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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spines and topology of thin Riemannian manifolds


Authors: Stephanie B. Alexander and Richard L. Bishop
Journal: Trans. Amer. Math. Soc. 355 (2003), 4933-4954
MSC (2000): Primary 53C21, 57M50
Published electronically: July 28, 2003
MathSciNet review: 1997598
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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 $L(3,\pm1)$, 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.


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Additional Information

Stephanie B. Alexander
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801
Email: sba@math.uiuc.edu

Richard L. Bishop
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801
Email: bishop@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03163-5
PII: S 0002-9947(03)03163-5
Keywords: Riemannian manifolds with boundary, collapse, $3$-manifolds, curvature bounds, inradius, stratification
Received by editor(s): June 4, 2001
Received by editor(s) in revised form: July 12, 2002
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society