Table of the homotopy groups pi_n+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,

n^k indicates the direct sum of k-copies of Z/nZ.

pi_n+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 pi_k(S²ⁿ⁺¹). Then pⁿ x=0.

Wu Theorem For any n>2, the homotopy group pi_n(S³) is isomorphic to the center of the group G(n) defined as follows:

Let x₁,...,x_n be letters and let w(x₁,...,x_n) denote a word in the free group F(x₁,...,x_n). Let 1 denote the identity of a group. The group G(n) is defined combinatorially by

generators: x₁,...,x_n;

relations:

₁

₂

₁

w(x₁,...,x_i-1,1,x_i+1,...,x_n)=1 for each i=1,2,...,n.

Note.

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.
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).

My Home page.

Table of the homotopy groups pin+k(Sn)

pin+k(Sn)

Table of the homotopy groups pi_n+k(Sⁿ)

pi_n+k(Sⁿ)