Homotopy groups of suspended classifying spaces: An experimental approach

When the results of a computer program are confronted to some theorems proved on a theoretical basis three situations can occur: there can be an agreement between both approaches, the computer program can obtain calculations not covered by the theorems, or a discrepancy can be found between both methods. In this note we report on a work where the three above mentioned situations happen. We have enhanced the Computer Algebra called Kenzo to deal with the computation of homotopy groups of suspended classifying spaces, a problem tackled by Mikhailov and Wu in a paper published in the journal Algebraic and Geometric Topology. Our experimental approach, based on completely different methods from those by Mikhailov and Wu, has allowed us in particular to detect an error in one of their published theorems. 1. Motivation and statement of the problem It is well-known that the suspension functor Σ applied to a topological space X shifts its homology groups, that is to say, Hn(ΣX) ∼= Hn−1(X). However, in the case of homotopy groups the situation is not so favorable and in general there is not a direct relation between π∗(ΣX) and π∗(X). Let us consider for instance a circle S, whose homotopy type is very simple. The homology groups of the suspension ΣS = S are obvious but the problem of computing the homotopy groups of S is a difficult problem in Algebraic Topology. Given a group G, its classifying space, that is to say, the Eilenberg-MacLane spaceK(G, 1) [10], has trivial homotopy groups: π1(K(G, 1)) ∼= G and πn(K(G, 1)) = 0 for each n 6= 1. But when applying the suspension functor Σ, the new homotopy groups π∗(ΣK(G, 1)) are in general unknown. In [11], several groups πn(ΣK(G, 1)) are obtained for some particular cases of G and n, making use of different results and techniques from group theory and homotopy theory. More concretely, the main results of the article by Mikhailov and Wu are descriptions of the groups π4(ΣK(A, 1)) and π5(ΣK(A, 1)) when A is any finitely generated Abelian group; as applications, they also determine πn(ΣK(G, 1)) with n = 4 and 5 for some nonAbelian groups, as G = Σ3 the 3-th symmetric group and G = SL(Z) the standard linear group, and π4(ΣK(A4, 1)) for A4 the 4-th alternating group. The goal of this work consists in trying to obtain a computer program calculating π∗(ΣK(G, 1)) for a general group G, which will allow one to obtain as particular computations some theoretical results of [11]. To this aim, we have developed 2010 Mathematics Subject Classification. 68W05,55-04,55Q99. Partially supported by Ministerio de Ciencia e Innovacion, Spain, project MTM2009-13842C02-01. c ©XXXX American Mathematical Society