Mappings of the sphere to a simply connected space

Fix an m ∈ ℕ, m ≥ 2. Let Y be a simply connected pointed CW-complex, and let B be a finite set of continuous mappings a: S^m → Y respecting the distinguished points. Let Γ(a) ⊂ S^m × Y be the graph of a, and we denote by [a] ∈ π_m(Y) the homotopy class of a. Then for some r ∈ ℕ depending on m only, there exist a finite set E ⊂ S^m × Y and a mapping k: E(r) = {F ⊂ E: |F| ≤ r} → π_m(Y) such that for each a ∈ B we have . Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 159–194.