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homotopy groups of spheres

Context

Homotopy theory

Stable Homotopy theory

Homotopy groups of spheres

Idea

The homotopy groups of spheres

The stable homotopy groups of the sphere spectrum

Tables

The first stable homotopy groups of the sphere spectrum 𝕊\mathbb{S}

k=k =0123456789101112131415\cdots
π k(𝕊)=\pi_k(\mathbb{S}) = \mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}0000 2\mathbb{Z}_2 240\mathbb{Z}_{240}( 2) 2(\mathbb{Z}_2)^2( 2) 3(\mathbb{Z}_2)^3 6\mathbb{Z}_6 504\mathbb{Z}_{504}00 3\mathbb{Z}_3( 2) 2(\mathbb{Z}_2)^2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2\cdots

The following tables for the p-primary components of π \pi_\bullet in low degrees are taken from (Hatcher), where in turn they were generated based on (Ravenel 86).

The horizontal index is the degree nn of the stable homotopy group π n\pi_n. The appearance of a string of kk connected dots vertically above index nn means that there is a direct summand primary group of order p kp^k. The bottom rows in each case are given by the image of the J-homomorphism. See example 1 below for illustration.

p=2p = 2-primary component

stable homotopy groups of spheres at 2

p=3p = 3-primary component

stable homotopy groups of spheres at 3

p=5p = 5-primary component

stable homotopy groups of spheres at 5

Example

The finite abelian group π 3(𝕊) 24\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24} decomposes into primary groups as 8 3\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3. Here 8=2 38 = 2^3 corresponds to the three dots above n=3n = 3 in the first table, and 3=3 13 = 3^1 to the single dot over n=3n = 3 in the second.

The finite abelian group π 7(𝕊) 24\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{24} decomposes into primary groups as 16 3 5\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5. Here 16=2 416 = 2^4 corresponds to the four dots above n=7n = 7 in the first table, and 3=3 13 = 3^1 to the single dot over n=7n = 7 in the second and 5=5 15 = 5^1 to the single dot over n=7n = 7 in the third table.

Properties

Basic properties

The Serre finiteness theorem

Theorem

The homotopy group π n+k(S k)\pi_{n+k}(S^k) is a finite group for k>0k \gt 0 except when n=2mn = 2m and k=2m1k = 2m -1 in which case

π 4m1(S 2m)F m \pi_{4m -1 }(S^{2m}) \simeq \mathbb{Z} \oplus F_m

for F mF_m a finite group.

(Serre 53)

J-homomorphism and Adams e-invariant

The following characterizes the image of the J-homomorphism

J:π (O)π (𝕊) J \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})

from the homotopy groups of the stable orthogonal group to the stable homotopy groups of spheres. This was first conjectured in (Adams 66) (since called the Adams conjecture) and then proven in (Quillen 71).

Remark

By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are

nmod8n\;mod\; 801234567
π n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}

Because all groups appearing here and in the following are cyclic groups, we instead write down the order

nmod8n\;mod\; 801234567
π n(O){\vert\pi_n(O)\vert}221\infty111\infty
Theorem

The stable homotopy groups of spheres π n(𝕊)\pi_n(\mathbb{S}) are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant.

Moreover,

  • for n=0modn = 0 \;mod \; and n=1mod8n = 1 \;mod \; 8 and nn positive the J-homomorphism is injective, hence its image is 2\mathbb{Z}_2,

  • for n=3mod8n = 3\; mod\; 8 and n=7mod8n = 7 \; mod \; 8 hence for n=4k1n = 4 k -1, the order of the image is equal to the denominator of B 2k/4kB_{2k}/4k, where B 2kB_{2k} is the Bernoulli number

  • for all other cases the image is necessarily zero.

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

References

General

Introductions and surveys include

  • Alex Writght, Homotopy groups of spheres: A very basic introduction (pdf)

  • Alan Hatcher, Stable homotopy groups of spheres (html)

A tabulation of stable homotopy groups of spheres is in

Original articles on basic properties include

  • Jean-Pierre Serre _ Groupes d’homotopie et classes de groupes abelien_, Ann. of Math. 58 (1953), 258–294.

See also

Image of the J-homomorphism

Discussion of the image of the J-homomorphism is due to

  • John Adams, On the groups J(X)J(X) IV, Topology 5: 21,(1966) Correction, Topology 7 (3): 331 (1968)
  • Daniel Quillen, The Adams conjecture, Topology. an International Journal of Mathematics 10: 67–80 (1971)

Formalization in homotopy type theory

For formalization in homotopy type theory see at

Revised on November 18, 2013 01:51:50 by Urs Schreiber (89.204.137.133)