ar X iv : m at h / 06 04 02 9 v 1 [ m at h . A T ] 3 A pr 2 00 6 SECONDARY HOMOTOPY GROUPS

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)-connected (n + 1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secondary operations termed Toda brackets. Such bracket operations are defined by pasting tracks where a track is a homotopy class of homotopies. Since Toda brackets play a crucial role in homotopy theory it seems feasible to investigate the algebraic nature of tracks. Therefore we shift focus from homotopy groups πnX to secondary homotopy groups πn,∗X = (πn,1X ∂ −→ πn,0X) defined in this paper. Here ∂ is a homomorphism of groups with Coker∂ = πnX and Ker ∂ = πn+1X . The groups πn,0X and πn,1X are defined directly by use of continuous maps f : S → X and tracks of such maps to the trivial map, so that πn,∗X is actually a functor in X . For n ≥ 2 the definition involves the new concept of Hopf invariant for tracks. We show that the homomorphism ∂ has additional algebraic structure, namely π1,∗X is a crossed module, π2,∗X is a reduced quadratic module and πn,∗X , n ≥ 3, is a stable quadratic module. Crossed modules were introduced by J. H. C. Whitehead in [Whi49] and, in fact, for a CW -complex X our secondary homotopy group π1,∗X is weakly equivalent to the crossed module π2(X,X ) −→ π1X 1 studied by [Whi49]. Similarly πn,∗X for n ≥ 2 is weakly equivalent to the quadratic modules obtained in [Bau91] in terms of the cell structure of X which can also be derived from the Kan loop simplicial group associated toX , see for example [Con84] and [BCC93]. The topological and functorial definition of secondary homotopy groups πn,∗X is crucial to understand new properties of these concepts in the literature. For 1991 Mathematics Subject Classification. 18D05, 55Q25, 55S45.