nLab homotopy groups of spheres

Theorems

Stable 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 $\pi_{n+k}(S^n)$ are the homotopy classes of maps $S^{n+k} \longrightarrow S^n$

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

For fixed $k$, the colimit over $n$ with respect to the suspension homomorphism

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

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

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

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

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

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

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

$k =$0123456789101112131415$\cdots$
$\pi_k(\mathbb{S}) =$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$$0$$0$$\mathbb{Z}_2$$\mathbb{Z}_{240}$$(\mathbb{Z}_2)^2$$(\mathbb{Z}_2)^3$$\mathbb{Z}_6$$\mathbb{Z}_{504}$$0$$\mathbb{Z}_3$$(\mathbb{Z}_2)^2$$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$$\cdots$

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

The horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a direct summand primary group of order $p^k$. The bottom rows in each case are given by the image of the J-homomorphism. See example 1 below for illustration (graphics taken from Hatcher’s website).

$p = 2$-primary component

$p = 3$-primary component

$p = 5$-primary component

Example

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

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

Properties

Basic properties

Theorem

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

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

for $F_m$ a finite group.

(Serre 53)

The following characterizes the image of the J-homomorphism

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

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

$n\;mod\; 8$01234567
$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$

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

$n\;mod\; 8$01234567
${\vert\pi_n(O)\vert}$221$\infty$111$\infty$
Theorem

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

Moreover,

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

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

• for all other cases the image is necessarily zero.

$n$012345678910111213141516
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\mathbb{Z}_{240}$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_6$$\mathbb{Z}_{504}$0$\mathbb{Z}_3$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\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.

Image of the J-homomorphism

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

• John Adams, On the groups $J(X)$ 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 May 13, 2016 14:22:48 by Urs Schreiber (131.220.184.222)