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homotopy groups of spheres

Context

Homotopy theory

Stable Homotopy theory

Homotopy groups of spheres


With all due respect to anyone interested in them, the stable homotopy groups of spheres are a mess. (J. F. Adams 74, p 204)

My initial inclination was to call this book The Music of the Spheres, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (D. Ravenel 86, preface)


Idea

The homotopy groups of spheres π n+k(S n)\pi_{n+k}(S^n) are the homotopy classes of maps S n+kS nS^{n+k} \longrightarrow S^n

π n+k(S n)[S n+k,S n]. \pi_{n+k}(S^n) \coloneqq [S^{n+k}, S^n] \,.

For fixed kk, the colimit over nn with respect to the suspension homomorphism

π n+k(S n)π n+k+1(S n+1) \pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})

over all π n+k(S n)\pi_{n+k}(S^n) (called the kk-stem) is called the stable homotopy groups of spheres (also: the “stable kk-stem”)

π k S=lim nπ n+k(S n). \pi_k^S = \coloneqq \underset{\longrightarrow}{\lim}_n \pi_{n+k}(S^n) \,.

In fact, by the Freudenthal suspension theorem, the value of the π n+k(S n)\pi_{n+k}(S^n) stabilizes for n>k+1n \gt k+1 (depend only on kk in this range), whence the name.

The stable homotopy groups of sphere are equivalently the homotopy groups of a spectrum for the sphere spectrum 𝕊\mathbb{S}

π k S=π k(𝕊). \pi_k^S = \pi_k(\mathbb{S}) \,.

The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of algebraic topology and stable homotopy theory were devised to compute more and more of the stable stems. This notably include the Adams spectral sequence, the Adams-Novikov spectral sequence.

Tables

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

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 show the p-primary decomposition of these and the following stable homotopy groups.

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 at fundamental theorem of finitely generated abelian grouops – Graphical representation for details on the notation used in these table, and see example 1 below for illustration.

(the following graphics are taken from Hatcher’s website, based on (Ravenel 86)

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

Serre finiteness theorem

Theorem

(Serre finiteness theorem)

The homotopy group π n+k(S k)\pi_{n+k}(S^k) is a finite group except

  1. for n=0n = 0 in which case π k(S k)=\pi_k(S^k) = \mathbb{Z};

  2. k=2mk = 2m and n=2m1n = 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)

Nishida nilpotence theorem

The Nishida nilpotence theorem

Relation to the framed bordism ring

By Thom's theorem, for any (B,f)-structure \mathcal{B}, there is an isomorphism (of commutative rings)

Ω π (M) \Omega^{\mathcal{B}}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(M\mathcal{B})

from the cobordism ring of manifolds with stable normal \mathcal{B}-structure to the homotopy groups of the universal \mathcal{B}-Thom spectrum.

Now for =Fr\mathcal{B} = Fr framing structure, then

MFr𝕊 M Fr \simeq \mathbb{S}

is equivalently the sphere spectrum. Hence in this case Thom's theorem states that there is an isomorphism

Ω frπ (𝕊) \Omega^{fr}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(\mathbb{S})

between the framed cobordism ring and the stable homotopy groups of spheres.

For discussion of computation of π (𝕊)\pi_\bullet(\mathbb{S}) this way, see for instance (Wang-Xu 10, section 2) and (Putnam).

For instance

  • Ω 0 fr=\Omega^{fr}_0 = \mathbb{Z} because there are two kk-framings on a single point, corresponding to π 0(O(k)) 2\pi_0(O(k)) \simeq \mathbb{Z}_2, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;

  • Ω 1 fr= 2\Omega^{fr}_1 = \mathbb{Z}_2 because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to π 1(O(k)) 2\pi_1(O(k)) \simeq \mathbb{Z}_2 and they are their own negatives.

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

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 August 21, 2016 12:52:19 by Anonymous Coward (45.78.13.229)