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 concept of cobordism sits at a subtle connection between differential topology/differential geometry and stable homotopy theory/higher category theory, this is the content of what is often called “cobordism theory”.
The insight goes back to the seminal thesis (Thom 54), which established that the Pontryagin-Thom construction exhibits the cobordism ring, whose elements are cobordism equivalence classes of manifolds, as the homotopy groups of a spectrum, whence called the Thom spectrum.
Via the Brown representability theorem, the full Thom spectrum represents a generalized cohomology theory, whence called cobordism cohomology theory. For every kind of topological structure carried by manifolds this has a variant, such as notably complex cobordism cohomology theory. These Thom spectra and their cobordism cohomology theories play a special role in stable homotopy theory, for instance for the concepts of orientation in generalized cohomology and the concept of genus.
A more recent refinement of this statement is (the proof of) the cobordism hypothesis which identified the (framed) (∞,n)-category of cobordisms as the free symmetric monoidal (∞,n)-category with duals on a single object.
For more introduction see at Introduction to Cobordism and Complex Oriented Cohomology.
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:
Original articles:
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, pdf)
Michael Atiyah, Thom complexes, Proc. London Math. Soc. (3), 11:291–310, 1961. 10
Textbook accounts:
Robert Stong, Notes on cobordism theory , Princeton Univ. Press (1968) (pdf)
Sergei Novikov, Section 4.3 in: Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)
Stanley Kochman, chapters I and IV of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)
Lecture notes include
Tom Weston, An introduction to cobordism theory (pdf)
John Francis, Topology of manifolds course notes (2010) (web), Lecture 2 Cobordisms (notes by Owen Gwilliam) (pdf), Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)
Cary Malkiewich, Unoriented cobordism and $M O$, 2011 (pdf)
Johannes Ebert, A lecture course on Cobordism Theory, 2012 (pdf)
This one here includes the connection to the (infinity,n)-category of cobordisms
For complex cobordism cohomology see there and see
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