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 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 original computation via Pontryagin's theorem in cobordism theory:
with a more comprehensive account in:
See also:
Review:
Daniel Freed, Karen Uhlenbeck, Appendix B of: Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)
Guozhen Wang, Zhouli Xu, Section 2.3 of: A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)
Andrew Putman, Section 5 of: Homotopy groups of spheres and low-dimensional topology (pdf, pdf)
Discussion in homotopy type theory:
Guillaume Brunerie: On the homotopy groups of spheres in homotopy type theory [arxiv:1606.05916]
Axel Ljungström, The Brunerie Number Is -2 (June 2022) [blog entry]
Last revised on April 21, 2024 at 08:48:06. See the history of this page for a list of all contributions to it.