From: Jie Wu, with an extra color (lime) indicating the stable part.

Table of the homotopy groups π_n+k(Sⁿ)

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

In the following table,

an integer n ≥ 1 indicates a cyclic group Z/nZ of order n (in particular, 1 denotes the trivial group).

"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.

π_n+k(Sⁿ)

n=1

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

π_k+1(S¹) → π_k+2(S²) → π_k+3(S³) → π_k+4(S⁴) → π_k+5(S⁵) → π_k+6(S⁶) → π_k+7(S⁷) → π_k+8(S⁸) → π_k+9(S⁹) → π_k+10(S¹⁰) → π_k+11(S¹¹) → π_k+12(S¹²) → π_k+13(S¹³) → π_k+14(S¹⁴) → π_k+15(S¹⁵) → π_k+16(S¹⁶) → π_k+17(S¹⁷) → π_k+18(S¹⁸) → π_k+19(S¹⁹) → π_k+20(S²⁰) → π_k^S

k=0

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

infty

k=1

1

infty

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

k=2

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

k=3

1

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

1

4+3

2

2²

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

k=5

1

2

2

2²

2

infty

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

k=6

1

2

3

8+3+3

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

k=7

1

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

1

3+5

2

2

2

8+2+3

2³

2⁴

2²

2²

2²

2²

2²

2²

2²

2²

2²

2²

2²

2²

2²

k=9

1

2

2²

2³

2³

2³

2⁴

2⁵

2⁴

infty+2³

2³

2³

2³

2³

2³

2³

2³

2³

2³

2³

2³

k=10

1

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

1

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

1

4+2²+3+7

2²

2⁶

2³

16+3+5

1

1

1

4+3

2

2²

2

1

1

1

1

1

1

1

1

k=13

1

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

1

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

1

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

1

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

1

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

1

4+2²+3

4+2²+3

8+4+2⁵+3²+5

8+2²+3

8+2²+3²

8+2²+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

1

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.

	n=1	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
	π_k+1(S¹) →	π_k+2(S²) →	π_k+3(S³) →	π_k+4(S⁴) →	π_k+5(S⁵) →	π_k+6(S⁶) →	π_k+7(S⁷) →	π_k+8(S⁸) →	π_k+9(S⁹) →	π_k+10(S¹⁰) →	π_k+11(S¹¹) →	π_k+12(S¹²) →	π_k+13(S¹³) →	π_k+14(S¹⁴) →	π_k+15(S¹⁵) →	π_k+16(S¹⁶) →	π_k+17(S¹⁷) →	π_k+18(S¹⁸) →	π_k+19(S¹⁹) →	π_k+20(S²⁰) →	π_k^S
k=0	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty	infty
k=1	1	infty	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
k=2	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
k=3	1	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	1	4+3	2	2²	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
k=5	1	2	2	2²	2	infty	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
k=6	1	2	3	8+3+3	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
k=7	1	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	1	3+5	2	2	2	8+2+3	2³	2⁴	2²	2²	2²	2²	2²	2²	2²	2²	2²	2²	2²	2²	2²
k=9	1	2	2²	2³	2³	2³	2⁴	2⁵	2⁴	infty+2³	2³	2³	2³	2³	2³	2³	2³	2³	2³	2³	2³
k=10	1	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	1	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	1	4+2²+3+7	2²	2⁶	2³	16+3+5	1	1	1	4+3	2	2²	2	1	1	1	1	1	1	1	1
k=13	1	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	1	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	1	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	1	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	1	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	1	4+2²+3	4+2²+3	8+4+2⁵+3²+5	8+2²+3	8+2²+3²	8+2²+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	1	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 πn+k(Sn)

πn+k(Sn)

Table of the homotopy groups π_n+k(Sⁿ)

π_n+k(Sⁿ)