of loop entanglement in a constrained liquid region

n

Homogeneous ENR's

-manifolds,

Piecewise Linear Topology

n\geq 6

Transversality in generalized manifolds

, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed

Embeddings with mapping cylinder neighborhoods

n

Manifold factors that are manifold quotients

-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology.

Embeddings in generalized manifolds

A constrained liquid region in a semicrystalline polymer has been modeled by random walk in the slab defined by two parallel planes a distance M units apart in space. Loops (walks beginning and ending on the same plane) play the role of strands of polymer that exit and reenter the same crystal while ties (walks that go from plane to opposite plane) play the role of strands that connect two crystals. A simulation has been designed to investigate the extent to which entanglement of loops may contribute to material properties. Results of the simulation indicate that several measures of entanglement (calculated from the matrix of linking numbers of a left loop with a complete distribution of right loops) approach positive constant values or increase without bound as M increases (while the effect of ties is known to be inversely proportional to M). A model of the simulation, in which random loops are replaced by loops with predictable geometry, yields asymptotic estimates of one such entanglement measure. The ...

Splitting manifolds as M × R where M has a k-fold end structure

Department of Mathematics, The Florida State University. Tallahassee, FL32306-3027, USA Received 15 August 1986 Revised 1 December 1986 We show that a homogeneous euclidean neighborhood retract (ENR) X is generalized manifold provided H,(X, X -x) is finitely generated for some (and, hence, every) x E X. As a consequence we obtain a partial answer of a question of Borsuk’s as to whether there exist homogeneous AR-spaces of finite dimension > 1. AMS (MOS) Subj. Class.: 54F40, 57P99 euclidean neighborhood retract