The authors have established a link between Riemannian geometry and the notion in PL-topology of collapse to a simple polyhedral spine. Following are their main results:
Theorem I. Suppose a Riemannian manifold \(M\) with connected boundary \(B\) satisifies \(|K_M|\leq 1\) and \(|K_B |\leq 1\), where \(K_M\) is the sectional curvature of the interior and \(K_B\) is normal curvature of the boundary. If \(M\) and \(B\) are simply connected, then \(M\) has inradius at least 0.108. More generally, if \(\pi_1(B)\) and \(\pi_1(M)\) are isomorphic under the inclusion map, then \(M\) has inradius at least 0.108.
Theorem 2. There exists a sequence of universal constants \(a_2<a_3\dots\) (independent of dimension \(n)\), such that a Riemannian manifold \(M\) with boundary \(B\) has curvature-normalized inradius less than \(a_k\), then the cut locus of \(B\) is a \(k\)-branched simple polyhedron of dimension \(n-1\), and is a spine of \(M\). Here \(a_2\approx 0.075\) and \(a_3\approx 0.108\).
Theorem 3. Suppose a manifold \(M\) with boundary \(B\) has a 3-branched simple polyhedron as spine, and \(H_1(B,\mathbb{Z}) =0\).
(a) If \(B\) is connected, then \(H_1(M,\mathbb{Z})\neq 0\).
(b) If \(M\) is compact, then \(H_1(M,\mathbb{Z})\) is a direct sum of copies of \(\mathbb{Z}\), \(\mathbb{Z}_2\), and \(\mathbb{Z}_3\), and depends only on the number of components of the boundary and a bipartite graph representing the combinatorial structure of the spine.