View source for Complex cobordism

In mathematics, '''complex cobordism''' is a [[generalized cohomology theory]] related to [[cobordism]] of [[manifold]]s. Its [[Spectrum (homotopy theory)|spectrum]] is denoted by MU. It is an exceptionally powerful [[cohomology]] 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]], that are easier to compute.

The generalized homology and cohomology complex cobordism theories were introduced by {{harvs|txt|last=Atiyah|first=Michael|authorlink=Michael Atiyah|year=1961}} using the [[Thom spectrum]].

==Spectrum of complex cobordism==

The complex bordism <math>MU^*(X)</math> of a space <math>X</math> is roughly the group of bordism classes of manifolds over <math>X</math> 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 space]]s as follows.

The space <math>MU(n)</math> is the [[Thom space]] of the universal <math>n</math>-plane bundle over the [[classifying space]] <math>BU(n)</math> of the [[unitary group]] <math>U(n)</math>. The natural inclusion from <math>U(n)</math> into <math>U(n+1)</math> induces a map from the double [[Suspension (topology)|suspension]] <math>\Sigma^2MU(n)</math> to <math>MU(n+1)</math>. Together these maps give the spectrum <math>MU</math>; namely, it is the [[homotopy colimit]] of <math>MU(n)</math>.

Examples: <math>MU(0)</math> is the sphere spectrum. <math>MU(1)</math> is the [[desuspension]] <math>\Sigma^{\infty -2} \mathbb{CP}^\infty</math> of <math>\mathbb{CP}^\infty</math>.

The [[nilpotence theorem]] states that, for any [[ring spectrum]] <math>R</math>, the kernel of <math>\pi_* R \to \operatorname{MU}_*(R)</math> consists of nilpotent elements.<ref>{{citation|url=http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf|first=Jacob|last=Lurie|author-link=Jacob Lurie|publisher=Harvard University|title=The Nilpotence Theorem (Lecture 25)|date=April 27, 2010|work=252x notes}}</ref> The theorem implies in particular that, if <math>\mathbb{S}</math> is the sphere spectrum, then for any <math>n>0</math>, every element of <math>\pi_n \mathbb{S}</math> is nilpotent (a theorem of [[Goro Nishida]]). (Proof: if <math>x</math> is in <math>\pi_n S</math>, then <math>x</math> is a torsion but its image in <math>\operatorname{MU}_*(\mathbb{S}) \simeq L</math>, the [[Lazard ring]], cannot be torsion since <math>L</math> is a polynomial ring. Thus, <math>x</math> must be in the kernel.)

==Formal group laws==
{{harvs|txt|last=Milnor|first=John|authorlink=John Milnor|year=1960}} and {{harvs|txt=yes|last=Novikov|first=Sergei|authorlink=Sergei Novikov (mathematician)|year1=1960|year2=1962}} showed that the coefficient ring <math>\pi_*(\operatorname{MU})</math> (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring <math>\Z[x_1,x_2,\ldots]</math> on infinitely many generators <math>x_i \in \pi_{2i}(\operatorname{MU})</math> of positive even degrees.

Write <math>\mathbb{CP}^{\infty}</math> 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 <math>\mu : \mathbb{CP}^{\infty} \times \mathbb{CP}^{\infty}\to \mathbb{CP}^{\infty}.</math> A '''complex orientation''' on an associative [[commutative ring spectrum]] ''E'' is an element ''x'' in <math>E^2(\mathbb{CP}^{\infty})</math> whose restriction to <math>E^2(\mathbb{CP}^{1})</math> 
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

:<math>E^*(\mathbb{CP}^\infty) = E^*(\text{point})[[x]]</math>

:<math>E^*(\mathbb{CP}^\infty)\times E^*(\mathbb{CP}^\infty) = E^*(\text{point})[[x\otimes1, 1\otimes x]]</math>

and <math>\mu^*(x) \in E^*(\text{point})[[x\otimes 1, 1\otimes x]]</math> is a [[formal group law]] over the ring <math>E^*(\text{point}) = \pi^*(E)</math>.

Complex cobordism has a natural complex orientation. {{harvs|txt|last=Quillen|first=Daniel|authorlink=Daniel Quillen|year=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<sup>*</sup>(point) to ''R'' such that ''F'' is the pullback of the formal group law of complex cobordism.

{{See also|complex-orientable cohomology theory}}

==Brown&ndash;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<sub>''p''</sub> of MU at a prime ''p'' splits as a sum of suspensions of a simpler cohomology theory called [[Brown&ndash;Peterson cohomology]], first described by {{harvtxt|Brown|Peterson|1966}}. In practice one often does calculations with Brown&ndash;Peterson cohomology rather than with complex cobordism. Knowledge of the Brown&ndash;Peterson cohomologies of a space for all primes ''p'' is roughly equivalent to knowledge of its complex cobordism.

==Conner–Floyd classes==

The ring <math>\operatorname{MU}^*(BU)</math> is isomorphic to the [[formal power series ring]] <math>\operatorname{MU}^*(\text{point})[[cf_1, cf_2, \ldots]]</math> where the elements cf are called Conner–Floyd classes. They are the analogues of Chern classes for complex cobordism. They were introduced by {{harvtxt|Conner|Floyd|1966}}.

Similarly <math>\operatorname{MU}_*(BU)</math> is isomorphic to the polynomial ring <math>\operatorname{MU}_*(\text{point})[[\beta_1, \beta_2, \ldots]]</math>

==Cohomology operations==

The [[Hopf algebra]] MU<sub>*</sub>(MU) is isomorphic to the polynomial algebra R[b<sub>1</sub>, b<sub>2</sub>, ...], where R is the reduced bordism ring of a 0-sphere.

The coproduct is given by

:<math>\psi(b_k) = \sum_{i+j=k}(b)_{2i}^{j+1}\otimes b_j</math>

where the notation ()<sub>2''i''</sub> means take the piece of degree 2''i''. This can be interpreted as follows. The map

:<math> x\to x+b_1x^2+b_2x^3+\cdots</math>

is a continuous automorphism of the [[ring of formal power series]] in ''x'', and the coproduct of MU<sub>*</sub>(MU) gives the composition of two such automorphisms.

==See also==

*[[Adams&ndash;Novikov spectral sequence]]
*[[List of cohomology theories]]
*[[Algebraic cobordism]]

== Notes ==
{{reflist}}

==References==

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*{{cite arXiv | last1=Morava | first1=Jack | author-link=Jack Morava| title=Complex cobordism and algebraic topology | eprint=0707.3216 | year=2007 | class=math.HO}}
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*{{springer|id=C/c022780|title=Cobordism|first=Yuli B.|last= Rudyak}}
*{{citation|first=Robert E.|last= Stong|author-link=Robert Evert Stong| title=Notes on cobordism theory|publisher= [[Princeton University Press]] |year=1968}}
*{{citation|mr=0061823|author-link=René Thom|last=Thom|first=René|title=Quelques propriétés globales des variétés différentiables|journal=[[Commentarii Mathematici Helvetici]]|volume=28|year=1954|pages=17–86|url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259|doi=10.1007/BF02566923|s2cid=120243638 }}

==External links==
*[https://archive.today/20121217235613/http://www.map.him.uni-bonn.de/Complex_bordism Complex bordism] at the manifold atlas
*{{nlab|id=cobordism+cohomology+theory|title=cobordism cohomology theory}}

[[Category:Algebraic topology]]