nLab
cobordism cohomology theory

Contents

Context

Cobordism theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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.

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

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. Sb. (N.S.) 57 (99) (1962), 407–442.

Textbook accounts include

The twisted and equivariant versions:

Last revised on June 19, 2019 at 04:32:16. See the history of this page for a list of all contributions to it.