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

Abstract

We show that if G is an upper semicontinuous decomposition of R n , n ≥ 4, into convex sets, then the quotient space R n /G is a codimension one manifold factor. In particular, we show that R n /G has the disjoint arc-disk property.Mathematics Subject Classification (2010). Primary 57N15, 57N75; Secondary 57P99, 53C70.

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

“…(7) Do (n − 2)-dimensional decompositions arising from a defining sequence of thickened (n − 2)-manifolds have the P-DADP? (8) Recently, we have proved in [16] that decomposition spaces resulting from decompositions of R n , n ≥ 4, into convex sets are topologically equivalent to R n . In fact, such spaces possess the DADP property.…”

“…16. A space X has the closed 0-stitched disks property if it has the 0-stitched disks property where "F σ -sets A and B" is replaced with "closed sets A and B" in Definition 3.14.…”

Survey on the Generalized R. L. Moore Problem

“…Resolvable generalized manifolds that possess the disjoint arc-disk property include spaces that arise from (n − 3)-dimensional or closed (n − 2)-dimensional decompositions [18]. Examples include decompositions of n-manifolds arising from a classical defining sequence [23, Proposition 9.1] and decompositions of R n≥4 into convex sets [39]. However, not all manifolds known to be codimension one manifold factors have the disjoint arc-disk property.…”