Computing the homology of groups: The geometric way

In this paper, we present several algorithms related with the computation of the homology of groups, from a geometric perspective (that is to say, carrying out the calculations by means of simplicial sets and using techniques of Algebraic Topology). More concretely, we have developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-MacLane spaces K(G,1) for different groups G, allowing one in particular to determine the homology groups of G. Our algorithms have been programmed as new modules for the Kenzo system, enhancing it with the following new functionalities: *construction of the effective homology of K(G,1) from a given finite type free resolution of the group G; *construction of the effective homology of K(A,1) for every finitely generated Abelian group A (as a consequence, the effective homology of K(A,n) is also available in Kenzo, for all [email protected]?N); *computation of homology groups of some 2-types; *construction of the effective homology for central extensions. In addition, an inverse problem is also approached in this work: given a group G such that K(G,1) has effective homology, can a finite type free resolution of the group G be obtained? We provide some algorithms to solve this problem, based on a notion of norm of a group, allowing us to control the convergence of the process when building such a resolution.

[1]  Francis Sergeraert,et al.  Constructive algebraic topology , 1999, SIGS.

[2]  Graham Ellis,et al.  Homology of 2-Types , 1992 .

[3]  Samuel Eilenberg,et al.  On Products of Complexes , 1953 .

[4]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[5]  V. K. A. M. Gugenheim,et al.  On the chain-complex of a fibration , 1972 .

[6]  Saunders MacLane,et al.  On the Groups H(Π, n), II: Methods of Computation , 1954 .

[7]  Ian J Leary.  The integral cohomology rings of some p-groups , 1991 .

[8]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[9]  Weishu Shih,et al.  Homologie des espaces fibrés , 1962 .

[10]  Julio Jesús Rubio García.  Integrating functional programming and symbolic computation , 1996 .

[11]  Ronald Brown,et al.  The twisted Eilenberg-Zilber Theorem , 2009 .

[12]  Julio Jesús Rubio García.  Erratum to “Integrating functional programming and symbolic computation” [Math. Comp. Simul. 42 (1996) 467–473] , 1997 .

[13]  A. Clément,et al.  Integral cohomology of finite postnikov towers , 2002 .

[14]  Francis Sergeraert,et al.  The Computability Problem in Algebraic Topology , 1994 .

[15]  Claude Irwin Palmer,et al.  Algebra and applications , 1918 .

[16]  Ana Romero,et al.  Interoperating between computer algebra systems: computing homology of groups with kenzo and GAP , 2009, ISSAC '09.

[17]  Julio Rubio,et al.  Constructive Homological Algebra and Applications , 2012, 1208.3816.

[18]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[19]  Samuel Eilenberg,et al.  On the Groups H(Π, n), I , 1953 .

[20]  Jon P. May.  Simplicial objects in algebraic topology , 1993 .