The generalized homology and cohomology complex cobordism theories were introduced by
Michael Atiyah (
1961) using the
Thom spectrum.
Spectrum of complex cobordism
The complex bordism of a space is roughly the group of bordism classes of manifolds over with a complex linear structure on the stable
normal bundle. Complex bordism is a generalized
homology theory, corresponding to a spectrum MU that can be described explicitly in terms of
Thom spaces as follows.
The space is the
Thom space of the universal -plane bundle over the
classifying space of the
unitary group. The natural inclusion from into induces a map from the double
suspension to . Together these maps give the spectrum ; namely, it is the
homotopy colimit of .
Examples: is the sphere spectrum. is the
desuspension of .
The
nilpotence theorem states that, for any
ring spectrum, the kernel of consists of nilpotent elements.[1] The theorem implies in particular that, if is the sphere spectrum, then for any , every element of is nilpotent (a theorem of
Goro Nishida). (Proof: if is in , then is a torsion but its image in , the
Lazard ring, cannot be torsion since is a polynomial ring. Thus, must be in the kernel.)
Formal group laws
John Milnor (
1960) and
Sergei Novikov (
1960,
1962) showed that the coefficient ring (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring on infinitely many generators of positive even degrees.
Write for infinite dimensional
complex projective space, which is the classifying space for complex line bundles, so that tensor product of line bundles induces a map A complex orientation on an associative
commutative ring spectrumE is an element x in whose restriction to
is 1, if the latter ring is identified with the coefficient ring of E. A spectrum E with such an element x is called a complex oriented ring spectrum.
Complex cobordism has a natural complex orientation.
Daniel Quillen (
1969) showed that there is a natural isomorphism from its coefficient ring to
Lazard's universal ring, making the formal group law of complex cobordism into the universal formal group law. In other words, for any formal group law F over any commutative ring R, there is a unique ring homomorphism from MU*(point) to R such that F is the pullback of the formal group law of complex cobordism.
Complex cobordism over the rationals can be reduced to ordinary cohomology over the rationals, so the main interest is in the torsion of complex cobordism. It is often easier to study the torsion one prime at a time by localizing MU at a prime p; roughly speaking this means one kills off torsion prime to p. The localization MUp of MU at a prime p splits as a sum of suspensions of a simpler cohomology theory called
Brown–Peterson cohomology, first described by
Brown & Peterson (1966). In practice one often does calculations with Brown–Peterson cohomology rather than with complex cobordism. Knowledge of the Brown–Peterson cohomologies of a space for all primes p is roughly equivalent to knowledge of its complex cobordism.
Conner–Floyd classes
The ring is isomorphic to the
formal power series ring where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by
Conner & Floyd (1966).
Similarly is isomorphic to the polynomial ring
Cohomology operations
The
Hopf algebra MU*(MU) is isomorphic to the polynomial algebra R[b1, b2, ...], where R is the reduced bordism ring of a 0-sphere.
The coproduct is given by
where the notation ()2i means take the piece of degree 2i. This can be interpreted as follows. The map
is a continuous automorphism of the
ring of formal power series in x, and the coproduct of MU*(MU) gives the composition of two such automorphisms.
Novikov, Sergei P. (1960), "Some problems in the topology of manifolds connected with the theory of Thom spaces", Soviet Math. Dokl., 1: 717–720. Translation of "О некоторых задачах топологии многообразий, связанных с теорией пространств Тома", Doklady Akademii Nauk SSSR, 132 (5): 1031–1034,
MR0121815,
Zbl0094.35902.
Novikov, Sergei P. (1962), "Homotopy properties of Thom complexes. (Russian)", Mat. Sb., New Series, 57: 407–442,
MR0157381