Spines and topology of thin Riemannian manifolds

Abstract:

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 $L(3,\pm 1)$, and $\ell$ handles chosen from $S^2\times S^1$ or $S^2\tilde \times S^1$, with $\beta$ 3-balls removed, where $p+t+\ell +\beta \ge 2$. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.