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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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  • [AB1] S. B. Alexander, R. L. Bishop, Thin Riemannian manifolds with boundary, Math. Ann. 311, 55-70 (1998). MR 99e:53037
  • [AB2] -, Spines and homology of thin Riemannian manifolds with boundary, Advances in Math 155, 23-48 (2000). MR 2002a:53039
  • [B] M. A. Buchner, The structure of the cut locus in dimension less than or equal to six, Compositio Math. 37 (1978), 103-119. MR 58:18549
  • [G] M. Gromov, Synthetic geometry in Riemannian manifolds in Proceedings of the International Congress of Mathematicians, Helsinki, 1978 (O. Lehto, ed.), vol. 1, Adacemia Scientarium Fennica, 1980, 415-419. MR 81g:53029
  • [H] W. Haken, Ein Verfahren zur Aufspaltung einer $3$-Mannigfaltigkeit in irreduzible $3$-Mannigfaltigkeiten, Math. Zeitschr. 76 (1961), 427-467. MR 25:4519c
  • [I] H. Ikeda, Acyclic fake surfaces which are spines of $3$-manifolds, Osaka J. Math. 9 (1972), 391-408. MR 50:5800
  • [K] H. Kneser, Geschlossen Flachen in dredimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker Vereinigung 38, 248-260 (1929).
  • [LF1] V. Lagunov, A. Fet, Extremal questions for surfaces of a given topological type, Siberian Math. J. 4 (1963), 145-176. (Russian; English redaction by R. L. Bishop available on request). MR 27:2918
  • [LF2] -, Extremal questions for surfaces of a given topological type, II, Siberian Math. J. 6 (1965), 1026-1036. (Russian; English translation by R. L. Bishop available on request). MR 33:6555
  • [MP] B. Martelli, C. Petronio, Three-manifolds having complexity at most $9$, Experimental Math. 10 (2001), 207-236. MR 2002f:57045
  • [Ma1] S. V. Matveev, Special spines of piecewise linear manifolds Mat. Sb. (N.S.). 92 (134) (1972), 282-293; English translation in Math. USSR Sbornik 21 (1973). MR 49:8027
  • [Ma2] -, Complexity theory of three-dimensional manifolds, Acta Applicandae Math. 19 (1990), 101-130. MR 92e:57029
  • [Ma3] -, Algorithmic Methods in 3-Manifold Topology, (book to appear).
  • [Mi] J.W. Milnor, A unique decomposition theorem for $3$-manifolds, 1-7 (1961). MR 25:5518
  • [RS] C. Rourke, B. Sanderson Introduction to Piecewise-Linear Topology, Springer-Verlag, Berlin, Heidelberg and New York, 1982. MR 83g:57009
  • [ST] H. Seifert, W. Threlfall, Lehrbuch der Topologie, Teubner, Stuttgart, 1934; English translation, A Textbook of Topology , Academic Press, New York, 1980. MR 82b:55001
  • [Th] W. Thurston; Silvio Levy, Ed., Three-Dimensional Geometry and Topology , Volume 1, Princeton Univ. Press, Princeton, NJ, 1997. MR 97m:57016
  • [W] A. D. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Ann. of Math. 87, 29-41 (1968). MR 36:4486

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Additional Information

Stephanie B. Alexander
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801

Richard L. Bishop
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801

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

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