Homotopy invariance of higher signatures and 3 -manifold groups

Michel Matthey; Hervé Oyono-Oyono; Wolfgang Pitsch

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 1, page 1-25
  • ISSN: 0037-9484

Abstract

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For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3 -manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3 -manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed.

How to cite

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Matthey, Michel, Oyono-Oyono, Hervé, and Pitsch, Wolfgang. "Homotopy invariance of higher signatures and $3$-manifold groups." Bulletin de la Société Mathématique de France 136.1 (2008): 1-25. <http://eudml.org/doc/272411>.

@article{Matthey2008,
abstract = {For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable $3$-manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable $3$-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed.},
author = {Matthey, Michel, Oyono-Oyono, Hervé, Pitsch, Wolfgang},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Baum-Connes conjecture; JSJ decomposition; Thurston geometrization conjecture},
language = {eng},
number = {1},
pages = {1-25},
publisher = {Société mathématique de France},
title = {Homotopy invariance of higher signatures and $3$-manifold groups},
url = {http://eudml.org/doc/272411},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Matthey, Michel
AU - Oyono-Oyono, Hervé
AU - Pitsch, Wolfgang
TI - Homotopy invariance of higher signatures and $3$-manifold groups
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 1
SP - 1
EP - 25
AB - For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable $3$-manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable $3$-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed.
LA - eng
KW - Baum-Connes conjecture; JSJ decomposition; Thurston geometrization conjecture
UR - http://eudml.org/doc/272411
ER -

