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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
     

The Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups

Author(s): James F. Davis; Kimberly Pearson
Journal: Proc. Amer. Math. Soc. 131 (2003), 3571-3578.
MSC (2000): Primary 53C21; Secondary 19L41, 19L64, 57R15, 55N15, 53C20
Posted: 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:

1.
E. Berkove, D. Juan-Pineda and K. Pearson, The lower algebraic $K$-theory of Fuchsian groups, Comment. Math. Helv. 76 (2001), 339-352. MR 2002c:19001

2.
B. Botvinnik and P.B. Gilkey, The eta invariant and the Gromov-Lawson conjecture for elementary abelian groups of odd order, Topology Appl. 80 (1997), 43-53. MR 99f:58194

3.
B. Botvinnik, P.B. Gilkey and S. Stolz, The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology, J. Differential Geom. 46 (1997), 374-405. MR 98i:58227

4.
P. Baum, A. Connes, N. Higson, Classifying space for proper actions and $K$-theory of group $C\sp *$-algebras, in $C\sp *$-algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math. 167, Amer. Math. Soc., 1994, 240-291. MR 96c:46070

5.
J.F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-theory, $K$-theory 15 (1998), 201-252. MR 99m:55004

6.
J.F. Davis and W. Lück, The $p$-chain spectral sequence, to appear in Math. Ann.

7.
J.F. Davis and W. Lück, Computations of $K$- and $L$-groups of group rings based on isomorphism conjectures, in preparation.

8.
T. tom Dieck, Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer, 1979. MR 82c:57025

9.
A. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 227 (1983), 275-284. MR 84f:57029

10.
M. Gromov and H.B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423-434. MR 81h:53036

11.
I. Hambleton and E.K. Pedersen, Identifying assembly maps in $K$- and $L$-theory, preprint.

12.
N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1-55. MR 50:11332

13.
M. Joachim and T. Schick, Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, in Geometry and Topology: Aarhus (1998), Contemp. Math. 258, Amer. Math. Soc., 2000, 213-226. MR 2002g:53079

14.
G.G. Kasparov, Lorentz groups: $K$-theory of unitary representations and crossed products, (Russian) Dokl. Akad. Nauk SSSR 275 (1984), 541-545. MR 85k:22015

15.
S. Katok, Fuchsian Groups, University of Chicago Press, 1992. MR 93d:20088

16.
S. Kwasik and R. Schultz, Positive scalar curvature and periodic fundamental groups, Comment. Math. Helv. 65 (1990), 271-286. MR 91k:57027

17.
H.B. Lawson and M-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. MR 91g:53001

18.
A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris, Sér. A-B 257 (1963), 7-9. MR 27:6218

19.
P.D. Mitchener, Symmetric $K$-theory spectra of $C^*$-categories, $K$-Theory 24 (2001), 157-201. MR 2002k:19007

20.
A.S. Miscenko and A.T. Fomenko,The index of elliptic operators over ${C}\sp{\ast} $-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 831-859. MR 81i:46075

21.
H. Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882), 1-62.

22.
J. Rosenberg, $C^*$-algebras, positive scalar curvature, and the Novikov conjecture, II, in Geometric Methods in Operator Algebras (Kyoto 1983), H. Araki and E.G. Effros, eds., Pitman Research Notes in Math, no. 123, Longman/Wiley, 1986, 341-374. MR 88f:58140

23.
J. Rosenberg, $C^*$-algebras, positive scalar curvature, and the Novikov conjecture, III, Topology 25 (1986), 319-336. MR 88f:58141

24.
J. Rosenberg and S. Stolz, Manifolds of positive scalar curvature, in Algebraic Topology and its Applications, MSRI Publications 27, Springer 1994, 241-267. MR 95b:55001
25.
J. Rosenberg and S. Stolz, A ``stable" version of the Gromov-Lawson conjecture, Contemporary Mathematics 181 (1995), 405-418. MR 96m:53042

26.
T. Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), 1165-1168. MR 99j:53049

27.
R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183. MR 80k:53064

28.
S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), 511-540. MR 93i:57033

29.
S. Stolz, Positive scalar curvature metrics -- existence and classification questions, Proceedings of the International Congress of Mathematicians (Zürich 1994), Birkhäuser, 1995, 625-636. MR 98h:53063

30.
R.W. Thomason, Beware the phony multiplication on Quillen's ${\mathcal{A}}^{-1}{\mathcal{A}}$, Proc. Amer. Math. Soc. 80 (1980), 569-573. MR 81k:18010

31.
H.C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87-102. MR 32:2483

32.
R. Wood, Banach algebras and Bott periodicity, Topology 4 (1965/1966), 371-389. MR 32:3062


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Additional Information:

James F. Davis
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: jfdavis@indiana.edu

Kimberly Pearson
Affiliation: Department of Mathematics and Computer Science, Valparaiso University, Valparaiso, Indiana 46383
Email: kpearson@valpo.edu

DOI: 10.1090/S0002-9939-03-06905-3
PII: S 0002-9939(03)06905-3
Keywords: Positive scalar curvature, $K$-theory, Fuchsian groups
Received by editor(s): January 16, 2002
Received by editor(s) in revised form: May 17, 2002
Posted: April 24, 2003
Communicated by: Paul Goerss
Copyright of article: Copyright 2003, American Mathematical Society


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