nLab third stable homotopy group of spheres



Stable Homotopy theory

Cobordism theory



The third stable homotopy group of spheres (the third stable stem) is the cyclic group of order 24:

(1)π 3 s /24 [h ] [1] \array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow& [1] }

where the generator [1]/24[1] \in \mathbb{Z}/24 is represented by the quaternionic Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4.

from SS21


As the third framed bordism group

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring Ω fr\Omega^{fr}_\bullet of stably framed manifolds (see at MFr), the generator (1) is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) \simeq Sp(1) )

π 3 s Ω 3 fr [h ] [S fr=1 3]. \array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3_{fr=1}] \,. }

Moreover, the relation 24[S Lie 3]024 \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).

Via the fourth (SU,fr)(SU,fr)-bordism group

Equivalently, the elements of π 3 sΩ 3 fr\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)

(2)0 Ω 8+4 SU i Ω 8+4 SU,fr Ω 8+3 fr π 8+3 s 12Td 12Td e 0 / = / \array{ 0 \to & \Omega^{SU}_{8\bullet+4} & \overset{i}{\longrightarrow} & \Omega^{SU,fr}_{8\bullet+4} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_{8\bullet + 3} & \simeq & \pi^s_{8\bullet+3} \\ & \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e_{\mathbb{R}}} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} }

which produces from half the Todd class of cobounding (SU,fr)(SU,fr)-manifolds the KO-theoretic Adams e-invariant e e_{\mathbb{R}} (Adams 66, p. 39) of the boundary manifold in Ω 8k+3 frπ 8k+3 s\Omega^{fr}_{8k + 3} \simeq \pi^s_{8k+3}. For k=0k = 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 e_{\mathbb{R}} takes the value [124]/\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z} on the quaternionic Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4 and hence reflects the full third stable homotopy group of spheres:

π 3 s e /24 / [h ] [124] \array{ \pi^s_3 & \underoverset{ \simeq }{ e_{\mathbb{R}} }{ \;\;\longrightarrow\;\; } & \mathbb{Z}/24 & \subset & \mathbb{Q}/\mathbb{Z} \\ [h_{\mathbb{H}}] &&\mapsto&& \left[\tfrac{1}{24}\right] }

while e e_{\mathbb{C}} sees only “half” of it (by Adams 66, Prop. 7.14).


The original computation:

  • Vladimir Abramovich Rokhlin, On a mapping of the (n+3)(n+3)-dimensional sphere into the nn-dimensional sphere, (Russian) Doklady Akad. Nauk SSSR (N.S.) 80, (1951). 541–544

with a mistake (in the unstable range) corrected in

  • Vladimir Abramovich Rokhlin, New results in the theory of four-dimensional manifolds, (Russian) Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.

French translations are in:

  • Lucien Guillou, Alexis Marin (eds.), A la Recherche de la Topologie Perdue: I. Du côté de chez Rohlin. II. Le côté de Casson, Progress in Mathematics 62, Birkhäuser Boston 1985 (ISBN:0817633294, 9780817633295)


  • Guozhen Wang, Zhouli Xu, Section 2.6 of: A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)

See also:

  • Mehmet Kirdar, On the First, the Second and the Third Stems of the Stable Homotopy Groups of Spheres [arXiv:2107.06103]

More on the computation via the framed cobordism ring and the K3-manifold giving the cobordism that witnesses the order of 24:

Via immersions of 3-spheres into Euclidean 4-space

  • A. Szűcs, Two Theorems of Rokhlin, Journal of Mathematical Sciences 113, 888–892 (2003) (doi:10.1023/A:1021208007146)

  • Tobias Ekholm, Masamichi Takase, Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions, Bulletin of the London Mathematical Society 43 (2011) 251–266 (arXiv:0903.0238)

Discussion of geometric string bordism in degreee 3 as a means to speak (via the Pontryagin-Thom theorem) about the third stable homotopy group of spheres:

Last revised on June 23, 2023 at 10:39:28. See the history of this page for a list of all contributions to it.