Computing the homology of groups: The geometric way

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• 作者：Ana Romero ^{ana.romero@unirioja.es} ; Julio Rubio ^{julio.rubio@unirioja.es}
• 关键词：Group homology ; Algebraic topology ; Effective homology ; Common Lisp
• 刊名：Journal of Symbolic Computation
• 出版年：2012
• 期刊代码：140_07477171
• 类别：cp
• 出版时间：July, 2012
• 卷：47
• 期：7
• 页码：752-770
• 文件大小：325 K

摘要

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 for different groups , allowing one in particular to determine the homology groups of .

Our algorithms have been programmed as new modules for the Kenzo system, enhancing it with the following new functionalities:

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construction of the effective homology of from a given finite type free resolution of the group ;

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construction of the effective homology of for every finitely generated Abelian group (as a consequence, the effective homology of is also available in Kenzo, for all );

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computation of homology groups of some -types;

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construction of the effective homology for central extensions.

In addition, an inverse problem is also approached in this work: given a group such that has effective homology, can a finite type free resolution of the group 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.