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 second stable homotopy group of spheres (the second stable stem) is the cyclic group of order 2:
Pontryagin had announced the computation in:
The explanation of the proof strategy via Pontryagin's theorem in cobordism theory appears in
but there was a mistake in the proof left, fixed in Pontryagin 50 and, independently and by different means, in Whitehead 50:
Lev Pontryagin, Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 (pdf)
George Whitehead, The $(n+2)$nd Homotopy Group of the $n$-Sphere, Annals of Mathematics Second Series, Vol. 52, No. 2 (Sep., 1950), pp. 245-247 (jstor:1969466)
A more comprehensive account of the computation and the cobordism theory behind it (Pontryagin's theorem) was then given in:
These historical references are also listed, with brief commentary, in the first part of:
For more historical commentary see p. 6 of Hopkins&Singer‘s Quadratic Functions in Geometry, Topology, and M-Theory Hopkins’s talk at Atiyah’s 80th Birthday conference, slide 8, 9:45.
Review:
Guozhen Wang, Zhouli Xu, Section 2.5 of: A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)
Andrew Putman, Section 10 of: Homotopy groups of spheres and low-dimensional topology (pdf, pdf)
See also:
Last revised on June 23, 2023 at 10:39:42. See the history of this page for a list of all contributions to it.