The Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups

Davis, James; Pearson, Kimberly

doi:10.1090/S0002-9939-03-06905-3

The Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups
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by James F. Davis and Kimberly Pearson

Proc. Amer. Math. Soc. 131 (2003), 3571-3578

DOI: https://doi.org/10.1090/S0002-9939-03-06905-3

Published electronically: April 24, 2003

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Abstract:

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature.

References

Ethan Berkove, Daniel Juan-Pineda, and Kimberly Pearson, The lower algebraic $K$-theory of Fuchsian groups, Comment. Math. Helv. 76 (2001), no. 2, 339–352. MR 1839350, DOI 10.1007/PL00000382
Boris Botvinnik and Peter B. Gilkey, The eta invariant and the Gromov-Lawson conjecture for elementary abelian groups of odd order, Topology Appl. 80 (1997), no. 1-2, 43–53. MR 1469465, DOI 10.1016/S0166-8641(97)00003-5
Boris Botvinnik, Peter Gilkey, and Stephan Stolz, The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology, J. Differential Geom. 46 (1997), no. 3, 374–405. MR 1484887
Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras, $C^\ast$-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, DOI 10.1090/conm/167/1292018
James F. Davis and Wolfgang Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-theory, $K$-Theory 15 (1998), no. 3, 201–252. MR 1659969, DOI 10.1023/A:1007784106877
J.F. Davis and W. Lück, The $p$-chain spectral sequence, to appear in Math. Ann.
J.F. Davis and W. Lück, Computations of $K$- and $L$-groups of group rings based on isomorphism conjectures, in preparation.
Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743, DOI 10.1007/BFb0085965
A. D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), no. 1, 275–284. MR 690052, DOI 10.1090/S0002-9947-1983-0690052-0
Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434. MR 577131, DOI 10.2307/1971103
I. Hambleton and E.K. Pedersen, Identifying assembly maps in $K$- and $L$-theory, preprint.
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
Michael Joachim and Thomas Schick, Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 213–226. MR 1778107, DOI 10.1090/conm/258/04066
G. G. Kasparov, Lorentz groups: $K$-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (1984), no. 3, 541–545 (Russian). MR 741223
Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
Sławomir Kwasik and Reinhard Schultz, Positive scalar curvature and periodic fundamental groups, Comment. Math. Helv. 65 (1990), no. 2, 271–286. MR 1057244, DOI 10.1007/BF02566607
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR 156292
Paul D. Mitchener, Symmetric $K$-theory spectra of $C^\star$-categories, $K$-Theory 24 (2001), no. 2, 157–201. MR 1869627, DOI 10.1023/A:1012773111555
A. S. Miščenko and A. T. Fomenko, The index of elliptic operators over $C^{\ast }$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 831–859, 967 (Russian). MR 548506
H. Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882), 1–62.
J. Rosenberg, $C^\ast$-algebras, positive scalar curvature and the Novikov conjecture. II, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 341–374. MR 866507
Jonathan Rosenberg, $C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III, Topology 25 (1986), no. 3, 319–336. MR 842428, DOI 10.1016/0040-9383(86)90047-9
G. E. Carlsson, R. L. Cohen, W. C. Hsiang, and J. D. S. Jones (eds.), Algebraic topology and its applications, Mathematical Sciences Research Institute Publications, vol. 27, Springer-Verlag, New York, 1994. MR 1268184, DOI 10.1007/978-1-4613-9526-3
Jonathan Rosenberg and Stephan Stolz, A “stable” version of the Gromov-Lawson conjecture, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 405–418. MR 1321004, DOI 10.1090/conm/181/02046
Thomas Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), no. 6, 1165–1168. MR 1632971, DOI 10.1016/S0040-9383(97)00082-7
R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700, DOI 10.1007/BF01647970
Stephan Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511–540. MR 1189863, DOI 10.2307/2946598
Stephan Stolz, Positive scalar curvature metrics—existence and classification questions, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 625–636. MR 1403963
R. W. Thomason, Beware the phony multiplication on Quillen’s ${\scr A}^{-1}{\scr A}$, Proc. Amer. Math. Soc. 80 (1980), no. 4, 569–573. MR 587929, DOI 10.1090/S0002-9939-1980-0587929-6
H. C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87–102. MR 185013, DOI 10.1007/BF01110157
R. Wood, Banach algebras and Bott periodicity, Topology 4 (1965/66), 371–389. MR 185598, DOI 10.1016/0040-9383(66)90035-8