×

zbMATH — the first resource for mathematics

On the Farrell-Jones conjecture for higher algebraic \(K\)-theory. (English) Zbl 1073.19002
Let \(\Gamma\) be a group and \(R\) any associative ring with unit. Furthermore, let \(E(VCyc)\) be a universal space for the family of virtually cyclic subgroups of \(\Gamma\) and \(\mathbb{H}^{\Gamma}(-,\mathbb{K}^{-\infty}_{R})\) denote the \(\Gamma\)-equivariant homology theory with coefficients in the \(K\)-theory spectrum of \(R\). The Farrell-Jones conjecture states that the assembly map \(\mathbb{A}_{VCyc}:\mathbb{H}^{\Gamma}_{n}(-,\mathbb{K}^{-\infty}_{R})\to K_{n}(R\Gamma)\) is an isomorphism for all \(n\in\mathbb{Z}\).
The authors verify this conjecture in the case \(\Gamma\) is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature.
In a previous work, A. Bartels, T. Farrell, L. Jones and H. Reich verified this conjecture for this class of groups and for all \(n\leq 1\), see [“On the isomorphism conjecture in algebraic \(K\)-theory”, Topology 43, 157–213, (2004; Zbl 1036.19003)]. They also verified the injectivity of the assembly map for all \(n\in\mathbb{Z}\). This work complements the above by proving the surjectivity of the assembly map for all \(n\geq 1\).

