• DOI:10.1112/JLMS/S1-30.2.154
• Corpus ID: 122480153

On the Homotopy Groups of the Union of Spheres

@article{Hilton1955OnTH,
  title={On the Homotopy Groups of the Union of Spheres},
  author={Peter J. Hilton},
  journal={Journal of The London Mathematical Society-second Series},
  year={1955},
  pages={154-172}
}

• P. Hilton
• Published 1 April 1955
• Mathematics
• Journal of The London Mathematical Society-second Series

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<b, and if b is defined as the Whitehead product, [c, d], of the basic products c, d, then c ̂ a. The basic products of weight w are then ordered arbitrarily among themselves and are greater than any product of lesser weight. It will be seen that a basic product of weight w is a string of symbols iVi...iVw, suitably bracketed, where l^v^k, j=l, ..., w. Suppose i,occurs wi times in this string. Then we will say that the basic product involves the sphere S{ wi times, and the height of the basic product is k defined as S r\,«;,.. t=i