On the Homotopy Groups of the Union of Spheres

Let 8( be a sphere of dimension r,+ l, rt^ 1, i = 1, ..., h, and let T be the union of the spheres Sv ..., Sk, with a single common point. Then T serves as a universal example for homotopy constructions (see [1]). The object of this paper is to compute the group irn{T), n > 1, as a direct sum of homotopy groups of spheres of appropriate dimensions^:. Each summand is embedded in TTn{T) by a certain multiple Whitehead product; the products which appear will be called basic products and will now be defined. Let TQ = SUi v SU2 v... v SUm, where 1 < % < u2 < ... < um < k. Then the injection 7rn(T0)->Trn(T) embeds 7rn(T0) univalently as a direct summand unrn(T). We will identify elements of 7rn(To) with their images in 7rn(T), and an element in the image of 7rn(T0) will be said to involve the spheres SUi, ..., 8Um. With these conventions, we define and order the basic products as follows. The basic products § of weight 1 are the elements tl3 ..., t,k, where i < h < ••• < k> i being the positive generator of 7rr.+1 (#,•), i = 1, ..., k. Now suppose the basic products of weight < w defined and ordered. Then a basic product of weight w > 1 is a Whitehead product [a, 6], where a is a basic product of weight u, b is a basic product of weight v, u-\-v = w, a

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