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| 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.
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| The generalized homology and cohomology complex cobordism theories were introduced by {{harvtxt|Atiyah|1961}} using the [[Thom spectrum]].
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| ==Spectrum of complex cobordism==
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| The complex bordism MU<sub>*</sub>(''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]]. 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.
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| The space MU(''n'') is the [[Thom space]] of the universal ''n''-plane bundle over the [[classifying space]] 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 (topology)|suspension]] S<sup>2</sup>MU(''n'') to MU(''n''+1). Together these maps give the spectrum MU; namely, it is the [[homotopy colimit]] of MU(''n'').
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| Examples: MU(0) is the sphere spectrum. MU(1) is the [[desuspension]] <math>\Sigma^{\infty -2} \mathbb{C}\mathbf{P}^\infty</math> of <math>\mathbb{C}\mathbf{P}^\infty</math>.
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| The [[nilpotence theorem]] states that, for any [[ring spectrum]] ''R'', the kernel of <math>\pi_* R \to \operatorname{MU}_*(R)</math> consists of nilpotent elements.<ref>http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf</ref> The theorem implies in particular that, if ''S'' is the sphere spectrum, then for any ''n'' > 0, every element of <math>\pi_n S</math> is nilpotent (a theorem of Nishida). (Proof: if ''x'' is in <math>\pi_n S</math>, then ''x'' is a torsion but its image in <math>\operatorname{MU}_*(S) \simeq L</math>, the [[Lazard ring]], cannot be torsion since ''L'' is a polynomial ring. Thus, ''x'' must be in the kernel.)
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| ==Formal group laws==
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| {{harvtxt|Milnor|1960}} and {{harvs|txt=yes|last=Novikov|year1=1960|year2=1962}} showed that the coefficient ring π<sub>*</sub>(MU) (equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring '''Z'''[''x''<sub>1</sub>, ''x''<sub>2</sub>,...] on infinitely many generators ''x''<sub>''i''</sub> ∈ π<sub>2''i''</sub>(MU) of positive even degrees.
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| Write '''CP'''<sup>∞</sup> 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 μ:'''CP'''<sup>∞</sup>× '''CP'''<sup>∞</sup> → '''CP'''<sup>∞</sup>.
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| A '''complex orientation''' on an associative [[commutative ring spectrum]] ''E'' is an element ''x'' in ''E''<sup>2</sup>('''CP'''<sup>∞</sup>) whose restriction to ''E''<sup>2</sup>('''CP'''<sup>1</sup>)
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| 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'''.
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| If ''E'' is a complex oriented ring spectrum, then
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| :<math>E^*(\mathbf{CP}^\infty) = E^*(\text{point})[[x]]</math>
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| :<math>E^*(\mathbf{CP}^\infty)\times E^*(\mathbf{CP}^\infty) = E^*(\text{point})[[x\otimes1, 1\otimes x]]</math>
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| and μ<sup>*</sup>(''x'') ∈ E<sup>*</sup>(point)<nowiki>[[</nowiki>''x''⊗1, 1⊗''x''<nowiki>]]</nowiki> is a [[formal group law]] over the ring E<sup>*</sup>(point) = π<sup>*</sup>(E). | |
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| Complex cobordism has a natural complex orientation.
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| {{harvtxt|Quillen|1969}} showed that there is a natural isomorphism from its coefficient ring to
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| [[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.
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| See also: [[complex-orientable cohomology theory]].
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| ==Brown–Peterson cohomology==
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| Complex cobordism over the rationals can be reduced to ordinary cohomology over the rationals, so the main
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| 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''.
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| 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–Peterson cohomology]], first described by {{harvtxt|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.
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| ==Conner–Floyd classes==
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| The ring MU<sup>*</sup>(BU) is isomorphic to the formal power series ring MU<sup>*</sup>(point)[[cf<sub>1</sub>, cf<sub>2</sub>, ...]] 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}}
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| Similarly MU<sub>*</sub>(BU) is isomorphic to the polynomial ring MU<sub>*</sub>(point)[β<sub>1</sub>, β<sub>2</sub>, ...]
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| ==Cohomology operations==
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| 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.
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| The coproduct is given by
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| :<math>\psi(b_k) = \sum_{i+j=k}(b)_{2i}^{j+1}\otimes b_j</math>
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| where the notation ()<sub>2''i''</sub> means take the piece of degree 2''i''.
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| This can be interpreted as follows. The map
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| :<math> x\rightarrow x+b_1x^2+b_2x^3+\cdots</math>
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| 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.
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| ==See also==
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| *[[Adams–Novikov spectral sequence]]
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| *[[List of cohomology theories]]
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| *[[Algebraic cobordism]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Adams | first1=J. Frank | title=Stable homotopy and generalised homology | url=http://books.google.com/books?id=6vG13YQcPnYC | publisher=[[University of Chicago Press]] | isbn=978-0-226-00524-9 | year=1974}}
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| *{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=Bordism and cobordism | doi=10.1017/S0305004100035064 | mr=0126856 | year=1961 | journal=Proc. Cambridge Philos. Soc. | volume=57 | pages=200–208 | issue=2}}
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| *{{citation|mr=0192494|last= Brown|first= Edgar H., Jr.|last2= Peterson|first2= Franklin P. |title=A spectrum whose Z<sub>p</sub> cohomology is the algebra of reduced ''p''<sup>th</sup> powers |journal= [[Topology]] |volume= 5 |year= 1966 |pages=149–154
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| |doi=10.1016/0040-9383(66)90015-2|issue=2}}.