MSC:
19D50 Computations of higher \(K\)-theory of rings
53C12 Foliations (differential geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429 – 461. · Zbl 0587.53045 · doi:10.2307/1971181 · doi.org
[2] Hyman Bass, Algebraic \?-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0174.30302
[3] Werner Ballmann, Misha Brin, and Patrick Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2) 122 (1985), no. 1, 171 – 203. , https://doi.org/10.2307/1971373 Werner Ballmann, Misha Brin, and Ralf Spatzier, Structure of manifolds of nonpositive curvature. II, Ann. of Math. (2) 122 (1985), no. 2, 205 – 235. · Zbl 0598.53046 · doi:10.2307/1971303 · doi.org
[4] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and \?-theory of group \?*-algebras, \?*-algebras: 1943 – 1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240 – 291. · Zbl 0830.46061 · doi:10.1090/conm/167/1292018 · doi.org
[5] Arthur Bartels, Tom Farrell, Lowell Jones, and Holger Reich, A foliated squeezing theorem for geometric modules, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 1 – 21. · Zbl 1049.57014 · doi:10.1142/9789812704443_0001 · doi.org
[6] Arthur Bartels, Tom Farrell, Lowell Jones, and Holger Reich, On the isomorphism conjecture in algebraic \?-theory, Topology 43 (2004), no. 1, 157 – 213. · Zbl 1036.19003 · doi:10.1016/S0040-9383(03)00032-6 · doi.org
[7] H. Bass, A. Heller, and R. G. Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 61 – 79. · Zbl 0248.18026
[8] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1 – 49. · Zbl 0191.52002
[9] K.-H. Brinkmann, Algebraische \(K\)-Theorie über Kettenkomplexe, Diplomarbeit, Bielefeld, 1979.
[10] Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. · Zbl 0309.53035
[11] M. Cárdenas and E. K. Pedersen, On the Karoubi filtration of a category, \?-Theory 12 (1997), no. 2, 165 – 191. · Zbl 0903.18005 · doi:10.1023/A:1007726201728 · doi.org
[12] James F. Davis and Wolfgang Lück, Spaces over a category and assembly maps in isomorphism conjectures in \?- and \?-theory, \?-Theory 15 (1998), no. 3, 201 – 252. · Zbl 0921.19003 · doi:10.1023/A:1007784106877 · doi.org
[13] Patrick Eberlein, Ursula Hamenstädt, and Viktor 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. · Zbl 0811.53038
[14] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45 – 109. · Zbl 0264.53026
[15] F. T. Farrell and L. E. Jones, \?-theory and dynamics. I, Ann. of Math. (2) 124 (1986), no. 3, 531 – 569. · Zbl 0653.58035 · doi:10.2307/2007092 · doi.org
[16] F. T. Farrell and L. E. Jones, \?-theory and dynamics. II, Ann. of Math. (2) 126 (1987), no. 3, 451 – 493. , https://doi.org/10.2307/1971358 F. T. Farrell and L. E. Jones, Foliated control with hyperbolic leaves, \?-Theory 1 (1987), no. 4, 337 – 359. · Zbl 0658.57020 · doi:10.1007/BF00539622 · doi.org
[17] F. T. Farrell and L. E. Jones, A topological analogue of Mostow’s rigidity theorem, J. Amer. Math. Soc. 2 (1989), no. 2, 257 – 370. , https://doi.org/10.1090/S0894-0347-1989-0973309-4 F. T. Farrell and L. E. Jones, Compact negatively curved manifolds (of dim \ne 3,4) are topologically rigid, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 10, 3461 – 3463. , https://doi.org/10.1073/pnas.86.10.3461 F. T. Farrell and L. E. Jones, Rigidity and other topological aspects of compact nonpositively curved manifolds, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 59 – 64. · Zbl 0676.53047
[18] F. T. Farrell and L. E. Jones, Stable pseudoisotopy spaces of compact non-positively curved manifolds, J. Differential Geom. 34 (1991), no. 3, 769 – 834. · Zbl 0749.53022
[19] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic \?-theory, J. Amer. Math. Soc. 6 (1993), no. 2, 249 – 297. · Zbl 0798.57018
[20] F. T. Farrell and L. E. Jones, 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. · Zbl 0796.53043
[21] Daniel Grayson, Higher algebraic \?-theory. II (after Daniel Quillen), Algebraic \?-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 217 – 240. Lecture Notes in Math., Vol. 551.
[22] Ernst Heintze and Hans-Christoph Im Hof, Geometry of horospheres, J. Differential Geom. 12 (1977), no. 4, 481 – 491 (1978). · Zbl 0434.53038
[23] Ian Hambleton and Erik K. Pedersen, Identifying assembly maps in \?- and \?-theory, Math. Ann. 328 (2004), no. 1-2, 27 – 57. · Zbl 1051.19002 · doi:10.1007/s00208-003-0454-5 · doi.org
[24] Nigel Higson, Erik Kjær Pedersen, and John Roe, \?*-algebras and controlled topology, \?-Theory 11 (1997), no. 3, 209 – 239. · Zbl 0879.19003 · doi:10.1023/A:1007705726771 · doi.org
[25] Wu Chung Hsiang, Geometric applications of algebraic \?-theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 99 – 118. · Zbl 0562.57001
[26] Max Karoubi, Foncteurs dérivés et \?-théorie, Séminaire Heidelberg-Saarbrücken-Strasbourg sur la Kthéorie (1967/68), Lecture Notes in Mathematics, Vol. 136, Springer, Berlin, 1970, pp. 107 – 186 (French).
[27] Jean-Louis Loday, \?-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309 – 377 (French). · Zbl 0362.18014
[28] 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. · Zbl 1120.19001
[29] Lawrence Perko, Differential equations and dynamical systems, 3rd ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001. · Zbl 0973.34001
[30] Erik Kjaer Pedersen and Lawrence R. Taylor, The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), no. 4, 887 – 896. · Zbl 0415.55010 · doi:10.2307/2373914 · doi.org
[31] Erik K. Pedersen and Charles A. Weibel, A nonconnective delooping of algebraic \?-theory, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 166 – 181. · Zbl 0591.55002 · doi:10.1007/BFb0074443 · doi.org
[32] Erik K. Pedersen and Charles A. Weibel, \?-theory homology of spaces, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 346 – 361. · doi:10.1007/BFb0085239 · doi.org
[33] Daniel Quillen, Higher algebraic \?-theory. I, Algebraic \?-theory, I: Higher \?-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85 – 147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004
[34] Frank Quinn, Ends of maps. II, Invent. Math. 68 (1982), no. 3, 353 – 424. · Zbl 0533.57008 · doi:10.1007/BF01389410 · doi.org
[35] Tammo tom Dieck, Orbittypen und äquivariante Homologie. I, Arch. Math. (Basel) 23 (1972), 307 – 317 (German). · Zbl 0252.55003 · doi:10.1007/BF01304886 · doi.org
[36] Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. · Zbl 0611.57002
[37] R. W. Thomason and Thomas Trobaugh, Higher algebraic \?-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247 – 435. · Zbl 0731.14001 · doi:10.1007/978-0-8176-4576-2_10 · doi.org
[38] Friedhelm Waldhausen, Algebraic \?-theory of generalized free products. I, II, Ann. of Math. (2) 108 (1978), no. 1, 135 – 204. , https://doi.org/10.2307/1971165 Friedhelm Waldhausen, Algebraic \?-theory of generalized free products. III, IV, Ann. of Math. (2) 108 (1978), no. 2, 205 – 256. · Zbl 0407.18009 · doi:10.2307/1971166 · doi.org
[39] Friedhelm Waldhausen, Algebraic \?-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318 – 419. · Zbl 0579.18006 · doi:10.1007/BFb0074449 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.