A characterization of manifold decompositions satisfying the disjoint triples property

Abstract:

A metric space $X$ satisfies the Disjoint Triples Property $({\text {D}}{{\text {D}}_{\text {3}}})$ if maps ${f_1}$, ${f_2}$ and ${f_3}$ from ${B^2}$ into $X$ are approximable by maps ${\tilde f_1}$, ${\tilde f_2}$ and ${\tilde f_3}$ with $\cap _{i = 1}^3{\tilde f_i}({B^2}) = \emptyset$. Those CE decompositions of manifolds satisfying ${\text {D}}{{\text {D}}_3}$ and yielding finite-dimensional nonmanifold decomposition spaces are shown to be precisely those intrinsically $0$-dimensional decompositions which yield nonshrinkable null cellular decompositions under amalgamation. This characterization results in another proof of the fact that ${E^n}/G \times {E^1}$ is secretly $0$-dimensional where $G$ is a CE usc decomposition of ${E^n}$, $n \geqslant 4$, with ${E^n}/G$ finite dimensional.