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cobordism cohomology theory

Contents

Contents

Idea

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 (MGM G for GG the trivial group) is stable cohomotopy (by the Pontryagin-Thom theorem).

See at those entries for more.

Examples

flavors of cobordism homology/cohomology theories and representing Thom spectra

bordism theoryMB\,M B (B-bordism):

equivariant bordism theory:

global equivariant bordism theory:


chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References

General

For complex cobordism theory see the references there.

Original articles include

  • John Milnor, On the cobordism ring ­Ω \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:

Textbook accounts:

The twisted and equivariant versions:

Relation to divisors

Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:

Last revised on November 25, 2020 at 10:08:21. See the history of this page for a list of all contributions to it.