The geometry of the Hilton splitting

One of the important theorems in homotopy theory is the Hilton Splitting Theorem which states: there is an isomorphism H=⊕γ∈ΓHγ from the mth homotopy group of the wedge of a number of spheres to the direct sum of the mth homotopy groups of some spheres, see [J. London Math. Soc. 30 (1955) 154]. In this paper we will construct geometrically all Hilton homomorphisms Hγ and prove a family of sharper symmetry relations between linking coefficients which desuspend and generalize the relations of Kervaire [Ann. of Math. 69 (1959) 335], Haefliger and Steer [Comment. Math. Helv. 39 (1965) 259].