Topology of combinatorial differential manifolds (Q1292670)

Combinatorial differential manifolds (CD manifolds) were introduced by \textit{R. MacPherson} [Combinatorial differential manifolds, in `Topological methods in modern mathematics', Proceedings of a symposium in honor of John Milnor's sixtieth birthday, 203-221 (1993; Zbl 0812.57019)]. Such a combinatorial object generalizes the notion of a smooth triangulation of a differential manifold. The tangent space is replaced by ``oriented matroids'' which roughly speaking are like a finite collection of vectors in some Euclidean space. It is not known if CD manifolds are topological manifolds. The author proves that under a certain condition (the oriented matroids are ``Euclidean'') a CD manifold is a topological manifold. Other results are obtained for ``matroid polytopes''.