ar X iv : m at h / 06 04 02 9 v 2 [ m at h . A T ] 1 7 O ct 2 00 6 SECONDARY HOMOTOPY GROUPS

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a 2-functor on the groupoid-enriched 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 , n ≥ 1. The adjective “secondary” in the title complements the word “group”, so secondary homotopy groups are secondary groups appearing in homotopy theory. The words “secondary groups” stand for a variety of dimension 2 generalizations of the notion of group, like Whitehead’s crossed modules [Whi49], or reduced or stable quadratic modules in the sense of [Bau91]. There is not a well-established terminology in the literature to designate all these 2-dimensional “groups” and we believe that “secondary groups” has the advantage of being new (so no confusion with older concepts is created) and short. 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 reduced CW -complex X with 1-skeleton X our secondary homotopy group 1991 Mathematics Subject Classification. 18D05, 55Q25, 55S45.