Coarse topology, enlargeability, and essentialness

Bernhard Hanke; Dieter Kotschick; John Roe; Thomas Schick

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 3, page 473-495
  • ISSN: 0012-9593

Abstract

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Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K -theory of the corresponding reduced C * -algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.

How to cite

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Hanke, Bernhard, et al. "Coarse topology, enlargeability, and essentialness." Annales scientifiques de l'École Normale Supérieure 41.3 (2008): 473-495. <http://eudml.org/doc/272174>.

@article{Hanke2008,
abstract = {Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the $K$-theory of the corresponding reduced $C^*$-algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.},
author = {Hanke, Bernhard, Kotschick, Dieter, Roe, John, Schick, Thomas},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {enlargeable manifold; universally enlargeable manifold; essential manifold; macroscopically large manifold; coarse homology; Baum-Connes assembly map; -theory enlargeability; scalar curvature},
language = {eng},
number = {3},
pages = {473-495},
publisher = {Société mathématique de France},
title = {Coarse topology, enlargeability, and essentialness},
url = {http://eudml.org/doc/272174},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Hanke, Bernhard
AU - Kotschick, Dieter
AU - Roe, John
AU - Schick, Thomas
TI - Coarse topology, enlargeability, and essentialness
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 3
SP - 473
EP - 495
AB - Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the $K$-theory of the corresponding reduced $C^*$-algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.
LA - eng
KW - enlargeable manifold; universally enlargeable manifold; essential manifold; macroscopically large manifold; coarse homology; Baum-Connes assembly map; -theory enlargeability; scalar curvature
UR - http://eudml.org/doc/272174
ER -

References

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  1. [1] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, in Colloque “Analyse et Topologie” en l’honneur de Henri Cartan (Orsay, 1974), Astérisque 32–33, 1976, 43–72. Zbl0323.58015MR420729
  2. [2] P. Baum, A. Connes & N. Higson, Classifying space for proper actions and K -theory of group C * -algebras, in Proceedings of a special session on C * -algebras:1943–1993 (San Antonio, TX, 1993), Contemp. Math. 167, 1994, 240–291. Zbl0830.46061
  3. [3] B. Blackadar, K -theory for operator algebras, second éd., Mathematical Sciences Research Institute Publications 5, Cambridge University Press, 1998. Zbl0913.46054MR1656031
  4. [4] J. Block & S. Weinberger, Aperiodic tilings, positive scalar curvature and amenability of spaces, J. Amer. Math. Soc.5 (1992), 907–918. Zbl0780.53031
  5. [5] I. Chatterji & G. Mislin, Atiyah’s L 2 -index theorem, Enseign. Math.49 (2003), 85–93. Zbl1044.58025
  6. [6] A. N. Dranishnikov, On hypersphericity of manifolds with finite asymptotic dimension, Trans. Amer. Math. Soc.355 (2003), 155–167. Zbl1020.53025MR1928082
  7. [7] W. Dwyer, T. Schick & S. Stolz, Remarks on a conjecture of Gromov and Lawson, in High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, 159–176. Zbl1052.57035
  8. [8] G. Gong & G. Yu, Volume growth and positive scalar curvature, Geom. Funct. Anal.10 (2000), 821–828. Zbl0979.53033
  9. [9] M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.É.S. 56 (1982), 5–99 (1983). Zbl0516.53046MR686042
  10. [10] M. Gromov, Large Riemannian manifolds, in Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math. 1201, Springer, 1986, 108–121. Zbl0601.53038MR859578
  11. [11] M. Gromov, Asymptotic invariants of infinite groups, in Geometric group theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, 1993, 1–295. Zbl0841.20039MR1253544
  12. [12] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, in Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math. 132, Birkhäuser, 1996, 1–213. Zbl0945.53022MR1389019
  13. [13] M. Gromov & H. B. J. Lawson, Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111 (1980), 209–230. Zbl0445.53025
  14. [14] M. Gromov & H. B. J. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.É.S. 58 (1983), 83–196. Zbl0538.53047
  15. [15] B. Hanke & T. Schick, Enlargeability and index theory, J. Differential Geom.74 (2006), 293–320. Zbl1122.58011
  16. [16] B. Hanke & T. Schick, Enlargeability and index theory: infinite covers, K -Theory 38 (2007), 23–33. Zbl1128.58012
  17. [17] N. Higson, E. K. Pedersen & J. Roe, C * -algebras and controlled topology, K -Theory 11 (1997), 209–239. Zbl0879.19003
  18. [18] N. Higson & J. Roe, Analytic K -homology, Oxford Mathematical Monographs, Oxford University Press, 2000. Zbl0968.46058
  19. [19] D. S. Kahn, J. Kaminker & C. Schochet, Generalized homology theories on compact metric spaces, Michigan Math. J.24 (1977), 203–224. Zbl0384.55001
  20. [20] J. Kaminker & C. Schochet, K -theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras, Trans. Amer. Math. Soc.227 (1977), 63–107. Zbl0368.46054
  21. [21] J. Milnor, On the Steenrod homology theory, in Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 226, Cambridge Univ. Press, 1995, 79–96. Zbl0954.55004MR1388297
  22. [22] J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics 90, American Math. Soc., 1996. Zbl0853.58003MR1399087
  23. [23] J. Roe, Comparing analytic assembly maps, Q. J. Math.53 (2002), 241–248. Zbl1014.46045MR1909514
  24. [24] J. Roe, Lectures on coarse geometry, University Lecture Series 31, American Math. Soc., 2003. Zbl1042.53027MR2007488
  25. [25] T. Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology37 (1998), 1165–1168. Zbl0976.53052MR1632971
  26. [26] R. Schultz (éd.), Group actions on manifolds, Contemporary Mathematics 36, Amer. Math. Soc., 1985. Zbl0549.00017MR780951
  27. [27] S. Stolz, Manifolds of positive scalar curvature, in Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, 661–709. Zbl1083.53036MR1937026
  28. [28] N. E. Wegge-Olsen, K -theory and C * -algebras: a friendly approach, Oxford Science Publications, Oxford University Press, 1993. Zbl0780.46038MR1222415
  29. [29] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math.139 (2000), 201–240. Zbl0956.19004MR1728880

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