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| *{{citation|mr=0216511|last=Conner|first= E. E.|last2= Floyd |title=The relation of cobordism to K-theories|series= Lecture Notes in Mathematics|volume= 28 |publisher=[[Springer-Verlag]] |publication-place= Berlin-New York |year=1966|doi=10.1007/BFb0071091|isbn=978-3-540-03610-4}}
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| *{{citation|authorlink=John Milnor |title=On the Cobordism Ring Ω∗ and a Complex Analogue, Part I
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| |first=J.|last= Milnor|journal=[[American Journal of Mathematics]] |volume= 82|issue= 3 |year= 1960|pages= 505–521|doi=10.2307/2372970|publisher=American Journal of Mathematics, Vol. 82, No. 3 |jstor=2372970}}
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| *{{cite arxiv | last1=Morava | first1=Jack | title=Complex cobordism and algebraic topology | eprint=0707.3216 | year=2007 | class=math.HO}}
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| *{{citation
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| | authorlink=Sergei Novikov (mathematician)
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| | last= Novikov
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| | first= S. P.
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| | title= Some problems in the topology of manifolds connected with the theory of Thom spaces
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| | journal= Soviet Math. Dokl.
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| | volume= 1 |year= 1960|pages= 717–720
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| }}. Translation of {{Citation
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| | title=О некоторых задачах топологии многообразий, связанных с теорией пространств Тома
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| | journal=[[Doklady Akademii Nauk SSSR]]
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| | volume=132
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| | issue=5
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| | pages=1031–1034
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| | postscript=.
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| | mr=0121815
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| | zbl=0094.35902
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| }}
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| *{{citation|mr=0157381|last=Novikov|first= S. P. |title=Homotopy properties of Thom complexes. (Russian) |journal= Mat. Sb. (N.S.) |volume= 57 |year= 1962|pages= 407–442}}
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| *{{citation |mr=0253350| authorlink=Daniel Quillen| last= Quillen|first= Daniel |title=On the formal group laws of unoriented and complex cobordism theory|journal= [[Bulletin of the American Mathematical Society]] |volume=75 |year=1969 |pages=1293–1298 |doi=10.1090/S0002-9904-1969-12401-8 |issue=6 }}.
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| *{{Citation | last1=Ravenel | first1=Douglas C. | title=Proceedings of the International Congress of Mathematicians (Helsinki, 1978) | url=http://mathunion.org/ICM/ICM1978.1/ | publisher=Acad. Sci. Fennica | location=Helsinki | isbn=978-951-41-0352-0 | mr=562646 | year=1980 | volume=1 | chapter=Complex cobordism and its applications to homotopy theory | pages=491–496}}
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| *{{citation|chapter= Complex cobordism theory for number theorists
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| |last= Ravenel |first= Douglas C.
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| |series=Lecture Notes in Mathematics
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| |publisher=Springer|publication-place= Berlin / Heidelberg
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| |issn = 1617-9692
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| |volume= 1326
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| |title=Elliptic Curves and Modular Forms in Algebraic Topology
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| |doi =10.1007/BFb0078042
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| |year=1988
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| |isbn =978-3-540-19490-3
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| |pages =123–133
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| }}
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| *{{citation
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| |last= Ravenel
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| |first= Douglas C.
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| |title= Complex cobordism and stable homotopy groups of spheres
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| |edition= 2nd
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| |url= http://www.math.rochester.edu/people/faculty/doug/mu.html
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| |publisher= AMS Chelsea
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| |year= 2003
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| |isbn= 978-0-8218-2967-7
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| |mr= 0860042
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| }}
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| *{{springer|id=C/c022780|title=Cobordism|author=Yu. B. Rudyak}}
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| * {{citation|first=R. E.|last= Stong|title=Notes on cobordism theory|publisher= [[Princeton University Press]] |year=1968}}
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| *{{citation|mr=0061823| authorlink=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://retro.seals.ch/digbib/view?did=c1:389597&sdid=c1:389960|doi=10.1007/BF02566923}}
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| ==External links==
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| *[http://www.map.him.uni-bonn.de/Complex_bordism Complex bordism] at the manifold atlas
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| *{{nlab|id=cobordism+cohomology+theory|title=cobordism cohomology theory}}
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| [[Category:Algebraic topology]]
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Myrtle Benny is how I'm called and I really feel comfy when individuals use the full name. Doing ceramics is what adore performing. He utilized to be unemployed but now he is a pc operator but his marketing by no means comes. South Dakota is exactly where I've usually been living.
My blog ... http://Ffnd.it/