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In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s. Its
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
is denoted by MU. It is an exceptionally powerful
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or
Morava K-theory In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is su ...
, that are easier to compute. The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum.


Spectrum of complex cobordism

The complex bordism MU^*(X) of a space X is roughly the group of bordism classes of manifolds over X with a complex linear structure on the stable
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemann ...
. Complex bordism is a generalized
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...
, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows. The space MU(n) is the Thom space of the universal n-plane bundle over the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
BU(n) of the unitary group U(n). The natural inclusion from U(n) into U(n+1) induces a map from the double suspension \Sigma^2MU(n) to MU(n+1). Together these maps give the spectrum MU; namely, it is the homotopy colimit of MU(n). Examples: MU(0) is the sphere spectrum. MU(1) is the
desuspension In topology, a field within mathematics, desuspension is an operation inverse to suspension. Definition In general, given an ''n''-dimensional space X, the suspension \Sigma has dimension ''n'' + 1. Thus, the operation of suspension crea ...
\Sigma^ \mathbb^\infty of \mathbb^\infty. The
nilpotence theorem In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, ca ...
states that, for any ring spectrum R, the kernel of \pi_* R \to \operatorname_*(R) consists of nilpotent elements.http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf The theorem implies in particular that, if \mathbb is the sphere spectrum, then for any n>0, every element of \pi_n \mathbb is nilpotent (a theorem of Goro Nishida). (Proof: if x is in \pi_n S, then x is a torsion but its image in \operatorname_*(\mathbb) \simeq L, the
Lazard ring In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universa ...
, cannot be torsion since L is a polynomial ring. Thus, x must be in the kernel.)


Formal group laws

and showed that the coefficient ring \pi_*(\operatorname) (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring \Z _1,x_2,\ldots/math> on infinitely many generators x_i \in \pi_(\operatorname) of positive even degrees. Write \mathbb^ 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 \mu : \mathbb^ \times \mathbb^\to \mathbb^. A complex orientation on an associative commutative ring spectrum ''E'' is an element ''x'' in E^2(\mathbb^) whose restriction to E^2(\mathbb^) 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. If ''E'' is a complex oriented ring spectrum, then :E^*(\mathbb^\infty) = E^*(\text) x :E^*(\mathbb^\infty)\times E^*(\mathbb^\infty) = E^*(\text) x\otimes1, 1\otimes x and \mu^*(x) \in E^*(\text) x\otimes 1, 1\otimes x is a formal group law over the ring E^*(\text) = \pi^*(E). Complex cobordism has a natural complex orientation. 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.


Brown–Peterson cohomology

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 MU''p'' of MU at a prime ''p'' splits as a sum of suspensions of a simpler cohomology theory called Brown–Peterson cohomology, first described by . 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 \operatorname^*(BU) is isomorphic to the formal power series ring \operatorname^*(\text) cf_1, cf_2, \ldots where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by . Similarly \operatorname_*(BU) is isomorphic to the polynomial ring \operatorname_*(\text) \beta_1, \beta_2, \ldots


Cohomology operations

The Hopf algebra MU*(MU) is isomorphic to the polynomial algebra R 1, b2, ... where R is the reduced bordism ring of a 0-sphere. The coproduct is given by :\psi(b_k) = \sum_(b)_^\otimes b_j where the notation ()2''i'' means take the piece of degree 2''i''. This can be interpreted as follows. The map : x\to x+b_1x^2+b_2x^3+\cdots 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.


See also

* Adams–Novikov spectral sequence * List of cohomology theories *
Algebraic cobordism In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by . An oriented cohomology theory on the category of smooth quasi-projective schemes Sm over a field '' ...


Notes


References

* * *. * * * *. Translation of * *. * * * * * *


External links


Complex bordism
at the manifold atlas *{{nlab, id=cobordism+cohomology+theory, title=cobordism cohomology theory Algebraic topology