References

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  1. [1] C. Anantharaman-Delaroche – « Amenability and exactness for dynamical systems and their C * -algebras », Trans. Amer. Math. Soc.354 (2002), p. 4153–4178. Zbl1035.46039MR1926869
  2. [2] M. T. Anderson – « Scalar curvature and geometrization conjectures for 3 -manifolds », Math. Sci. Res. Inst. Publ.30 (1997), p. 49–82. Zbl0890.57026MR1452867
  3. [3] M. F. Atiyah – Elliptic operators, discrete groups and von Neumann algebras, Astérisque, vol. 32-33, Soc. Math. France, 1976. Zbl0323.58015MR420729
  4. [4] M. F. Atiyah & I. M. Singer – « The index of elliptic operators. I », Ann. of Math. (2) 87 (1968), p. 484–530. Zbl0164.24001MR236950
  5. [5] P. Baum & A. Connes – « Geometric K -theory for Lie groups and foliations », Enseign. Math. (2) 46 (2000), p. 3–42. Zbl0985.46042MR1769535
  6. [6] P. Baum, A. Connes & N. Higson – « Classifying space for proper actions and K -theory of group C * -algebras », Contemp. Math.167 (1994), p. 240–291. Zbl0830.46061MR1292018
  7. [7] P. Baum, S. Millington & R. Plymen – « Local-global principle for the Baum-Connes conjecture with coefficients », K -Theory 28 (2003), p. 1–18. Zbl1034.46073MR1988816
  8. [8] C. Béguin, H. Bettaieb & A. Valette – « K -theory for C * -algebras of one-relator groups », K -Theory 16 (1999), p. 277–298. Zbl0932.46063MR1681180
  9. [9] M. E. B. Bekka, P.-A. Cherix & A. Valette – « Proper affine isometric actions of amenable groups », in Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, 1995, p. 1–4. Zbl0959.43001MR1388307
  10. [10] F. Bonahon – « Geometric structures on 3 -manifolds », in Handbook of geometric topology (R. Daverman et al., éds.), Elsevier, 2002, p. 93–164. Zbl0997.57032MR1886669
  11. [11] P.-A. Chérix, M. Cowling, P. Jolissaint, P. Julg & A. Valette – Groups with the Haagerup property, Gromov’s a-T-menability, Progress in Math., vol. 197, Birhäuser, 2001. Zbl1030.43002MR1852148
  12. [12] J. Cuntz – « K -theoretic amenability for discrete groups », J. reine angew. Math. 344 (1983), p. 180–195. Zbl0511.46066MR716254
  13. [13] M. Dehn – « Über die Topologie des dreidimensionalen Raumes », Math. Ann.69 (1910), p. 137–168. Zbl41.0543.01MR1511580JFM41.0543.01
  14. [14] A. Dold – « Lectures on algebraic topology », in Classics in Mathematics, Springer, 1972, Die Grundlehren der mathematischen Wissenschaften, Band 200, p. 377. Zbl0434.55001MR415602
  15. [15] K. J. Dykema – « Exactness of reduced amalgamated free product C * -algebras », Forum Math.16 (2004), p. 161–180. Zbl1050.46040MR2039095
  16. [16] D. B. A. Epstein – « Periodic flows on 3 -manifolds », Ann. of Math.95 (1972), p. 66–82. Zbl0231.58009MR288785
  17. [17] K. Fujiwara – « 3-manifold groups and property T of Kazhdan », Proc. Japan Acad. Ser. A Math. Sci.75 (1999), p. 103–104. Zbl0957.57004MR1729853
  18. [18] M. Gromov – « Geometric reflections on the Novikov conjecture », in Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, 1995, p. 164–173. Zbl0966.53028MR1388301
  19. [19] J. Hempel – 3 -Manifolds, Annals of Math Studies, vol. 86, Princeton University Press, 1976, Ann. of Math. Studies, No. 86. Zbl0345.57001MR415619
  20. [20] N. Higson – « Bivariant K -theory and the Novikov conjecture », Geom. Funct. Anal.10 (2000), p. 563–581. Zbl0962.46052MR1779613
  21. [21] N. Higson & G. Kasparov – « Operator K -theory for groups which act properly and isometrically on Hilbert space », Electron. Res. Announc. Amer. Math. Soc.3 (1997), p. 131–142. Zbl0888.46046MR1487204
  22. [22] —, « E -theory and K K -theory for groups which act properly and isometrically on Hilbert space », Invent. Math.144 (2001), p. 23–74. Zbl0988.19003MR1821144
  23. [23] F. Hirzebruch – Topological methods in algebraic geometry, Classics in Mathematics, Springer, 1995. Zbl0843.14009MR1335917
  24. [24] W. H. Jaco – « Finitely presented subgroups of three-manifold groups », Invent. Math.13 (1971), p. 335–346. Zbl0232.57003MR300279
  25. [25] W. H. Jaco & P. B. Shalen – Seifert fibered spaces in 3 -manifolds, Mem. Amer. Math. Soc., vol. 21, 1979. Zbl0415.57005MR539411
  26. [26] M. Jankins & W. D. Neumann – « Lectures on Seifert manifolds », in Brandeis Lecture Notes, Brandeis Lecture Notes, vol. 2, Brandeis University, 1983. MR741334
  27. [27] K. Johannson – « Homotopy equivalence of 3 -manifolds with boundaries », in Springer Lecture Notes in Math., vol. 761, 1979. Zbl0412.57007MR551744
  28. [28] F. E. A. Johnson & J. P. Walton – « Parallelizable manifolds and the fundamental group », Mathematika 47 (2000), p. 165–172 (2002). Zbl1016.57004MR1924495
  29. [29] P. Julg – « K -théorie équivariante et produits croisés », C. R. Acad. Sci. Paris Sér. I Math.292 (1981), p. 629–632. Zbl0461.46044MR625361
  30. [30] P. Julg & G. Kasparov – « Operator K -theory for the group SU ( n , 1 ) », J. reine angew. Math. 463 (1995), p. 99–152. Zbl0819.19004MR1332908
  31. [31] M. Kapovich – Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser, 2001. Zbl0958.57001MR1792613
  32. [32] G. Kasparov – « Lorentz groups: K -theory of unitary representations and crossed products », Dokl. Akad. Nauk SSSR275 (1984), p. 541–545. Zbl0584.22004MR741223
  33. [33] —, « Equivariant K K -theory and the Novikov conjecture », Invent. Math.91 (1988), p. 147–201. Zbl0647.46053MR918241
  34. [34] E. Kirchberg & S. Wassermann – « Permanence properties of C * -exact groups », Doc. Math.4 (1999), p. 513–558. Zbl0958.46036MR1725812
  35. [35] H. Kneser – « Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten », Jahresbericht der Deut. Math. Verein.38 (1929), p. 248–260. JFM55.0311.03
  36. [36] V. Lafforgue – « K -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes », Invent. Math.149 (2002), p. 1–95. Zbl1084.19003MR1914617
  37. [37] H. B. Lawson, Jr. & M.-L. Michelsohn – Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, 1989. Zbl0688.57001MR1031992
  38. [38] A. Markov – « The insolubility of the problem of homeomorphy », Dokl. Akad. Nauk SSSR121 (1958), p. 218–220. Zbl0092.00702MR97793
  39. [39] V. Mathai – « The Novikov conjecture for low degree cohomology classes », Geom. Dedicata99 (2003), p. 1–15. Zbl1029.19006MR1998926
  40. [40] J. W. Milnor – « A unique factorization theorem for 3 -manifolds », Amer. J. Math.84 (1962), p. 1–7. Zbl0108.36501MR142125
  41. [41] G. Mislin & A. Valette – « Proper group actions and the Baum-Connes conjecture », in Advanced Course in Mathematics, CRM Barcelona, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, 2003. Zbl1028.46001MR2027168
  42. [42] A. S. Miščenko – « Infinite-dimensional representations of discrete groups, and higher signatures ( Russian ) », Izv. Akad. Nauk SSSR Ser. Mat.38 (1974), p. 81–106. Zbl0299.57010MR362407
  43. [43] H. Oyono-Oyono – « La conjecture de Baum-Connes pour les groupes agissant sur les arbres », C. R. Acad. Sci. Paris Sér. I Math.326 (1998), p. 799–804. Zbl0918.46062MR1648575
  44. [44] —, « Baum-Connes conjecture and extensions », J. reine angew. Math. 532 (2001), p. 133–149. Zbl0973.46064MR1817505
  45. [45] —, « Baum-Connes conjecture and group actions on trees », K -Theory 24 (2001), p. 115–134. Zbl1008.19001MR1869625
  46. [46] N. Ozawa – « Amenable actions and exactness for discrete groups », C. R. Acad. Sci. Paris Sér. I Math.330 (2000), p. 691–695. Zbl0953.43001MR1763912
  47. [47] M. Puschnigg – « The Kadison-Kaplansky conjecture for word-hyperbolic groups », Invent. Math.149 (2002), p. 153–194. Zbl1019.22002MR1914620
  48. [48] P. Scott – « The geometries of 3 -manifolds », Bull. London Math. Soc.15 (1983), p. 401–487. Zbl0561.57001MR705527
  49. [49] J-P. Serre – Arbres, amalgames, SL 2 , Astérisque, vol. 46, 1983. Zbl0369.20013
  50. [50] W. P. Thurston – Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, 1997, Edited by Silvio Levy. Zbl0873.57001MR1435975
  51. [51] J. L. Tu – « The Baum-Connes conjecture and discrete group actions on trees », K -theory 17 (1999), p. 303–318. Zbl0939.19002MR1706113
  52. [52] A. Valette – « Introduction to the Baum-Connes conjecture », in Lectures in Mathematics, ETH, Zürich (1999), Birkhäuser, 2002. Zbl1136.58013MR1907596

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