Path concordances as detectors of codimension-one manifold factors

Abstract

We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.Comment: This is the version published by Geometry & Topology Monographs on 22 April 200

Detecting codimension one manifold factors with topographical techniques

“…In particular, equate (f i , τ i ) as a topographical map pair with F i = f i × τ i as a concordance. The fact that (d) and (e) are equivalent was the main result established in [4]. The fact that (c) implies (a) is trivial since D c ⊂ D.…”

“…A codimension one manifold factor is a space X such that X × R is a manifold. The famous Cell-like Approximation Theorem of Edwards [1,4,7,8] characterizes the manifolds of dimension n ≥ 5 as precisely the finite-dimensional resolvable generalized manifolds with the disjoint disk property. In the same vein, it has been shown that codimension one manifold factors of dimension n ≥ 4 are precisely the finite-dimensional resolvable generalized manifolds with the disjoint concordances property.…”

Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties

“…Several techniques have by now been developed for detecting codimension one manifold factors of dimension n ≥ 4. In particular, a resolvable generalized manifold X of dimension n ≥ 4 is known to be a codimension one manifold factor in the case it has one of the following general position properties: the disjoint arc-disk property [5], the disjoint homotopies property [11], or the disjoint topographies (or disjoint concordance) property [7,14]. The disjoint arc-disk property, satisfied by manifolds of dimension n ≥ 4, is the most natural first guess as a general position property to detect codimension one manifold factors.…”

“…However, although sufficient, it is not necessary (for examples, see [3,9,11]). On the other hand, the disjoint topographies (or disjoint concordance) property is a necessary and sufficient condition for resolvable spaces of dimension n ≥ 4 to be codimension one manifold factors [7,14]. It is still unknown whether or not the disjoint homotopies property likewise provides such a characterization.…”

Decompositions of $${\mathbb {R}^n, n \ge 4}$$ , into Convex Sets Generate Codimension 1 Manifold Factors

“…For resolvable generalized manifolds, we have the following very useful approximate lifting theorem, which follows from [11, Theorem 17.1 and Corollary 16.12B]: General position properties are very useful in detecting codimension one manifold factors [12,16,17,18,20]. For our results, we shall only need to employ the following:…”