cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
normally framed submanifolds$\leftrightarrow$ Cohomotopy
normally oriented submanifolds$\leftrightarrow$ maps to Thom space
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\,M B$ (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
The universal Thom spectrum (see there for more) of the orthogonal group. (…) Abstractly, this is the homotopy colimit of the J-homomorphism in Spectra:
By Thom's theorem the stable homotopy groups of $M O$ form the bordism ring of unoriented manifolds
Moreover, this is the polynomial algebra
Due to (Thom 54). See for instance (Kochman 96, theorem 3.7.6)
The corresponding statement for MU is considerably more subtle, see Milnor-Quillen theorem on MU.
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\,M B$ (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86
Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)
Textbook accounts:
Robert Stong, Chapter VI of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)
Stanley Kochman, section 1.5 and section 3.7 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Review:
Discussion of MO-bordism with MSO-boundaries:
In the incarnation of $MO$ as a symmetric spectrum:
In the incarnation as an orthogonal spectrum (in fact as an equivariant spectrum in global equivariant stable homotopy theory):
Last revised on January 25, 2021 at 10:25:55. See the history of this page for a list of all contributions to it.