Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(e) ISSN 1061-0022(p)

     
     

Topological and geometric properties of graph-manifolds

Author(s): S. Buyalo; P. Svetlov
Translated by: the authors
Original publication: Algebra i Analiz, tom 16 (2004), vypusk 2.
Journal: St. Petersburg Math. J. 16 (2005), 297-340.
MSC (2000): Primary 57N10
Posted: March 9, 2005
MathSciNet review: 2068341
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: This is a unified exposition of results (obtained by different authors) on the existence of $\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds. The basis for unification is provided by the notion of compatible cohomology classes and by a certain difference equation on the graph of a graph-manifold (the BKN-equation). Criteria for seven different properties of graph-manifolds are given at three levels: at the level of compatible cohomology classes; at the level of solutions of the BKN-equation; and in terms of spectral properties of operator invariants of a graph-manifold.

References:

[B1]
S. V. Buyalo, Collapsing manifolds of nonpositive curvature. I, II, Algebra i Analiz 1 (1989), no. 5, 74-94; no. 6, 70-97; English transl., Leningrad Math. J. 1 (1990), 1135-1156; 1371-1399. MR 1036838 (91i:53055a); MR 1047962 (91i:53055b)

[B2]
-, Metrics of nonpositive curvature on graph-manifolds and electromagnetic fields on graphs, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), 28-72; English transl., J. Math. Sci. (N. Y.) 119 (2004), no. 2, 141-164. MR 1879256 (2003a:58037)

[BK1]
S. V. Buyalo and V. L. Kobel'skii, Geometrization of graph-manifolds. I. Conformal geometrization, Algebra i Analiz 7 (1995), no. 2, 1-45; English transl., St. Petersburg Math. J. 7 (1996), no. 2, 185-216. MR 1347511 (97k:57016)

[BK2]
-, Geometrization of graph-manifolds. II. Isometric geometrization, Algebra i Analiz 7 (1995), no. 3, 96-117; English transl., St. Petersburg Math. J. 7 (1996), no. 3, 387-404. MR 1353491 (97k:57017)

[BK3]
-, Geometrization of infinite graph-manifolds, Algebra i Analiz 8 (1996), no. 3, 56-77; English transl., St. Petersburg Math. J. 8 (1997), no. 3, 413-427. MR 1402288 (97g:53042)

[BK4]
-, Generalized graphmanifolds of nonpositive curvature, Algebra i Analiz 11 (1999), no. 2, 64-87; English transl., St. Petersburg Math. J. 11 (2000), no. 2, 251-268. MR 1702579 (2001f:53062)

[CE]
J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland Math. Library, vol. 9, North-Holland Publishing Co., Amsterdam-Oxford, 1975. MR 0458335 (56:16538)

[E]
P. Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), 435-476. MR 0577132 (82m:53040)

[GW]
D. Gromoll and J. A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545-552. MR 0281122 (43:6841)

[JS]
W. H. Jaco and P. B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220. MR 0539411 (81c:57010)

[JLLP]
A. Jacques, C. Lenormand, A. Lentin, and J.-F. Perrot, Un résultat extrémal en théorie des permutations, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A446-A448. MR 0229711 (37:5285)

[J]
K. Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Math., vol. 761, Springer-Verlag, Berlin-New York, 1979. MR 0551744 (82c:57005)

[KL]
M. Kapovich and B. Leeb, Actions of discrete groups on nonpositively curved spaces, Math. Ann. 306 (1996), 341-352. MR 1411351 (98j:57022)

[L]
B. Leeb, $3$-manifolds with (out) metrics of nonpositive curvature, Invent. Math. 122 (1995), 277-289. MR 1358977 (97g:57015)

[LW]
J. Luecke and Y. Wu, Relative Euler number and finite covers of graph manifolds, Geometric Topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2.1, Amer. Math. Soc., Providence, RI, 1997, pp. 80-103. MR 1470722 (98h:57036)

[LY]
H. B. Lawson and S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geom. 7 (1972), 211-228. MR 0334083 (48:12402)

[N1]
W. D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299-344. MR 0632532 (84a:32015)

[N2]
-, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997), 355-378. MR 1415593 (98d:57007)

[N3]
-, Immersed and virtually embedded $\pi_1$-injective surfaces in graph manifolds, Algebr. Geom. Topol. 1 (2001), 411-426 (electronic). MR 1852764 (2002f:57047)

[RW]
J. H. Rubinstein and S. Wang, $\pi_1$-injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998), 499-515. MR 1639876 (99h:57039)

[Sv1]
P. V. Svetlov, Graph-manifolds of nonpositive curvature are virtually fibered over the circle, Algebra i Analiz 14 (2002), no. 5, 188-201; English transl., St. Petersburg Math. J. 14 (2003), no. 5, 847-856. MR 1970339 (2004f:53041)

[Sv2]
-, Homological and geometric properties of graph-manifolds, Candidate Dissertation, St. Petersburg, 2002. (Russian)

[Sc]
P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487. MR 0705527 (84m:57009)

[Sch]
V. Schroeder, Rigidity of nonpositively curved graphmanifolds, Math. Ann. 274 (1986), 19-26. MR 0834102 (87h:53054)

[T]
W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357-381. MR 0648524 (83h:57019)

[W]
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II, Invent. Math. 4 (1967), 87-117. MR 0235576 (38:3880)

[WSY]
S. Wang and F. Yu, Graph manifolds with non-empty boundary are covered by surface bundles, Math. Proc. Cambridge Philos. Soc. 122 (1997), 447-455. MR 1466648 (98k:57039)

[WYY]
Y. Wang and F. Yu, When closed graph manifolds are finitely covered by surface bundles over $S^1$, Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 1, 11-20. MR 1701130 (2000d:57032)

Similar Articles:

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 57N10

Retrieve articles in all Journals with MSC (2000): 57N10

Additional Information:

S. Buyalo
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: sbuyalo@pdmi.ras.ru

P. Svetlov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: svetlov@pdmi.ras.ru

DOI: 10.1090/S1061-0022-05-00852-6
PII: S 1061-0022(05)00852-6
Keywords: Immersed and embedded surfaces, compatible cohomology classes, BNK-equation
Received by editor(s): 2/SEP/2002
Posted: March 9, 2005
Additional Notes: Supported by CRDF (grant no. RM1-2381-ST-02) and by RFBR (grant no. 02-01-00090).
Copyright of article: Copyright 2005, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia