complex analysis

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# Complex analysis
* table of contents
{: toc}

## Idea

__Complex analysis__ is the mathematical [[analysis]] of [[complex number|complex]]-valued [[analytic function|analytic]] (typically) functions of a complex variable, of several complex variables, or on a complex [[analytic manifold]].

Since a complex number can be understood as a pair of [[real numbers]], this would naively reduce to analysis of pairs of functions of an even number of real variables; however by complex analysis we mean mathematical analysis which takes into account limits and derivative which do not depend on the real line in a complex plane on which we approach a point. This leads to the notions of [[holomorphic function]], [[meromorphic function]], etc. which are the main subject of complex analysis.  However, there are connections to such real-analytic notions as [[harmonic analysis]], and the geometric approach to complex analysis builds on the theory of [[smooth functions]].

## Related entries

* [[Weierstrass preparation theorem]], [[several complex variables]], [[spectral theory]], [[Riemann surface]]

## References

* Wikipedia (English): [Complex analysis](http://en.wikipedia.org/wiki/Complex_analysis), [Several complex variables](http://en.wikipedia.org/wiki/Several_complex_variables)

* [[John B. Conway]], _Functions of one complex variable_, Springer 1978; _Functions of one complex variable II_, GTM 159

* Lars V. Ahlfors, _Complex analysis_, McGraw-Hill, 1966.

* Raghavan Narasimhan, _Complex analysis in one variable_, Birkhäuser, 1985.

* Lars Hörmander, _An introduction to complex analysis in several variables_, North-Holland

* Steven G. Krantz, _Geometric function theory: explorations in complex analysis_, Birkhäuser

* [[Robert C. Gunning]], Hugo Rossi, _Analytic functions of several complex variables_, AMS Chelsea Publishing

* Douglas N. Arnold, _Complex analysis_, lecture notes, [pdf](http://www.ima.umn.edu/~arnold/502.s97/complex.pdf)

* Elias M. Stein, Rami Shakarchi, _Complex analysis_, Princeton University Press 2003, 2012

Via [[constructive analysis]]:

* [[Errett Bishop]], Chapter 5 of: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]], Chapter 5 of: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

category: analysis

[[!redirects complex analysis]]


complex analytic space

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# Contents
* table of contents
{: toc}

## Idea

The notion of _complex analytic space_ is the notion of _[[analytic space]]_ in [[complex geometry]]; the generalization of the notion of [[complex manifold]] to spaces with [[singularity|singularities]].

## Definition

A _complex analytic test space_ is a common vanishing locus of a set of [[holomorphic functions]] $\mathbb{C}^n \to \mathbb{C}$. This is naturally a [[locally ringed space]] over the [[complex numbers]] $\mathbb{C}$. A complex analytic space is a [[locally ringed space]] over $\mathbb{C}$ that is locally isomorphic to such a complex analytic test space.

## Properties

### Local contractibility

A _smooth_ complex analytic space is locally isomorphic to a [[polydisc]] and hence locally contractible. See also ([Berkovich, p.2](#Berkovich)).

### GAGA theorems

Comparison to complex [[algebraic varieties]] ([[GAGA]]):

* [[comparison theorem (étale cohomology)]]

* [[Chow's theorem]]

## Related concepts

* [[complex analytic topology]]

* [[comparison theorem (etale cohomology)]]

* [[real analytic space]]

## References

Introductions include

* Brian Osserman, _Complex varieties and the analytic topology_ ([pdf](http://www.math.ucdavis.edu/~osserman/classes/248B-W10/analytic-top.pdf))

Generalization of smooth complex analytic spaces to smooth $p$-adic analytic spaces is discussed in

* [[Vladimir Berkovich]], _Smooth $p$-adic analytic spaces ([[Berkovich spaces]]) are locally contractible_ ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Inven_1999_137_contra.pdf))
 {#Berkovich}

Discussion in [[higher geometry]]/[[higher algebra]] ([[derived complex analytic spaces]]) is in

* {#Lurie08} [[Jacob Lurie]], section 4.4. of _[[Structured Spaces]]_, 2008

* [[Jacob Lurie]], sections 11 and 12 of _[[Closed Immersions]]_

[[!redirects complex analytic space]]
[[!redirects complex analytic spaces]]

[[!redirects complex analytic variety]]
[[!redirects complex analytic varieties]]


complex analytic stack

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#Contents#
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## Idea

A [[stack]] over the [[site]] of [[complex manifolds]], hence a [[1-truncated]] [[complex analytic ∞-stack]].

Beware that standard terminology such as for _[[algebraic stack]]_ might suggest that a complex analytic stack is a [[geometric stack]] over the complex analytic site, but usage as in ([Behrend-Noohi 05](#BehrendNoohi05)) differs from that would-be convention.

## Examples

For $G$ a [[complex Lie group]] then its [[delooping]] in complex analytic stack is the [[moduli stack]] $\mathbf{B}G$ for holomorphic $G$-[[principal bundles]]. For instance for [[holomorphic vector bundles]] if $G$ is the [[general linear group]] $GL(n,\mathbb{C})$.

