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:
    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).


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