Collapses, products and LC manifolds

The quantum physicists Durhuus and Jonsson (1995) [9] introduced the class of “locally constructible” (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with one facet removed.