## Related concepts

* [[complex orbifold]]

* [[algebraic stack]], [[smooth stack]], [[topological stack]],

* [[higher complex analytic geometry]], [[complex analytic ∞-stack]]

* [[holomorphic line 2-bundle]]

## References

* {#BehrendNoohi05} [[Kai Behrend]], [[Behrang Noohi]], section 3.2 of _Uniformization of Deligne-Mumford curves_ ([arXiv:0504309](http://arxiv.org/abs/math/0504309))

* [[Jacob Lurie]], sections 11 and 12 of _[[Closed Immersions]]_

[[!redirects complex analytic stacks]]

[[!redirects complex analytic groupoid]]
[[!redirects complex analytic groupoids]]


complex analytic topology

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#Contents#
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## Idea

The _analytic topology_ on an  [[complex analytic space]] is the one given by [[covering]] the space by affine opens equipped with the standard [[topology]] induced from that of the [[complex numbers]] $\mathbb{C}^n$.

## Properties

### Riemann existence theorem

* [[Riemann existence theorem]]

### Comparison theorem

[[Chow's theorem]] states that a projective [[complex analytic variety]], hence a closed analytic subspace of a complex [[projective space]], is also an [[algebraic variety]] over the [[complex numbers]]. In general this is not true.

The [[comparison theorem (étale cohomology)]] relates the [[étale cohomology]] of a [[complex variety]] with the [[ordinary cohomology]] of its complex analytic topological space.

For more along such lines see at _[[GAGA]]_.

## Related concepts

* [[étale topology]]

* [[analytification]], [[GAGA]]

* [[analytic continuation]]

* [[higher complex analytic geometry]]

## References

* [[Brian Osserman]], _Complex varieties and the analytic topology_ ([pdf](http://www.math.ucdavis.edu/~osserman/math/analytic.pdf))

* [[James Milne]], section 21 of _[[Lectures on Étale Cohomology]]_

[[!redirects analytic topology]]
[[!redirects complex analytic site]]
[[!redirects analytic site]]


complex analytic ∞-groupoid

> under construction

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#Contents#
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## Idea

A _complex analytic $\infty$-groupoid_ is an [[∞-groupoid]] equipped with [[higher geometry|geometric]] structure in the sense of [[complex analytic geometry]], such that [[complex analytic spaces]] constitute a [[full subcategory]] of the [[n-truncated object in an (∞,1)-topos|0-truncated]] complex analytic $\infty$-groupoids. Hence a complex analytic $\infty$-groupoid is an [[(∞,1)-sheaf]]/[[∞-stack]] on the [[site]] of [[complex manifolds]] (or some of its [[dense subsites]]). This is directly analogous to how [[(∞,1)-sheaves]] over the [[site]] of [[smooth manifolds]] may be regarded as _[[smooth ∞-groupoids]]_.

## Definition

+-- {: .num_defn #TheSites}
###### Definition

Write $CplxMfd$ for the [[category]] of [[complex manifolds]], regarded as a [[site]] with the standard [[Grothendieck topology]].

Write

$$
  \mathbb{C}Disc \hookrightarrow SteinSp \hookrightarrow CplxMfd
$$

for the [[full subcategories]] of [[Stein spaces]] and of complex [[polydiscs]], respectively, regarded as [[sites]] by equipping them with the induced [[coverages]].

=--

+-- {: .num_prop}
###### Proposition

The inclusions in def. \ref{TheSites} exhibit [[dense subsite]] inclusions.

=--

+-- {: .num_defn }
###### Definition

Write

$$
  \mathbb{C}Analytic\infty Grpd \coloneqq Sh_\infty(CplxMfd)
$$

for the [[hypercomplete (∞,1)-topos|hypercomplete]] [[(∞,1)-sheaf (∞,1)-topos]]  over the sites of complex manifolds def. \ref{TheSites}.

=--

+-- {: .num_remark }
###### Remark

By the discussion at _[[model structure on simplicial presheaves]]_
this means that $\mathbb{C}Analytic\infty Grpd$ is equivalently the [[simplicial localization]] of any of the hypercomplete local model structures on simplicial (pre-)sheaves, such as the Joyal [[model structure on simplicial sheaves]].

=--

This is considered in ([Hopkins-Quick 12, section 2.1](#HopkinsQuick12)).

## Properties

### Cohesion
 {#Cohesion}

+-- {: .num_prop #Cohesion}
###### Proposition

The [[global section geometric morphism]]

$$
  \Gamma
  \;\colon\;
  \mathbb{C}Analytic\infty Grpd
  \longrightarrow
  \infty Grpd
$$

exhibits a [[cohesive (∞,1)-topos]].

=--

+-- {: .proof}
###### Proof

We discuss the existence of the extra left adjoint $\Pi$ (the [[shape modality]]). This proceeds essentially as in the discussion of the [[cohesion]] of [[Smooth∞Grpd]] (see there) only that where there one may choose [[good open covers]] here we have to choose genuine [[split hypercovers]] by [[polydiscs]]. The rest of the proof is verbatim as for [[Smooth∞Grpd]].

To start with, since the [[hypercompletion]] depends only on the underlying [[sheaf topos]] of a site we may represent

$$
  \mathbb{C}Analytic\infty Grpd
  \simeq
  L_{lwhe} sPSh(\mathbb{C}Discs)_{proj, loc}
$$

by the [[simplicial localization]] of the [[local model structure on simplicial presheaves]] over $\mathbb{C}Disc$, [[Bousfield localization of model categories|localized]] (by the [theorem of descent recognition along hypercovers](hypercover#DescentFromDescentAlongHypercovers)) at the [[hypercovers]] $U_\bullet\to X$ as seen over $CplxMfd$ restricted to $X$ a [[polydisc]] (this roundabout way since $\mathbb{C}Disc$ does not carry a [[Grothendieck topology]] but just a [[coverage]]).

Now before [[Bousfield localization of model categories|Bousfield localization]] we have a simplicially enriched [[Quillen adjunction]]

$$
  sPSh(\mathbb{C}Disc)_{proj, loc}
  \stackrel{\overset{\underset{\to}{\lim}}{\longrightarrow}}{\underset{\Delta}{\longleftarrow}}
  sSet_{Quillen}
$$

which is simplicial-degree wise just the defining [[adjunction]] of the [[colimit]] functor $\underset{\to}{\lim}$ [[left adjoint]] to the [[constant presheaf]] functor.

To see that this descends to a Quillen adjunction on the local model structure, by the [recognition theorem for simplicial Quillen adjunctions](Quillen+adjunction#RecognitionOfSimplicialQuillenAdjunctions) we hence need to check that $\Delta$ preserves fibrant objects with respect to the local model structure, hence that any constant simplicial presheaf $\Delta S$ for $S$ a [[Kan complex]] already satisfies [[descent]] with respect to [[hypercovers]] as seen over $CplxMfd$.

By the [theorem of descent recognition along hypercovers](hypercover#DescentFromDescentAlongHypercovers) $\Delta S$ satisfies descent precisely if for each [[hypercover]] $U_\bullet \to X$ of any [[polydisc]] $X$. the induced morphism of [[derived hom-spaces]]

$$
  RHom(X, \Delta S) \longrightarrow RHom(U_\bullet, \Delta S)
$$

in the global [[model structure on simplicial presheaves]] is a [[weak equivalence]]. This $RHom$ in turn may be computed as the $sSet$-enriched [[hom object]] out of a cofibrant resolution $\hat X \to X$ and $\hat U_\bullet \to U_\bullet$, respectively. By the [recognition of cofibrant objects in the projective model structure](model%20structure%20on%20simplicial%20presheaves#CofibrantObjects) the [[representable functor|representable]] $X$ is already cofibrant and a sufficient condition for a resolution $\hat U_\bullet \to U_\bullet$ to be cofibrant is that it is a [[split hypercover]].

Since $\mathbb{C}Disc \hookrightarrow CplxMfd$ is a [[dense subsite]], we may always choose such a split hypercover such that $\hat U_\bullet$ consists simplicial-degreewise of [[coproducts]] of [[polydiscs]]: by the proposition at _[hypercover -- Existence of split refinements](hypercover#ExistenceOfSplitRefinements)_

Since the [[colimit]] of a [[representable functor]] is the point, this means that with such a choice $\underset{\to}{\lim} \hat U_\bullet$ is the [[simplicial set]] obtained by replacing in the hypercover by polydiscs each polydisc by a point. Forgetting the [[complex structure]] on all manifolds involved, one sees that this is precisely the "[[etale homotopy type]]" of $X$ as seen by the site [[CartSp]] of [[Cartesian spaces]], by the discussion at _[[Smooth∞Grpd]]_, and this comes out as the ordinary [[homotopy type]] of the underlying [[topological space]] of $X$. But this is of course [[contractible homotopy type|contractible]].

In conclusion this means that

$$
  \begin{aligned}
     RHom(U_\bullet, \Delta S)
     &\simeq
     sPSh(\mathbb{C}Disc)(\hat U_\bullet, \Delta S)
     \\
     & \simeq
     sSet(Sing(X), S)
     \\
     & \simeq
     sSet(\ast, S)
     \\
     &\simeq S
  \end{aligned}
$$

Therefore the [[descent]] condition for $\Delta S$ is satisfied. (This is also the statement of ([Hopkins-Quick 12, lemma 2.3, prop. 2.4, lemma 2.5, prop. 2.6](#HopkinsQuick12))).

From here on the argument for the [[cohesion]] of $\mathbb{C}Analytic \infty Grpd$ proceeds as at _[[∞-cohesive site]]_ (which might just as well be adapted to the hyper-discussion here).

=--

+-- {: .num_cor #ShapeOfComplexManifold}
###### Corollary

For $X\in CplxMfd \hookrightarrow \mathbb{C}Analytic\infty Grpd$ a [[complex manifold]], then the [[shape modality|shape]] $\Pi(X)$ is the [[homotopy type]] of its underlying [[topological space]].

=--

In ([Hopkins-Quick 12](#HopkinsQuick12))) this is part of prop. 2.6.

