quaternionic projective line$\,\mathbb{H}P^1$
homotopy theory, (β,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directedβ¦
models: topological, simplicial, localic, β¦
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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)
With all due respect to anyone interested in them, the stable homotopy groups of spheres are a mess. (J. F. Adams 74, p 204)
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 βstable $k$-stemβ)
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 β Graphical representation for details on the notation used in these table, and see example below for illustration.
(the following graphics are taken from Hatcher, 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 $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240}$ 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.
See at Hopf degree theorem.
The first stable homotopy group of spheres (the first stable stem) is the cyclic group of order 2:
where the generator $[1] \in \mathbb{Z}/2$ is represented by the complex Hopf fibration $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$.
Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ of stably framed manifolds (see at MFr), the generator (1) is represented by the 1-sphere (with its left-invariant framing induced from the identification with the Lie group U(1))
Moreover, the relation $2 \cdot [S^1_{Lie}] \,\simeq\, 0$ is represented by the bordism which is the complement of 2 open balls inside the 2-sphere.
The second stable homotopy group of spheresβ¦
The third stable homotopy group of spheres (the third stable stem) is the cyclic group of order 24:
where the generator $[1] \in \mathbb{Z}/24$ is represented by the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.
Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ of stably framed manifolds (see at MFr), the generator (2) is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) $\simeq$ Sp(1) )
Moreover, the relation $24 \cdot [S^3_{Lie}] \,\simeq\, 0$ is represented by the bordism which is the complement of 24 open balls inside the K3-manifold (e.g. Wang-Xu 10, Sec. 2.6, Bauer 10, SP 17).
Equivalently, the elements of $\pi_3^s \,\simeq\, \Omega^{fr}_3$ are detected by half the Todd classes of cobounding manifolds with special unitary group-tangential structure on their stable tangent bundle (elements of the MSUFr-bordism ring):
We have the following short exact sequence of the MSU-, MSUFr- and MFr-bordism rings (Conner-Floyd 66, p. 104)
which produces from half the Todd class of cobounding $(SU,fr)$-manifolds the KO-theoretic Adams e-invariant $e_{\mathbb{R}}$ (Adams 66, p. 39) of the boundary manifold in $\Omega^{fr}_{8k + 3} \simeq \pi^s_{8k+3}$. For $k = 0$ this detects the third stable homotopy group of spheres, by the following:
(Adams 66, Example 7.17 and p. 46)
In degree 3, the KO-theoretic e-invariant $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ and hence reflects the full third stable homotopy group of spheres:
while $e_{\mathbb{C}}$ sees only βhalfβ of it (by Adams 66, Prop. 7.14).
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, see Ravenel 86, Chapter I, Lemma 1.1.8)
The Nishida nilpotence theoremβ¦
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 β 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 group | string group | fivebrane group | ninebrane group | |||||||||||||
higher versions | special orthogonal group | spin group | string 2-group | fivebrane 6-group | ninebrane 10-group | |||||||||||||
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 Wright, 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, pdf)
Andrew Putman, Homotopy groups of spheres and low-dimensional topology ([pdf]https://www3.nd.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf), pdf)
Allen Hatcher, Pictures of stable homotopy groups of spheres (html)
Mark Mahowald, Doug Ravenel, Towards a Global Understanding of the Homotopy Groups of Spheres, in: Samuel Gitler (ed.): The Lefschetz Centennial Conference: Proceedings on Algebraic Topology II, Contemporary Mathematics volume 58, AMS 1987 (pdf, pdf, ISBN:978-0-8218-5063-3)
Haynes Miller, Doug Ravenel, Mark Mahowaldβs 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 β A Computer-Assisted Approach, Lecture Notes in Mathematics, 1990
Doug Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies 128, Princeton University Press (1992) [ISBN:9780691025728, pdf, webpage]
Stanley Kochmann, section 5 of of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Daniel Isaksen, Guozhen Wang, Zhouli Xu, Stable homotopy groups of spheres, PNAS October 6, 2020 117 (40) 24757-2476 (arXiv:2001.04247, doi:10.1073/pnas.2012335117)
Daniel Isaksen, Guozhen Wang, Zhouli Xu, More stable stems (arXiv:2001.04511)
A tabulation of stable homotopy groups of spheres is in
Original articles on basic properties:
Early computation of unstable homotopy groups of spheres $\pi_{n+k}(S^k)$ up to $n\leq 19$:
See also
Wikipedia, Homotopy groups of spheres
MO, Computational complexity of computing homotopy groups of spheres
Prasit Bhattacharya, Irina Bobkova, J.D. Quigley. New infinite families in the stable homotopy groups of spheres (2024). (arXiv:2404.10062).
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: 67β80 (1971)
For formalization in homotopy type theory see
formalization of the homotopy groups of spheres in homotopy type theory
Guillaume Brunerie, On the homotopy groups of spheres in homotopy type theory (arXiv:1606.05916)
Last revised on April 30, 2024 at 18:48:00. See the history of this page for a list of all contributions to it.