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Homological realization of prescribed abelian groups via K-theory

Published online by Cambridge University Press:  10 April 2007

A. J. BERRICK  and
M. MATTHEY
A. J. BERRICK
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore. e-mail: [email protected]
M. MATTHEY
Affiliation:
Late of: University of Lausanne, IGAT (Institute for Geometry, Algebra and Topology), Bâtiment BCH, EPFL, CH-1015 Lausanne, Switzerland.
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Abstract

Using algebraic and topological K-theory together with complex C*-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and isomorphic to two arbitrarily prescribed abelian groups in dimensions two and three.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 249 - 258
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1] Arlettaz, D.. The Hurewicz homomorphism in algebraic K-theory. J. Pure Appl. Algebra 71 (1991), 1–12.CrossRefGoogle Scholar
[2] Baumslag, G., Dyer, E. and Heller, A.. The topology of discrete groups. J. Pure Appl. Algebra 16 (1980), 1–47.CrossRefGoogle Scholar
[3] Baumslag, G., Dyer, E. and Miller, C. F. III. On the integral homology of finitely presented groups. Topology 22 (1983), 27–46.CrossRefGoogle Scholar
[4] Berrick, A. J.. An Approach to Algebraic K-theory. Pitman Research Notes Math. 56 (London, 1982).Google Scholar
[5] Berrick, A. J.. Two functors from abelian groups to perfect groups. J. Pure Appl. Algebra 44 (1987), 35–43.CrossRefGoogle Scholar
[6] Berrick, A. J.. Remarks on the structure of acyclic groups. Bull. London Math. Soc. 22 (1990), 227–232.CrossRefGoogle Scholar
[7] Berrick, A. J.. Torsion generators for all abelian groups. J. Algebra 139 (1991), 190–194.CrossRefGoogle Scholar
[8] Berrick, A. J.. A topologist's view of perfect and acyclic groups. In Invitations to Geometry and Topology, ed. Bridson, M. R. and Salamon, S. M.. Oxford Graduate Texts in Math. 5 (Oxford University Press, 2002), chapter 1, 1–28.CrossRefGoogle Scholar
[9] Berrick, A. J. and Kropholler, P. H.. Groups with infinite homology. In Cohomological Methods in Homotopy Theory (BCAT98, Bellaterra 1998). Progr. Math. 196 (2001), 27–33.Google Scholar
[10] Berrick, A. J. and Miller, C. F. III. Strongly torsion generated groups. Math. Proc. Camb. Phil. Soc. 111 (1992), 219–229.CrossRefGoogle Scholar
[11] Claborn, L.. Every abelian group is a class group. Pacific J. Math. 118 (1966), 219–222.CrossRefGoogle Scholar
[12] Eilenberg, S. and Lane, S. Mac. On the groups H(Π,n). II. Methods of computation. Ann. of Math. 60 (1954), 49–139.CrossRefGoogle Scholar
[13] Evens, L.. The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101 (1961), 224–239.CrossRefGoogle Scholar
[14] Gersten, S. M.. K 3 of a ring is H 3 of the Steinberg group. Proc. Amer. Math. Soc. 37 (1973), 366–368.Google Scholar
[15] Harpe, P. de la and Skandalis, G.. Sur la simplicité essentielle du groupe des inversibles et du groupe unitaire dans une C*-algèbre simple. J. Func. Anal. 62 (1985), 354–378.CrossRefGoogle Scholar
[16] Giffen, C. H.. Contributed problems. In Proc. Northwestern Univ. Homotopy Theory Conf., Contemp. Math. 19 (1982), 447.Google Scholar
[17] Hickin, K. K.. Universal locally finite central extensions of groups. Proc. London Math. Soc. (3) 52 (1986), 53–72.CrossRefGoogle Scholar
[18] Higson, N.. Algebraic K-theory of stable C*-algebras. Adv. Math. 67 (1988), 1–140.CrossRefGoogle Scholar
