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third stable homotopy group of spheres

Contents

Context

Stable Homotopy theory

Cobordism theory

Contents

Idea

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

Properties

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:

Proposition

(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).

References

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)

Review:

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

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)

Last revised on February 7, 2021 at 10:55:18. See the history of this page for a list of all contributions to it.