### Oka principle
 {#OkaPrinciple}

Discussion of the [[Oka principle]] in terms of $\mathbb{C}Analytic\infty Grpd$ is in ([Larusson 01](#Larusson01)).

+-- {: .num_defn #OkaManifold}
###### Definition

Say that a [[complex manifold]] $X$ is an _[[Oka manifold]]_ if for every [[Stein manifold]] $\Sigma$ the canonical inclusion

$$
  Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)
$$

from the [[mapping space]] of [[holomorphic functions]] to that of [[continuous functions]] (both equipped with the [[compact-open topology]]) is a [[weak homotopy equivalence]].

=--

+-- {: .num_theorem}
###### Theorem

This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies [[descent]] with respect to [[finite set|finite]] [[covers]].

=--

([Larusson 01, theorem 2.1](#Larusson01))

+-- {: .num_remark}
###### Remark

By corollary \ref{ShapeOfComplexManifold}, in terms of [[cohesion]], prop. \ref{Cohesion}, definition \ref{OkaManifold} should (...check...) read

$$
  \Pi[\Sigma,X] \simeq \flat [\Pi(\Sigma), \Pi(X)]
  \,.
$$

=--

### Complex analytic differential generalized cohomology
 {#DifferentialCohomology}

By prop. \ref{Cohesion} $\mathbb{C}Analytic \infty Grpd$ is [[cohesive (infinity,1)-topos|cohesive]] and hence by the discussion at
_[[differential cohomology hexagon]]_ the objects  $\hat E \in Stab(\mathbb{C}Analytic \infty Grpd)$ (hence the [[sheaves of spectra]] on $\mathbb{C}Mfd$ ) qualify as [[differential cohomology]] refinements of the  [[cohomology theories]] [[Brown representability theorem|represented]] by the [[shape modality|shapes]] $E\coloneqq \Pi \hat E \in $ [[Spectra]].

Discussion of such complex analytic differential generalized cohomology is in ([Hopkins-Quick 12, section 4](#HopkinsQuick12)),.

## Examples

### Multiplicative group and holomorphic line $n$-bundles

The [[multiplicative group]] is a canonical [[∞-group]] object

$$
  \mathbb{G}_m \in Grp(\mathbb{C}Analytic\infty Grpd)
$$

given as an [[(∞,1)-presheaf]] by the assignment

$$
  \mathbb{G}_m \;\colon\; \Sigma \mapsto \mathcal{O}_\Sigma^\times
$$

that sends a [[Stein manifold]] to the multiplicative [[abelian group]] of non-vanishing [[holomorphic functions]] on it.

The [[delooping]] $\mathbf{B}\mathbb{G}_m$ is the universal [[moduli stack]] for [[holomorphic line bundles]] (the [[Picard stack]]) and the double delooping $\mathbf{B}^2 \mathbb{G}_m$ that for [[holomorphic line 2-bundles]] (the [[Brauer stack]]).

## Related concepts

* [[smooth ∞-groupoid]]

* [[complex analytic stack]]

* [[holomorphic line n-bundle]]

## References

* {#Larusson01} [[Finnur Lárusson]], _Excision for simplicial sheaves on the Stein site and Gromov's Oka principle_ ([arXiv:math/0101103](http://arxiv.org/abs/math/0101103))

* [[Jacob Lurie]], section 4.4. of _[[Structured Spaces]]_

* [[Jacob Lurie]], sections 11 and 12 of _[[Closed Immersions]]_

* {#HopkinsQuick12} [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_, Journal of Topology, **8** 1 (2015) ([arXiv:1212.2173](http://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021))

[[!redirects complex analytic ∞-groupoids]]

[[!redirects complex analytic infinity-groupoid]]
[[!redirects complex analytic infinity-groupoids]]

[[!redirects ComplexAnalytic∞Grpd]]
[[!redirects ComplexAnalyticInfinityGrpd]]

[[!redirects complex analytic ∞-stack]]
[[!redirects complex analytic ∞-stacks]]
[[!redirects complex analytic infinity-stack]]
[[!redirects complex analytic infinity-stacks]]


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This entry collects pointers related to the book

* [[Doug Ravenel]]

  **Complex cobordism and stable homotopy groups of spheres**

  Academic Press Orland (1986)

  reprinted as:

  AMS Chelsea Publishing, Volume 347 (2004)

  [ISBN:978-0-8218-2967-7](https://bookstore.ams.org/chel-347-h)

  ([webpage](http://www.math.rochester.edu/people/faculty/doug/mu.html), [pdf](https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf))

on [[stable homotopy theory]] in general and in particular the computation of the [[homotopy groups of spheres]] via the [[Adams-Novikov spectral sequence]] and its use of [[complex cobordism cohomology theory]].

