On algebraic characterizations for finiteness of the dimension of EG

Iranian Journal of Numerical Analysis and Optimization

مقاله 2، دوره 1، شماره 1، فروردین 2008، صفحه 1-22   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v1i1.616

نویسنده

Olympia Talelli^*

Department of Mathematics, University of Athens Panepistemiopolis, 15784 Athens - Greece

چکیده

Certain algebraic invariants of the integral group ring ZG of a group G were introduced and investigated in relation to the problem of extending the Farrell-Tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension. It turns out that the finiteness of these invariants of a group G implies the existence of a generalized Farrell-Tate cohomology for G which is computed via complete resolutions. In this article we present these algebraic invariants and their basic properties and discuss their relationship to the generalized Farrell-Tate cohomology. In addition we present the status of conjecture which claims that the finiteness of these invariants of a group G is equivalent to the existence of a finite dimensional model for EG, the classifying space for proper actions.

کلیدواژه‌ها

Farrell-Tate cohomology؛ virtual cohomological dimension؛ complete resolution؛ finitistic dimension of the integral group ring؛ classifying space for proper action

مراجع

[1] Benson, D.J. and Carlson, J., Products in negative cohomology, J. Pure Appl. Algebra 82(1992), 107–130.
[2] Brown, K.S., Cohomology of Groups, Springer Graduate Text in Math. 87, 1982.
[3] Cornick, J. and Kropholler, P.H., On complete resolutions, Topology and its Applications 78(1997), 235–250.
[4] Cornick J. and Kropholler, P.H., Homological finiteness conditions for mod-ules over group algebras, J. London Math. Soc. 58(1998), 49–62.

[5] Dembegioti, F. and Talelli, O., An integral homological characterization of finite groups, to appear in J. of Algebra.
[6] Dembegioti, F. and Talelli, O., On a relation between certain cohomological invariants, preprint 2007.
[7] Dicks, W., Leary, I.J., Kropholler P.H. and Tomas, S., Classifying spaces for proper actions of locally finite groups, J. Group Theory 5(4)(2002), 453–480.
[8] Dyer, J.L., On the residual finiteness of generalized free products, Trans. Amer Math. Soc. bf133(1968), 131–143.
[9] Farrell, F.T., An extension of Tate cohomology to a class of infinite groups, J. Pure Appl. Algebra 10(1977), 153–161.
[10] Gedrich, T.V. and Gruenberg, K.W., Complete cohomological functors on groups, Topology and its Application 25(1987), 203–223.
[11] Goichot, F., Homologie de Tate-Vogel equivariant, J. Pure Appl. Algebra 82(1992), 39–64.
[12] Ikenaga, B.M., Homological dimension and Farrell cohomology, Journal of Algebra 87(1984), 422–457.
[13] Ivanov, S.V., Relation modules and relation bimodules of groups, semigroups and associative algebras, Internat. J. Algebra Comput. 1(1991), No. 1, 89–114.
[14] Kropholler, P.H., On groups of type F P∞, J. Pure Appl. Algebra 90(1993), 55–67.
[15] Kropholler, P.H. and Mislin, G., Groups acting on finite dimensional spaces with finite stabilizers, Comment. Math. Helv. 73(1998), 122–136.
[16] Kropholler, P.H. and Talelli, O., On a property of fundamental groups of graphs of finite groups, J. Pure Appl. Algebra 74(1991), 57–59.

[17] L¨uck, W., The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149(2000), 177–203.
[18] L¨uck, W., Survey on Classifying Spaces for Families of subgroups, Preprint-reihe SFB 478-Geometrische Strukturen in der Mathematik 308(2004).
[19] Mislin, G., Tate cohomology for arbitrary groups via satelites, Topology and its applications 56(1994), 293–300.
[20] Mislin, G., On the classifying space for proper actions, In cohomological Methods in Homotopy Theory, Progress in Mathematics 196, Birkh¨auser, 2001.
[21] Mislin, G. and Talelli, O., On groups which act freely and properly on finite dimensional homotopy spheres Computational and Geometric Aspects of Modern Algebra. LMS Lecture Note Series (275) Cambridge Univ., Press 2000.
[22] Nucinkis, B.E.A., Is there an easy algebraic characterization of universal proper G-spaces? Manuscripta Math. Vol. 2 (3)(2000), 335–345.
[23] Schneebeli, H.R., On virtual properties of group extensions, Math. Z. 159(1978), 159–167.
[24] Talelli, O., Periodicity in group cohomology and complete resolutions, Bull. London Math. Soc. 37(2005), 547–554.
[25] Talelli, O., On complete resolutions, to appear in the LMS Durham Sympo-sium on Geometry and Cohomology in Group Theory, July 2003.
[26] Talelli, O., On groups of type Φ, Archiv der Mathematik 89(1)(2007), 24–32.
[27] Wall, C.T.C., Periodic projective resolutions, Proc. London Math. Soc. 39(1979), 509–533.