Homotopy Groups of Spheres

This table gives π_{r}(S^{n}) for a range of values of r and n. In the table, k denotes the cyclic group Z/k and + denotes direct sum. The groups B, C, D, E, F denote respectively the direct sum of 2, 3, 4, 5, 6 copies of Z/2.

See Composition methods in homotopy groups of spheres by H. Toda, Annals of Math. Studies 49 (Princeton University Press, 1962) for this table and much more.

n = 1 n = 2 n = 3  n = 4  n = 5 n = 6 n = 7  n = 8  n = 9 n = 10 n = 11 n = 12
r = 1 Z  0  0  0  0  0  0  0  0  0  0  0
r = 2 0  Z  0  0  0  0  0  0  0  0  0  0
r = 3 0  Z  Z  0  0  0  0  0  0  0  0  0
r = 4 0  2  2  Z  0  0  0  0  0  0  0  0
r = 5 0  2  2  2  Z  0  0  0  0  0  0  0
r = 6 0  12  12  2  2  Z  0  0  0  0  0  0
r = 7 0  2  2  Z+12  2  2  Z  0  0  0  0  0
r = 8 0  2  2  2+2  24  2  2  Z  0  0  0  0
r = 9 0  3  3  2+2  2  24  2  2  Z  0  0  0
r = 10 0  15  15  24+3  2  0  24  2  2  Z  0  0
r = 11 0  2  2  15  2  Z  0  24  2  2  Z  0
r = 12 0  2+2 2+2  2  30  2  0  0  24  2  2  Z
r = 13 0  12+2 12+2  2+2+2  2  60  2  0  0  24  2  2
r = 14 0  84+B 84+B 120+12+2  C  24+2  120  2  0  0  24  2
r = 15 0  2+2 2+2  84+E  72+2  C  C  Z+120  2  0  0  24
r = 16 0  6  6  F  504+B 72+2  D  D  240  2  0  0
r = 17 0  30  30  24+6+2  C  504+4 24+2  E  C  240  2  0
r = 18 0  30  30  2520+6+2  6+2  240 504+2 24+24+2  D  2+2  240  2
r = 19 0  6+2 6+2  30  6+2  6  0  504+2  24+2 Z+C 2+2  240
r = 20 0  12+B 12+B  6+6+2  30+2 12+2  6  0  504+2 12+2  C  2+2
r = 21 0  12+B 12+B 24+12+4+B 2+2  60+6 24+4  6+2  0  504  6+2  C
r = 22 0 132+2 132+2 120+12+E 4+B 504+B 120+C 240+24+4  6  12  504  6
r = 23 0  ?  ?  132+E  24+B  D  D  120+E  16+4  6  2  Z+504

110.615–6 Algebraic Topology JMB File: hpysph, Revision B; 23 Oct 2003