> {#InitialInclination} My initial inclination was to call this book [[The Music of the Spheres]], but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (preface to the first edition)

See also

* [[Frank Adams]], _[[Stable homotopy and generalised homology]]_, 1974

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, 1999

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, 2010

#Contents#
* table of contents
{:toc}

## Chapter 1. An introduction to the Homotopy Groups of Spheres

### 1. Classical theorems Old and New

* [[homotopy group]]

* [[Hurewicz theorem]]

* [[suspension]], [[reduced suspension]]

* [[Freudenthal suspension theorem]]

* [[stable homotopy groups of spheres]]

* [[Hopf fibration]]

* [[Serre finiteness theorem]]

* [[Nishida nilpotence theorem]]

* [[Bott periodicity theorem]]

* [[J-homomorphism]]

* [[Adams conjecture]]

### 2. Methods of computing $\pi_\ast(S^n)$

* [[Postnikov tower]]

* [[Serre spectral sequence]]

* [[Adams spectral sequence]]

* [[Steenrod algebra]]

* [[parallelizable sphere]]

* [[Hopf invariant one theorem]]

* [[complex cobordism cohomology]]

* [[cobordism ring]]

* [[Adams-Novikov spectral sequence]]

### 3. The Adams-Novikov $E_2$-term, Formal Group Laws, and the Greek Letter Construction

* [[formal group law]]

* [[Lazard ring]]

* [[Greek letter construction]]

### 4. More formal group law theory, Morava's point of view, and the Chromatic Spectral Sequence

* [[Brown-Peterson cohomology]]

* [[p-typical formal group law]]

* [[height of a formal group]]

* [[chromatic spectral sequence]]

### 5. Unstable homotopy groups and the EHP spectral sequence

* [[EHP spectral sequence]]

## Chapter 2. Setting up the Adams Spectral sequence

### 1. The classical Adams spectral sequence

* [[exact couple]]

* [[spectral sequence]]

* [[Adams resolution]]

* [[Adams spectral sequence]]

* [[Eilenberg-MacLane spectrum]], [[ordinary cohomology]]

* [[inverse limit]]

* [[p-completion]]

### 2. The Adams spectral sequence based on a generalized homology theory

* [[commutative Hopf algebroid]] structure on [[Steenrod algebra]]

### 3. The smash product pairing and the Generalized connecting homomorphism

* [[connecting homomorphism]]

## Chapter 3. The Classical Adams Spectral Sequence

* [[classical Adams spectral sequence]]

### 1. The Steenrod algebra and some easy calculuation

* [[Hopf algebra]] structure on [[Steenrod algebra]]

* [[cobar complex]], [[cosimplicial object]]

### 2. The May spectral sequence

* [[May spectral sequence]]

* [[Massey product]]

### 3. The Lambda Algebra

* [[Lambda algebra]]

* [[EHP spectral sequence]]

### 4. Some general properties of $Ext$

* [[periodicity theorem]]

### 5. Survey and further reading

## Chapter 4. $B P$-Theory and the Adams-Novikov Spectral Sequence

* [[Adams-Novikov spectral sequence]]

### 1. Quillen's theorem and the structure of $BP_\bullet(BP)$

* [[Quillen's theorem on MU]]

* [[Landweber-Novikov theorem]]

* [[Brown-Peterson spectrum]]

* [[p-typical formal group]]

* [universal complex orientation on MU](cobordism+cohomology+theory#UniversalComplexOrientation), [[complex orientation and MU]]

* [[Adams-Quillen theorem]]

### 2. A survey of $B P$-theory

* [[cobordism group]]

* [[Thom spectrum]]

### 3. Some calculations in $B P_\bullet(B P)$

### 4. Beginning calculations with the Adams-Novikov Spectral Sequence

* [[BP]]-[[Adams-Novikov spectral sequence]]

## Chapter 5. The Chromatic Spectral Sequence

### 1. The algebraic construction

* [[chromatic spectral sequence]]

### 2. $Ext^1(B P_\bullet/I_n)$ and Hopf Invariant One

* [[Hopf invariant one]]

### 3. $Ext(M^1)$ and the $J$-Homomorphism

* [[J-homomorphism]]

### 4. $Ext^2$ and the Thom Reduction

### 5. Periodic families in $Ext^2$

### 6. Elements in $Ext^3$ and Beyond

## Chapter 6. Morava Stabilizer Algebras

* [[Morava stabilizer group]]

### 1. The Change-of-Rings Isomorphism

### 2. The Structure of $\Sigma(n)$

### 3. The Cohomology of $\Sigma(n)$

### 4. The Odd Primary Kervaire Invariant Elements

### 5. The Spectra $T(m)$

* [[Ravenel's spectra]]

## Chapter 7. Computing Stable Homotopy Groups with the Adams-Novikov Spectral Sequence

## Appendix 1. Hopf Algebras and Hopf Algebroids

### 1. Basic definitions

* [[Hopf algebra]]

* [[commutative Hopf algebroid]]

* [[module]], [[comodule]]

* [[cotensor product]]

* [[change of rings theorem]]

### 2. Homological algebra

* [[derived functor]]

* [[Ext]] and [[Cotor]] for [[comodules]]

### 3. Some spectral sequences

* [[spectral sequence]]

* [[Cartan-Eilenberg spectral sequence]]

### 4. Massey products

* [[Massey product]]

### 5. Algebraic Steenrod operations

* [[Steenrod algebra]] on [[Cotor]]-groups of [[comodules]] over [[commutative Hopf algebroids]]

## Appendix 2. Formal Group Laws

* [[formal group law]]

## Appendix 3. Table of homotopy groups of spheres

category: reference


complex cohomology

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## Idea

By _complex cohomology_ one usually means [[ordinary cohomology]] with [[complex number]] [[coefficients]], denoted $H^\bullet\big(-, \mathbb{C}\big)$.

