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Effective homology of a classifying space

Sergeraert defines in [9] a method to solve the main problems of Constructivism in Algebraic Topology. The first resulting computer program, E.A.T., showed this method is a usable concrete framework for a Constructive Algebraic Topology implementation [8].Rubio and Sergeraert solved in this way an important but particular problem: computing the homology of iterated loop spaces. The application field of the EAT program is rather limited, despite its interesting results. Implementation of mathematical categories in Lisp has met real difficulties and it looked unreasonable to plan a general Constructive Algebraic Topology program along the same line.The OOP Lisp implementation CLOS (Common Lisp Object System) provides a powerful means of implementing these categories. The new program Kenzo also implements Sergeraert's method for Constructive Algebraic Topology, but the scope is considerably larger [11]. In particular the Kenzo program allows one to compute the first homotopy groups of an arbitrary simplicial set. It is the first computer program able to solve this problem; new homotopy groups have been calculated in this way.