cobordism cohomology theory


Cobordism theory



Special and general types

Special notions


Extra structure





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 MR-theory.

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


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

For complex cobordism theory see the references there.

Revised on February 18, 2016 08:17:29 by Urs Schreiber (