En utilisant des méthodes de topologie à grande échelle, on prouve que les classes fondamentales des variétés agrandissables ne s’annulent pas, ni dans l’homologie rationnelle de leurs groupes fondamentaux, ni dans la -théorie des -algèbres réduites correspondantes. Nos résultats ne dépendent pas de la conjecture de Baum-Connes, et confirment de façon indépendante certaines conséquences de cette conjecture.
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 -theory of the corresponding reduced -algebras. Our proofs do not depend on the Baum-Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.
@article{ASENS_2008_4_41_3_473_0,
author = {Hanke, Bernhard and Kotschick, Dieter and Roe, John and Schick, Thomas},
title = {Coarse topology, enlargeability, and essentialness},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
volume = {41},
year = {2008},
pages = {473-495},
doi = {10.24033/asens.2073},
mrnumber = {2482205},
zbl = {1169.53032},
language = {en},
url = {http://dml.mathdoc.fr/item/ASENS_2008_4_41_3_473_0}
}
Hanke, Bernhard; Kotschick, Dieter; Roe, John; Schick, Thomas. Coarse topology, enlargeability, and essentialness. Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008) pp. 473-495. doi : 10.24033/asens.2073. http://gdmltest.u-ga.fr/item/ASENS_2008_4_41_3_473_0/
[1] , 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. | MR 420729 | Zbl 0323.58015
[2] , & , Classifying space for proper actions and -theory of group -algebras, in Proceedings of a special session on -algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math. 167, 1994, 240-291. | Zbl 0830.46061
[3] , -theory for operator algebras, second éd., Mathematical Sciences Research Institute Publications 5, Cambridge University Press, 1998. | MR 1656031 | Zbl 0913.46054
[4] & , Aperiodic tilings, positive scalar curvature and amenability of spaces, J. Amer. Math. Soc. 5 (1992), 907-918. | Zbl 0780.53031
[5] & , Atiyah’s -index theorem, Enseign. Math. 49 (2003), 85-93. | Zbl 1044.58025
[6] , On hypersphericity of manifolds with finite asymptotic dimension, Trans. Amer. Math. Soc. 355 (2003), 155-167. | MR 1928082 | Zbl 1020.53025
[7] , & , Remarks on a conjecture of Gromov and Lawson, in High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, 159-176. | Zbl 1052.57035
[8] & , Volume growth and positive scalar curvature, Geom. Funct. Anal. 10 (2000), 821-828. | Zbl 0979.53033
[9] , Volume and bounded cohomology, Publ. Math. I.H.É.S. 56 (1982), 5-99 (1983). | Numdam | MR 686042 | Zbl 0516.53046
[10] , Large Riemannian manifolds, in Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math. 1201, Springer, 1986, 108-121. | MR 859578 | Zbl 0601.53038
[11] , 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. | MR 1253544 | Zbl 0841.20039
[12] , 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. | MR 1389019 | Zbl 0945.53022
[13] & , Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111 (1980), 209-230. | Zbl 0445.53025
[14] & , Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.É.S. 58 (1983), 83-196. | Numdam | Zbl 0538.53047
[15] & , Enlargeability and index theory, J. Differential Geom. 74 (2006), 293-320. | Zbl 1122.58011
[16] & , Enlargeability and index theory: infinite covers, -Theory 38 (2007), 23-33. | Zbl 1128.58012
[17] , & , -algebras and controlled topology, -Theory 11 (1997), 209-239. | Zbl 0879.19003
[18] & , Analytic -homology, Oxford Mathematical Monographs, Oxford University Press, 2000. | Zbl 0968.46058
[19] , & , Generalized homology theories on compact metric spaces, Michigan Math. J. 24 (1977), 203-224. | Zbl 0384.55001
[20] & , -theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras, Trans. Amer. Math. Soc. 227 (1977), 63-107. | Zbl 0368.46054
[21] , 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. | MR 1388297 | Zbl 0954.55004
[22] , Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics 90, American Math. Soc., 1996. | MR 1399087 | Zbl 0853.58003
[23] , Comparing analytic assembly maps, Q. J. Math. 53 (2002), 241-248. | MR 1909514 | Zbl 1014.46045
[24] , Lectures on coarse geometry, University Lecture Series 31, American Math. Soc., 2003. | MR 2007488 | Zbl 1042.53027
[25] , A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), 1165-1168. | MR 1632971 | Zbl 0976.53052
[26] (éd.), Group actions on manifolds, Contemporary Mathematics 36, Amer. Math. Soc., 1985. | MR 780951 | Zbl 0549.00017
[27] , 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. | MR 1937026 | Zbl 1083.53036
[28] , -theory and -algebras: a friendly approach, Oxford Science Publications, Oxford University Press, 1993. | MR 1222415 | Zbl 0780.46038
[29] , The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), 201-240. | MR 1728880 | Zbl 0956.19004