Hence, with the pertinent conditions on the [[domain]] [[space]] $X$ satisfied, its complex cohomology $H^\bullet\big(-, \mathbb{C}\big)$ is what is computed by the [[Cech cohomology]] or [[singular cohomology]] or [[sheaf cohomology]] of $X$ with [[coefficients]] in $\mathbb{C}$.

### Properties

* [[de Rham theorem]]

* [[PL de Rham theorem]]

## Related concepts

* [[Dolbeault cohomology]]

* [[complex vector space]]

* [[integral cohomology]], [[rational cohomology]], [[real cohomology]]


complex conjugation

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## Idea

_Complex conjugation_ is the operation on [[complex numbers]] which reverses the sign of the [[imaginary part]], hence the [[function]]

$$
  \array{
    \mathbb{C}
    &
      \overset{
        \;\;\;
        (-)^\ast
        \;\;\;
      }{
        \longrightarrow
      }
    &
    \mathbb{C}
    \\
    a + \mathrm{i} b &\mapsto&  a - \mathrm{i} b
  }
  \phantom{AAAAAA}
  for\;\; a,b \in \mathbb{R}
  \,.
$$

More generally, the [[anti-involution]] on any [[star-algebra]] may be referred to as _conjugation_. For instance one speaks of _quaternionic conjugation_ for the analogous operation on [[quaternions]]:

$$
  \array{
    \mathbb{H}
    &
      \overset{
        \;\;\;
        (-)^\ast
        \;\;\;
      }{
        \longrightarrow
      }
    &
    \mathbb{H}
    \\
    a
      + \mathrm{i} b
      + \mathrm{j} c
      + \mathrm{k} d
    &\mapsto&
    a
      - \mathrm{i} b
      - \mathrm{j} c
      - \mathrm{k} d
  }
  \phantom{AAAAAA}
  for\;\; a, b, c, d \in \mathbb{R}
  \,.
$$

## Related concepts

* [[real part]], [[imaginary part]]

* [[anti-involution]]

* [[anti-linear map]], [[anti-dual linear space]]

* [[automorphism of the complex numbers]]

For an unrelated (or vaguely related) notion with a similar name see at _[[conjugacy class]]_ and _[[adjoint action]]_.

[[!redirects complex conjugations]]

[[!redirects complex conjugate]]
[[!redirects complex conjugates]]

[[!redirects quaternion conjugation]]
[[!redirects quaternion conjugations]]

[[!redirects quaternionic conjugation]]
[[!redirects quaternionic conjugations]]

[[!redirects star-conjugation]]
[[!redirects star-conjugations]]

[[!redirects star conjugation]]
[[!redirects star conjugations]]


complex connection

Let $\xi = (p:E\to B)$ be a complex-analytic $G$-principal bundle where $G$ is a complex Lie group. The complex-analytic [[Ehresmann connection]] is the analytic field of horizontal subspaces, which is $G$-equivariant. One can consider the underlying real principal bundle $\xi_{\mathbf{R}}$. The operator of complex structure becomes an  automorphism $I:\xi_{\mathbf{R}}\to \xi_{\mathbf{R}}$ of the smooth real $G_{\mathbf{R}}$-bundle $\xi_{\mathbf{R}}$. One can consider the differential
$$
(d I)_p T_p E\to T_p E
$$
which on each vertical subspace $(T_p^V E)_{\mathbf{R}}$ is an operator of the complex structure on the fiber.

For a field $H$ of horizontal subspaces on $\xi_{\mathbf{R}}$ the following are equivalent:

(i) $H$ is a connection on $\xi$

(ii) $I^* H = H$

(iii) $H_p = (d I)_p H_p$

One can characterize complex connections also by conditions on a [[covariant derivative]] on $\xi_{\mathbf{R}}$.

* [[M M Postnikov]], _Lectures on geometry_, vol. III, lec. 10


complex curve

#Contents#
* table of contents
{:toc}

## Idea

A _complex curve_ is a _[[curve]]_ in [[complex analytic geometry]], hence a [[complex analytic space]] of complex [[dimension]] 1 (hence of real dimension 2).

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[Riemann surface]]

* [[algebraic curve]]

* [[arithmetic curve]]

* [[Teichmüller space]]

* [[moduli stack of curves]]

* [[complex surface]]

[[!redirects complex curves]]


complex geometry

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=--
#### Analytic geometry
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In **complex geometry** one studies _[[complex manifolds]]_ as a special case of how in general [[differential geometry]] one studies more generally [[smooth manifolds]].

In _complex [[analytic geometry]]_ one studies, more generally, [[complex analytic spaces]].

