cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
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 (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) )
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 “half” of it (by Adams 66, Prop. 7.14).
The original computation:
with a mistake (in the unstable range) corrected in
French translations are in:
Review:
More on the computation via the framed cobordism ring and the K3-manifold giving the cobordism that witnesses the order of 24:
Tilman Bauer, answer to: third stable homotopy group of spheres via geometry?, 2010 (MO:a/44885)
Chris Schommer-Pries, answer to: Nilpotence of the stable Hopf map via framed cobordism, 2017 (MO:a/218053)
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:
Domenico Fiorenza, Eugenio Landi, Integrals detecting degree 3 string cobordism classes [arXiv:2209.12933]
Domenico Fiorenza, String bordism invariants in dimension 3 from $U(1)$-valued TQFTs, talk at QFT and Cobordism, CQTS (Mar 2023) [web]
Last revised on March 16, 2023 at 10:28:08. See the history of this page for a list of all contributions to it.