[19] Hilton, P. J. and Stammbach, U.. A Course in Homological Algebra, 2nd ed. Graduate Texts in Math. 4 (Springer 1997).CrossRefGoogle Scholar
[20] Inassaridze, H.. Algebraic K-theory of normed algebras. K-Theory 21 (2000), 25–56.CrossRefGoogle Scholar
[21] Inassaridze, H. and Kandelaki, T.. K-theory of stable generalized operator algebras. K-Theory 27 (2002), 103–110.CrossRefGoogle Scholar
[22] Inassaridze, H. and Kandelaki, T.. Smooth K-theory of locally convex algebras. Preprint (2005), available at http://www.math.uiuc.edu/K-theory/0746/.Google Scholar
[23] Karoubi, M.. K-théorie algébrique de certaines algèbres d'opérateurs. Lecture Notes in Math. 725 (Springer, 1979), 254–290.CrossRefGoogle Scholar
[24] Karoubi, M.. Homologie des groupes discrets associés à des algèbres d'opérateurs J. Operator Theory 15 (1986), 109–161.Google Scholar
[25] Leedham–Green, C. R.. The class group of Dedekind domains. Trans. Amer. Math. Soc. 163 (1972), 493–500.CrossRefGoogle Scholar
[26] Loday, J.-L.. K-théorie algébrique et représentations de groupes. Ann. Sci. École. Norm. Sup. 9 (1976), 309–377.CrossRefGoogle Scholar
[27] Magurn, B. A.. An Algebraic Introduction to K-Theory. Encyclopedia of Math. and its Applications 87 (Cambridge University Press, 2002).CrossRefGoogle Scholar
[28] Malle, G. and Matzat, B. H.. Inverse Galois Theory. Springer Monographs in Math. (Springer, 1999).CrossRefGoogle Scholar
[29] Milgram, R. J.. The cohomology of the Mathieu group M 23 . J. Group Theory 3 (2000), 7–26.CrossRefGoogle Scholar
[30] Miller, H.. The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120 (1984), 39–87.CrossRefGoogle Scholar
[31] Milnor, J.. Introduction to Algebraic K-Theory. Ann. of Math. Stud. 72 (Princeton University Press, 1971).Google Scholar
[32] Neumann, B. H.. A note on algebraically closed groups. J. London Math. Soc. 27 (1952), 247–249.CrossRefGoogle Scholar
[33] Phillips, R. E.. Existentially closed locally finite central extensions, multipliers and local systems. Math. Z. 187 (1984), 383–392.CrossRefGoogle Scholar
[34] Rørdam, M., Larsen, F. and Laustsen, N.. An Introduction to K-Theory for C*-Algebras. London Math. Soc. Student Texts 49 (Cambridge University Press, 2000).CrossRefGoogle Scholar
[35] Rosenberg, J.. Algebraic K-Theory and Its Applications. Graduate Texts in Math. 147 (Springer, 1994).CrossRefGoogle Scholar
[36] Scott, W. R.. Algebraically closed groups. Proc. Amer. Math. Soc. 2 (1951), 118–121.CrossRefGoogle Scholar
[37] Suslin, A. A. and Wodzicki, M.. Excision in algebraic K-theory and Karoubi's conjecture. Proc. Nat. Acad. Sci. U.S.A. 87 (1990), 9582–9584.CrossRefGoogle ScholarPubMed
[38] Suslin, A. A. and Wodzicki, M.. Excision in algebraic K-theory. Ann. of Math. (2) 136 (1992), 51–122.CrossRefGoogle Scholar
[39] Swan, R. G.. The nontriviality of the restriction map in the cohomology of groups. Proc. Amer. Math. Soc. 11 (1960), 885–887.Google Scholar
[40] Switzer, R. M.. Algebraic Topology – Homotopy and Homology. Die Grundlehren der Math. Wissenschaften, Band 212 (Springer, 1975).CrossRefGoogle Scholar
[41] Tapia, J.. K-théorie algébrique négative et K-théorie topologique de l'algèbre de Fréchet des opérateurs régularisants. Ann. Sci. École Norm. Sup. 30 (1997), 241–277.CrossRefGoogle Scholar
[42] Wegge-Olsen, N. E.. K-Theory and C*-Algebras. A Friendly Approach. Oxford Science Publications (Oxford University Press, 1993).CrossRefGoogle Scholar