## Properties

### GAGA principle and relation to algebraic geometry

[[complex analytic geometry]] is closely related to [[algebraic geometry]] over the [[complex numbers]]. See at _[[GAGA]]_ for more on this.

## Related concepts

* [[analytic geometry]], [[GAGA]]

* [[complex analytic topology]]

* [[complex manifold]], [[almost complex manifold]]

  [[complex surface]], [[complex curve]]

  [[Kähler manifold]], [[hyper-Kähler manifold]]

  [[Calabi-Yau manifold]]

* [[Picard-Lefschetz theory]]

* [[Hodge theory]]

* [[Arakelov geometry]]

* [[higher complex analytic geometry]], [[derived complex analytic space]]

## References

* {#Huybrechts04} [[Daniel Huybrechts]],  _Complex geometry: An introduction_, Universitext, Springer (2004) [[doi:10.1007/b137952](https://doi.org/10.1007/b137952), [pdf](http://www.im.ufrj.br/~andrew/ensino/VC20-1/Huybrechts-Complex_Geometry.pdf)]

* [[Claire Voisin]], _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

[[!redirects complex geometries]]

[[!redirects complex analytic geometry]]
[[!redirects complex-analytic geometry]]


complex geometry - contents

[[geometry]], [[complex numbers]], [[complex line]]

**[[complex geometry]]**

* [[complex manifold]], [[complex structure]]

* [[complex analytic space]]

### Variants

* [[generalized complex geometry]]

* [[complex supermanifold]]

### Structures

* [[Dolbeault complex]], [[holomorphic de Rham complex]]

* [[Hodge structure]], [[Hodge filtration]]

* [[Hodge-filtered differential cohomology]]

### Examples

* $dim = 1$: [[Riemann surface]], [[super Riemann surface]]

* [[Calabi-Yau manifold]]

  * $dim = 2$: [[K3 surface]]

* [[generalized Calabi-Yau manifold]]


complex Lie algebra

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#Contents#
* table of contents
{:toc}

## Definition

A _complex Lie algebra_ is a [[Lie algebra]] [[internalization|internal]] to [[complex vector spaces]].

## Related concepts

* [[complex Lie group]]

## References

* Adam Kaye, _Complex Lie algebras_ ([pdf](http://www-personal.umich.edu/~adamkaye/papers/Kaye.pdf))

* [[Terry Tao]], _[Notes on the classification of complex Lie algebras](http://terrytao.wordpress.com/2013/04/27/notes-on-the-classification-of-complex-lie-algebras/)_

[[!redirects complex Lie algebras]]


complex Lie group

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#### Complex geometry
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _complex Lie group_ is a [[Lie group]] that is a [[group object]] not just [[internalization|internal]] to [[smooth manifolds]] but in fact to [[complex manifolds]]. Hence it is a [[complex manifold]] $G$ equipped with a [[group]] structure such that both the multiplication map $G \times G \to G$ as well as the [[inverse]] map $G \to G$ are [[holomorphic functions]].

## Properties

### Relation to almost complex structure

One can also consider groups in [[almost complex manifolds]]. But every _almost complex Lie group_ is automatically also a complex Lie group.

### Complexification of compact Lie groups

For $K$ a [[compact Lie group]] there is a unique [[connected topological space|connected]] complex Lie group $G$ such that

1. the [[Lie algebra]] of $G$ is the [[complexification]] of the Lie algebra of $K$:

   $$
     Lie(G) \simeq Lie(K) \otimes_{\mathbb{R}} \mathbb{C}
     \,,
   $$

1. $K$ is a [[maximal compact subgroup]] of $G$.

This $G$ is called the **complexification** of $K$.

### Oka property

\begin{prop}\label{CosetSpacesOfComplexLieGroupsAreOkaManifolds}
**([[coset spaces]] of [[complex Lie groups]] are [[Oka manifolds]])** \linebreak
Every [[complex Lie group]] and every [[coset space]] ([[homogeneous space]]) of complex Lie groups is an [[Oka manifolds]].
\end{prop}
(review in [Forstnerič & Lárusson 11, p. 9](Oka+manifold#ForstnericLarusson11), [Forstnerič 2013, Thm. 2.6](Oka+manifold#Forstneric13))

## Related concepts

* [[complex Lie algebra]]

* [[relation between compact Lie groups and reductive algebraic groups]]

## References

* [[Dietmar Salamon]], *Notes on complex Lie groups*, 2019 ([pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/cx-lie.pdf), [[Salamon_ComplexLieGroups.pdf:file]])

See also

* Wikipedia, *[Complex Lie group](https://en.wikipedia.org/wiki/Complex_Lie_group)*

[[!redirects complex Lie groups]]


complex line

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#### Complex geometry
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# Contents
* table of contents
{: toc}

## Idea

A _complex [[line]]_ is a [[complex vector space]] of [[dimension]] 1 (of _complex dimension_ that is, as a $\mathbb{C}$-vector space, meaning that as a vector space over the [[real numbers]] it is a [[plane]]).

