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)
The homotopy groups of spheres $\pi_{n+k}(S^n)$ are the homotopy classes of maps $S^{n+k} \longrightarrow S^n$
For fixed $k$, the colimit over $n$ with respect to the suspension homomorphism
over all $\pi_{n+k}(S^n)$ (called the $k$-stem) is called the stable homotopy groups of spheres (also: the “stable $k$-stem”)
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 $\mathbb{S}$
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.
The first few stable homotopy groups of the sphere spectrum $\mathbb{S}$ are
$k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$\pi_k(\mathbb{S}) =$ | $\mathbb{Z}$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $\mathbb{Z}/{24}$ | $0$ | $0$ | $\mathbb{Z}/2$ | $\mathbb{Z}/{240}$ | $(\mathbb{Z}/2)^2$ | $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/6$ | $\mathbb{Z}/{504}$ | $0$ | $\mathbb{Z}/3$ | $(\mathbb{Z}/2)^2$ | $\mathbb{Z}/{480} \oplus \mathbb{Z}/2$ | $\cdots$ |
The following tables show the p-primary decomposition of these and the following stable homotopy groups.
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 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)
$p = 2$-primary component
$p = 3$-primary component
$p = 5$-primary component
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.
The homotopy group $\pi_{n+k}(S^k)$ is a finite group except
for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$;
$k = 2m$ and $n = 2m -1$ in which case
for $F_m$ a finite group.
(Serre 53)
The Nishida nilpotence theorem…
By Thom's theorem, for any (B,f)-structure $\mathcal{B}$, there is an isomorphism (of commutative rings)
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 $\mathcal{B} = Fr$ framing structure, then
is equivalently the sphere spectrum. Hence in this case Thom's theorem states that there is an isomorphism
between the framed cobordism ring and the stable homotopy groups of spheres.
For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance (Wang-Xu 10, section 2) and (Putnam).
For instance
$\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, 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;
$\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.
The following characterizes the image of the J-homomorphism
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).
By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
$\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |
Because all groups appearing here and in the following are cyclic groups, we instead write down the order
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |
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.
$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Whitehead tower of orthogonal group | orientation | spin | string | fivebrane | ninebrane | |||||||||||||
homotopy groups of stable orthogonal group | $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ |
stable homotopy groups of spheres | $\pi_n(\mathbb{S})$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | 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 | $\mathbb{Z}_{24}$ | 0 | 0 | 0 | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{504}$ | 0 | 0 | 0 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |
Introductions and surveys include
Alex Writght, Homotopy groups of spheres: A very basic introduction (pdf)
Guozhen Wang, Zhouli Xu A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)
Andrew Putman, Homotopy groups of spheres and low-dimensional topology (pdf)
Alan Hatcher, Stable homotopy groups of spheres (html)
Mark Mahowald, Doug Ravenel, Towards a Global Understanding of the Homotopy Groups of Spheres (pdf)
Haynes Miller, Doug Ravenel, Mark Mahowald’s work on the homotopy groups of spheres (pdf)
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, since 1986 (web)
Stanley Kochmann, Stable Homotopy Groups of Spheres – A Computer-Assisted Approach, Lecture Notes in Mathematics, 1990
Stanley Kochmann, section 5 of of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
A tabulation of stable homotopy groups of spheres is in
Original articles on basic properties include
See also
Wikipedia, Homotopy groups of spheres
MO, Computational complexity of computing homotopy groups of spheres
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)
For formalization in homotopy type theory see at
Guillaume Brunerie, On the homotopy groups of spheres in homotopy type theory (arXiv:1606.05916)