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Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000{000
S 0002-9947(XX)0000-0
SPINES AND TOPOLOGY OF THIN RIEMANNIAN
MANIFOLDS
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. Consider Riemannian manifolds M for which the sectional curva-
ture 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 suÆciently small
inradius (i.e. all points are suÆciently 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 strati cation 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
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