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:
This entry is about the articles
Classification of continuous maps of a complex into a sphere, Communication I
Doklady Akademii Nauk SSSR 19(3) (1938), 147-149
Classification of continuous maps of a complex into a sphere,
Communication II
Doklady Akademii Nauk SSSR 19(5) (1938), 361-363
(this article contains a famous mistake, see also p. 6 of Hopkins, Singer‘s Quadratic Functions in Geometry, Topology, and M-Theory and Hopkins’s talk at Atiyah’s 80th Birthday conference, slide 8, 9:45)
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)
(this article fixes the mistake)
(all three of which are available in English translation in Gamkrelidze 86)
or rather about their joint comprehensive exposition in
Smooth manifolds and their applications in homotopy theory
Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955
AMS Translation Series 2, Vol. 11, 1959
doi:10.1142/9789812772107_0001
pdf)
on differential topology, establishing the Pontryagin isomorphism between unstable Cohomotopy and cobordism classes of normally framed submanifolds, and applying it to the computation of the lowest couple of stable homotopy groups of spheres (the first and the second).
Together with
due to which Pontryagin’s construction came to be mainly known as the Pontryagin-Thom construction, this lays the foundations of cobordism theory as such and as a tool in stable homotopy theory.
Last revised on February 4, 2021 at 17:03:41. See the history of this page for a list of all contributions to it.