Computing Spectral Sequences A. Romero a,⋆, J. Rubio a,⋆, F. Sergeraert b aDepartamento de Matem´aticas y Computaci´on. Universidad de La Rioja, Spain. bInstitut Fourier. Universit´eJoseph Fourier. Grenoble, France Abstract In this paper, a set of programs enhancing the Kenzo system is presented. Kenzo is a Common Lisp program designed for computing in Algebraic Topology, in partic- ular it allows the user to calculate homology and homotopy groups of complicated spaces. The new programs presented here entirely compute Serre and Eilenberg- r r Moore spectral sequences, in particular the Ep,q and dp,q for arbitrary r. They also r ∞ determine when Ep,q = Ep,q and describe the filtration of the target homology ∞ groups Hp+q by the Ep,q’s. Key words: Symbolic Computation, Spectral Sequences, Serre Spectral Sequence, Eilenberg-Moore Spectral Sequence, Constructive Algebraic Topology, Common Lisp. 1 Introduction arXiv:cs/0602064v1 [cs.SC] 17 Feb 2006 The computation of homology groups of topological spaces is one of the first problems in Algebraic Topology, and these groups can be difficult to reach, for example when loop spaces or classifying spaces are involved. The Kenzo program [1], a Symbolic Computation system devoted to Algebraic Topology, allows the computation of homology groups of a large class of spaces, namely the members of the class SS EH, the class of the simplicial sets with effective homology; for example this class is closed under the constructors loop spaces, ⋆ Partially supported by SEUI-MEC, project TIC2002-01626. Email addresses:
[email protected] (A. Romero),
[email protected] (J.