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 for which the sectional curvature of and second fundamental form of the boundary are bounded above by one in absolute value. Previously we proved that if has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of exhibits canonical branching behavior of arbitrarily low branching number. In particular, if is *thin* in the sense that its inradius is less than a certain universal constant (known to lie between and ), then 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 when is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When is -dimensional and compact, has complexity in the sense of Matveev, and is a connected sum of copies of the real projective space , copies chosen from the lens spaces , and handles chosen from or , with 3-balls removed, where . Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

**[AB1]**Stephanie B. Alexander and Richard L. Bishop,*Thin Riemannian manifolds with boundary*, Math. Ann.**311**(1998), no. 1, 55–70. MR**1624263**, 10.1007/s002080050176**[AB2]**Stephanie B. Alexander and Richard L. Bishop,*Spines and homology of thin Riemannian manifolds with boundary*, Adv. Math.**155**(2000), no. 1, 23–48. MR**1789847**, 10.1006/aima.2000.1923**[B]**Michael A. Buchner,*The structure of the cut locus in dimension less than or equal to six*, Compositio Math.**37**(1978), no. 1, 103–119. MR**0501100****[G]**M. Gromov,*Synthetic geometry in Riemannian manifolds*, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 415–419. MR**562635****[H]**Wolfgang Haken,*Ein Verfahren zur Aufspaltung einer 3-Mannigfaltigkeit in irreduzible 3-Mannigfaltigkeiten*, Math. Z.**76**(1961), 427–467 (German). MR**0141108****[I]**Hiroshi Ikeda,*Acyclic fake surfaces which are spines of 3-manifolds*, Osaka J. Math.**9**(1972), 391–408. MR**0353316****[K]**H. Kneser,*Geschlossen Flachen in dredimensionalen Mannigfaltigkeiten*, Jahresbericht der Deutschen Mathematiker Vereinigung**38**, 248-260 (1929).**[LF1]**V. N. Lagunov and A. I. Fet,*Extremal problems for surfaces of prescribed topological type (1)*, Sibirsk. Mat. Ž.**4**(1963), 145–176 (Russian). MR**0152947****[LF2]**V. N. Lagunov and A. I. Fet,*Extremal problems for a surface of given topological type. II*, Sibirsk. Mat. Ž.**6**(1965), 1026–1036 (Russian). MR**0198397****[MP]**Bruno Martelli and Carlo Petronio,*Three-manifolds having complexity at most 9*, Experiment. Math.**10**(2001), no. 2, 207–236. MR**1837672****[Ma1]**S. V. Matveev,*Special skeletons of piecewise linear manifolds*, Mat. Sb. (N.S.)**92(134)**(1973), 282–293, 344 (Russian). MR**0343285****[Ma2]**S. V. Matveev,*Complexity theory of three-dimensional manifolds*, Acta Appl. Math.**19**(1990), no. 2, 101–130. MR**1074221****[Ma3]**-, Algorithmic Methods in 3-Manifold Topology, (book to appear).**[Mi]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****[RS]**Colin Patrick Rourke and Brian Joseph Sanderson,*Introduction to piecewise-linear topology*, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR**665919****[ST]**Herbert Seifert and William Threlfall,*Seifert and Threlfall: a textbook of topology*, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With “Topology of 3-dimensional fibered spaces” by Seifert; Translated from the German by Wolfgang Heil. MR**575168****[Th]**William P. Thurston,*Three-dimensional geometry and topology. Vol. 1*, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR**1435975****[W]**Alan D. Weinstein,*The cut locus and conjugate locus of a riemannian manifold*, Ann. of Math. (2)**87**(1968), 29–41. MR**0221434**

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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:
https://doi.org/10.1090/S0002-9947-03-03163-5

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