# Table of the homotopy groups pin+k(Sn)

From Toda's book: Composition Methods in Homotopy Groups of Spheres

In the following table,

• an integer n indicates a cyclic group Z/nZ of order n,
• "infty" indicates the infinite cyclic group Z,
• the symbol "+" indicates the direct sum of the (abelian) groups,
• nk indicates the direct sum of k-copies of Z/nZ.

• ## pin+k(Sn)

 k\n n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=17 n=18 n=19 n=20 n>k+1 k=1 infty 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 k=2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 k=3 2 4+3 infty+4+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 8+3 k=4 4+3 2 22 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 k=5 2 2 22 2 infty 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 k=6 2 3 8+3+3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 k=7 3 3+5 3+5 2+3+5 4+3+5 8+3+5 infty+8+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 16+3+5 k=8 3+5 2 2 2 8+2+3 23 24 23 22 22 22 22 22 22 22 22 22 22 22 22 k=9 2 22 23 23 23 24 25 24 infty+23 23 23 23 23 23 23 23 23 23 23 23 k=10 22 4+2+3 8+4+2+32+5 8+2+9 8+2+9 8+3+2 82+2+32 8+2+3 4+2+3 22+3 2+3 2+3 2+3 2+3 2+3 2+3 2+3 2+3 2+3 2+3 k=11 4+2+3 4+22+3+7 4+25+3+7 8+22+9+7 8+4+9+7 8+2+9+7 8+2+9+7 8+2+9+7 8+9+7 8+9+7 infty+8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 8+9+7 k=12 4+22+3+7 22 26 23 16+3+5 0 0 0 4+3 2 22 2 0 0 0 0 0 0 0 0 k=13 22 2+3 8+22+32 22+3 2+3 2+3 22+3 2+3 2+3 22+3 22+3 2+3 infty+3 3 3 3 3 3 3 3 k=14 2+3 2+3+5 8+22+9+3+5+7 22+3 4+2+3 8+4+3 16+8+4+32+5 16+4 16+2 16+2 16+4+2+3 16+2 8+2 4+2 22 22 22 22 22 22 k=15 2+3+5 2+3+5 2+3+5 22+3+5 4+2+32+5 8+23+3+5 8+25+3+5 16+23+3+5 16+22+3+5 16+2+3+5 16+2+3+5 32+2+3+5 32+2+3+5 32+2+3+5 infty+32+2+3+5 32+2+3+5 32+2+3+5 32+2+3+5 32+2+3+5 32+2+3+5 k=16 2+3+5 22+3 23+32 22 8+22+9+7 24 27 24 16+2+3+5 2 2 2 8+2+3 23 24 23 22 22 22 22 k=17 22+3 4+22+3 8+42+22+32 4+22 24 24 25+3 24 23 23 24 24 24 25 26 25 infty+24 24 24 24 k=18 4+22+3 4+22+3 8+4+25+32+5 8+22+3 8+22+32 8+2+3 82+2+9+3+7 8+2+3 8+22+3 8+4+2 32+42+2+3+5 82+2 82+2 82+2 83+2+3 82+2 8+4+2 8+22 8+2 8+2 k=19 4+22+3 4+2+3+11 4+25+3+11 8+2+3+11 32+8+3+11 8+2+3+11 8+2+3+11 8+2+3+11 8+2+32+11 8+23+3+11 8+25+3+11 8+23+3+11 8+4+2+3+11 8+22+3+11 8+22+3+11 8+22+3+11 8+2+3+11 8+2+3+11 infty+8+2+3+11 8+2+3+11

• Table of the homotopy groups of the suspensions of the (real) projective plane.

• Cohen-Moore-Neisendorfer Theorem Let p be an odd prime and let x be any element in the p-primary torsion component of pik(S2n+1). Then pn x=0.

Wu Theorem For any n>2, the homotopy group pin(S3) is isomorphic to the center of the group G(n) defined as follows:
Let x1,...,xn be letters and let w(x1,...,xn) denote a word in the free group F(x1,...,xn). Let 1 denote the identity of a group. The group G(n) is defined combinatorially by
• generators: x1,...,xn;
• relations:
(1) the product element x1x2...xn;
(2) the words w(x1,...,xn) that satisfy:
w(x1,...,xi-1,1,xi+1,...,xn)=1 for each i=1,2,...,n.

Note.
1. The second relation above consists of all those words that will collapses to the identity if one of the generators is replaced by the identity.
2. There is a braid group action on G(n) induced by the canonical braid group action on free groups. The center of G(n), that is the n-th homotopy group of S3, is the fixed set of the pure braid group action on G(n).