Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

On the Farrell-Jones conjecture for higher algebraic $K$-theory

Author(s): Arthur Bartels; Holger Reich
Journal: J. Amer. Math. Soc. 18 (2005), 501-545.
MSC (2000): Primary 19D50; Secondary 53C12
Posted: March 30, 2005
MathSciNet review: 2138135
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We prove the Farrell-Jones Conjecture for the algebraic $K$-theory of a group ring $R \Gamma$ in the case where the group $\Gamma$ is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring $R$ is an arbitrary associative ring with unit and the result applies to all dimensions.

References:

[AS85]
M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429-461. MR 0794369 (87a:58151)

[Bas68]
H. Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491 (40:2736)

[BBE85]
W. Ballmann, M. Brin, and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2) 122 (1985), no. 1, 171-203. MR 0799256 (87c:58092a)

[BCH94]
P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and $K$-theory of group $C\sp \ast$-algebras, $C\sp \ast$-algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291. MR 1292018 (96c:46070)

[BFJR03]
A. Bartels, F. T. Farrell, L. E. Jones, and H. Reich, A foliated squeezing theorem for geometric modules, High-dimensional manifold topology, World Sci. Publishing, River Edge, NJ, 2003, pp. 1-21. MR 2048714 (2004m:57046)

[BFJR04]
-, On the isomorphism conjecture in algebraic $K$-theory, Topology 43 (2004), no. 1, 157-213. MR 2030590 (2004m:19004)

[BHS64]
H. Bass, A. Heller, and R. G. Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. (1964), no. 22, 61-79. MR 0174605 (30:4806)

[BO69]
R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664 (40:4891)

[Bri79]
K.-H. Brinkmann, Algebraische $K$-Theorie über Kettenkomplexe, Diplomarbeit, Bielefeld, 1979.

[CE75]
J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematical Library, Vol. 9. MR 0458335 (56:16538)

[CP97]
M. Cárdenas and E. K. Pedersen, On the Karoubi filtration of a category, $K$-Theory 12 (1997), no. 2, 165-191. MR 1469141 (98g:18012)

[DL98]
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), no. 3, 201-252. MR 1659969 (99m:55004)

[EHS93]
P. Eberlein, U. Hamenstädt, and V. Schroeder, Manifolds of nonpositive curvature, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 179-227. MR 1216622 (94d:53060)

[EO73]
P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. MR 0336648 (49:1421)

[FJ86]
F. T. Farrell and L. E. Jones, $K$-theory and dynamics. I, Ann. of Math. (2) 124 (1986), no. 3, 531-569. MR 0866708 (88f:57062)

[FJ87]
-, $K$-theory and dynamics. II, Ann. of Math. (2) 126 (1987), no. 3, 451-493. MR 0916716 (89f:57029a)

[FJ89]
-, A topological analogue of Mostow's rigidity theorem, J. Amer. Math. Soc. 2 (1989), no. 2, 257-370. MR 0973309 (90h:57023a)

[FJ91]
-, Stable pseudoisotopy spaces of compact non-positively curved manifolds, J. Differential Geom. 34 (1991), no. 3, 769-834. MR 1139646 (93b:57020)

[FJ93a]
-, Isomorphism conjectures in algebraic $K$-theory, J. Amer. Math. Soc. 6 (1993), no. 2, 249-297. MR 1179537 (93h:57032)

[FJ93b]
-, Topological rigidity for compact non-positively curved manifolds, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 229-274. MR 1216623 (94m:57067)

[Gra76]
D. Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 217-240. Lecture Notes in Math., Vol. 551. MR 0574096 (58:28137)

[HIH77]
E. Heintze and H.-C. Im Hof, Geometry of horospheres, J. Differential Geom. 12 (1977), no. 4, 481-491 (1978). MR 0512919 (80a:53051)

[HP04]
I. Hambleton and E. K. Pedersen, Identifying assembly maps in $K$- and $L$-theory, Math. Ann. 328 (2004), no. 1-2, 27-57. MR 2030369 (2004j:19001)

[HPR97]
N. Higson, E. K. Pedersen, and J. Roe, $C\sp \ast$-algebras and controlled topology, $K$-Theory 11 (1997), no. 3, 209-239. MR 1451755 (98g:19009)

[Hsi84]
W. C. Hsiang, Geometric applications of algebraic $K$-theory, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Warsaw, 1983) (Warsaw), PWN, 1984, pp. 99-118. MR 0804679 (87g:57034)

[Kar70]
M. Karoubi, Foncteurs dérivés et $K$-théorie, Séminaire Heidelberg-Saarbrücken-Strasbourg sur la Kthéorie (1967/68), Lecture Notes in Mathematics, Vol. 136, Springer, Berlin, 1970, pp. 107-186. MR 0265435 (42:344)

[Lod76]
J.-L. Loday, $K$-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309-377. MR 0447373 (56:5686)

[LR04]
W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones Conjectures in ${K}$- and ${L}$-Theory, arXiv:math.KT/0402405, to appear in Handbook of $K$-Theory, Eds. E. Friedlander, D. Grayson, Springer, 2004.

[Per01]
L. Perko, Differential equations and dynamical systems, third ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001. MR 1801796 (2001k:34001)

[PT78]
E. K. Pedersen and L. R. Taylor, The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), no. 4, 887-896. MR 0509078 (80g:55030)

[PW85]
E. K. Pedersen and C. A. Weibel, A nonconnective delooping of algebraic $K$-theory, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 166-181. MR 0802790 (87b:18012)

[PW89]
-, $K$-theory homology of spaces, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 346-361. MR 1000388 (90m:55007)

[Qui73]
D. Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85-147. Lecture Notes in Math., Vol. 341. MR 0338129 (49:2895)

[Qui82]
F. Quinn, Ends of maps. II, Invent. Math. 68 (1982), no. 3, 353-424. MR 84j:57011

[tD72]
T. tom Dieck, Orbittypen und äquivariante Homologie. I, Arch. Math. (Basel) 23 (1972), 307-317. MR 0310919 (46:10017)

[tD87]
-, Transformation groups, de Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 0889050 (89c:57048)

[TT90]
R. W. Thomason and T. Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)

[Wal78]
F. Waldhausen, Algebraic $K$-theory of generalized free products. I, II, Ann. of Math. (2) 108 (1978), no. 1, 135-204. MR 0498807 (58:16845a)

[Wal85]
-, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318-419. MR 0802796 (86m:18011)

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 19D50, 53C12

Retrieve articles in all Journals with MSC (2000): 19D50, 53C12

Additional Information:

Arthur Bartels
Affiliation: Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: bartelsa@math.uni-muenster.de

Holger Reich
Affiliation: Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: reichh@math.uni-muenster.de

DOI: 10.1090/S0894-0347-05-00482-0
PII: S 0894-0347(05)00482-0
Keywords: $K$-theory, group rings, controlled algebra
Received by editor(s): August 5, 2003
Posted: March 30, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia