cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
flavors of cobordism homology/cohomology theories and representing Thom spectra
In the context of cobordism theory, a generalized cohomology theory represented by a Thom spectrum is called a cobordism cohomology theory. (Dually, the corresponding generalized homology theory is called bordism homology theory.)
By default “cobordism cohomology” usually refers to what is represented by MO. The cohomology represented by MU is complex cobordism cohomology. Both are unified by the equivariant cohomology theory called MR-theory. The periodic cohomology theory version is denoted MP.
On the other hand, framed cobordism cohomology theory ($M G$ for $G$ the trivial group) is stable cohomotopy (by the Pontryagin-Thom theorem).
See at those entries for more.
flavors of cobordism homology/cohomology theories and representing Thom spectra
bordism theory$\,M B$ (B-bordism):
the refinement of cobordism cohomology theory to differential cohomology is differential cobordism cohomology.
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
For complex cobordism theory see the references there.
Original articles include
John Milnor, On the cobordism ring $\Omega^\bullet$ and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
Sergei Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).
Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)
Survey:
Victor Buchstaber, Cobordisms in problems of algebraic topology, J Math Sci 7, 629–653 (1977) (doi:10.1007/BF01084983)
Peter Landweber, A survey of bordism and cobordism, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 2 September 1986 , pp. 207-223 (doi:10.1017/S0305004100066032)
Textbook accounts:
Robert Stong, Notes on Cobordism theory, 1968 (toc pdf, publisher page)
Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)
The twisted and equivariant versions:
James Cruickshank, Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)
James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (arXiv:10.1016/S0166-8641(02)00183-9)
Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:
Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467-493 (doi:10.1090/S0894-0347-97-00232-4)
Last revised on November 25, 2020 at 10:08:21. See the history of this page for a list of all contributions to it.