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.
See at those entries for more.
|chromatic level||complex oriented cohomology theory||E-∞ ring/A-∞ ring||real oriented cohomology theory|
|0||ordinary cohomology||Eilenberg-MacLane spectrum||HZR-theory|
|0th Morava K-theory|
|1||complex K-theory||complex K-theory spectrum||KR-theory|
|first Morava K-theory|
|first Morava E-theory|
|2||elliptic cohomology||elliptic spectrum|
|second Morava K-theory|
|second Morava E-theory|
|algebraic K-theory of KU|
|3 …10||K3 cohomology||K3 spectrum|
|th Morava K-theory|
|th Morava E-theory||BPR-theory|
|algebraic K-theory applied to chrom. level||(red-shift conjecture)|
|complex cobordism cohomology||MU||MR-theory|
Original articles include
John Milnor, On the cobordism ring 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.