Collapses, products and LC manifolds

TL;DR

This paper explores the properties of locally constructible manifolds in higher dimensions, demonstrating that unlike in 2 and 3 dimensions, some LC d-manifolds are not spheres, by analyzing collapses of product manifolds.

Contribution

It extends the understanding of LC manifolds to higher dimensions, showing they are not all spheres, unlike the 2- and 3-dimensional cases.

Findings

LC d-manifolds for d>3 can be non-spherical

Collapse techniques for product manifolds are effective in higher dimensions

All LC 2- and 3-manifolds are spheres, but this does not hold for higher dimensions

Abstract

Durhuus and Jonsson (1995) 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 exactly one facet removed.