Spines and topology of thin Riemannian manifolds

Consider Riemannian manifolds M for which the sectional curvature of M and second fundamental form of the boundary B are bounded above by one in absolute value. Previously we proved that if M has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of B exhibits canonical branching behavior of arbitrarily low branching number. In particular, if M is thin in the sense that its inradius is less than a certain universal constant ( known to lie between : 108 and : 203), then M collapses to a triply branched simple polyhedral spine. We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of M when B is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When M is 3-dimensional and compact, M has complexity 0 in the sense of Matveev, and is a connected sum of p copies of the real projective space P-3, t copies chosen from the lens spaces L(3; +/- 1), and l handles chosen from S-2 x S-1 or S-2 (x) over tilde S-1, with beta 3-balls removed, where p + t + l + beta greater than or equal to 2. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.