In particular the [[complex plane]] $\mathbb{C}$ itself is a complex line; conversely, any two complex lines are [[isomorphic]] (to each other and to the complex plane).  However, there are many such isomorphisms; the [[automorphism group]] is $\mathbb{C} \setminus \{0\}$.

## Related concepts

* [[real line]], [[continuum]]

* [[complex vector space]]

  * **complex line**

* [[complex vector bundle]]

  * [[complex line bundle]]

[[!redirects complex lines]]


complex manifold

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#### Differential geometry
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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **complex manifold** is a [[manifold]] holomorphically modeled on [[polydiscs]] $D$ in $\mathbb{C}^n$ ([[complexified]] $n$-dimensional [[cartesian space]]):

* a [[smooth manifold]] locally isomorphic to $D \hookrightarrow \mathbb{C}^n$ whose transition functions are [[holomorphic functions]];

* equivalently: a [[smooth manifold]] equipped with an integrable [[almost complex structure]];

* equivalently a smooth [[complex analytic space]].

## Properties

### Covers

+-- {: .num_prop}
###### Proposition

Every complex manifold admits a [[good open cover]] in $Disk_{cmpl}$.

=--

For instance ([Maddock, lemma 3.2.8](#Maddock)).

## Examples

### Complex 1-dimensional: Riemann surfaces

A complex manifold of complex [[dimension]] 1 is called a  _[[Riemann surface]]_.

### Calabi-Yau manifolds

A complex manifold whose [[canonical bundle]] is trivializable is a  _[[Calabi-Yau manifold]]_. In complex dimension 2 this is a _[[K3 surface]]_.

### Other examples

* [[Riemann sphere]]

* [[Stein manifolds]]

## Related concepts

* [[complex analytic space]]

* [[domain of holomorphy]], [[Stein manifold]], [[Oka manifold]]

* [[Dolbeault cohomology]], [[Bott-Chern cohomology]]

* [[holomorphic symplectic manifold]]

* [[complex Lie group]]

* [[complex orbifold]]

* [[complex analytic stack]]

* [[complex analytic ∞-groupoid]]

* [[hypercomplex manifold]]

* [[smooth manifold]], [[topological manifold]]

## References

Textbook accounts:

* {#Voisin02} [[Claire Voisin]], section 2 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#Forstneric11} [[Franc Forstnerič]], Section 1.5 in: _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011 ([doi:10.1007/978-3-642-22250-4](https://link.springer.com/book/10.1007/978-3-642-22250-4))

  > (in the context of the [[Oka principle]])

See also:

* Wikipedia, *[Complex manifold](https://en.wikipedia.org/wiki/Complex_manifold)*

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 8 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

Lectures notes:

* Stefan Vandoren, _Lectures on Riemannian Geometry, Part II:
Complex Manifolds_ ([pdf](http://www.phys.uu.nl/~vandoren/MRIlectures.pdf))

* {#Maddock} Zachary Maddock, _Dobeault cohomology_ ([pdf](http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf))

[[!redirects complex manifolds]]

[[!redirects complex analytic manifold]]
[[!redirects complex analytic manifolds]]

[[!redirects complex-analytic manifold]]
[[!redirects complex-analytic manifolds]]


complex n-dimensional space

Given a positive integer $n$, the __complex $n$-dimensional space__ $\mathbb{C}^n$ is the set of $n$-tuples of complex numbers considered as a $n$-dimensional topological vector over the field of complex numbers $\mathbb{C}$ (such a Hausdorff topology is unique). For a continuous map from a complex $n$-dimensional space to a complex $m$-dimensional space this structure is enough to define complex [[differentiability]] of a map, hence the complex analytic structure is induced.

[[!redirects Complex n-dimensional space]]


complex number

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#Contents#
* table of contents
{:toc}

## Definition

A __complex number__ is a [[number]] of the form $a + \mathrm{i} b$, where $a$ and $b$ are [[real numbers]] and $\mathrm{i}^2 = - 1$ is an [[imaginary unit]].  The set of complex numbers (in fact a [[field]] and [[topological vector space]]) is denoted $\mathbf{C}$ or $\mathbb{C}$.

This can be thought of as:

*  the vector space $\mathbb{R}^2$ made into an [[associative algebra|algebra]] by the rule
   $$ (a, b) \cdot (c, d) = (a c - b d, a d + b c) ;$$
*  the subalgebra of those $2$-by-$2$ real [[matrix|matrices]] of the form
   $$ \left(\array { a & b \\ - b & a } \right);$$
*  the [[polynomial]] ring $\mathbb{R}[\mathrm{x}]$ modulo $\mathrm{x}^2 + 1$;
*  the result of applying the [[Cayley–Dickson construction]] to $\mathbb{R}$;
*  the $2$-dimensional [[normed division algebra]] over $\mathbb{R}$;
*  the [[Clifford algebra]] $Cl_{0,1}(\mathbb{R})$;
*  the elliptic $2$-dimensional algebra of [[hypercomplex numbers]];
*  the [[complexification]] of $\mathbb{R}$;
*  In mathematics with [[weak countable choice]], the [[algebraic closure]] of $\mathbb{R}$ as a field;

We think of $\mathbb{R}$ as a [[subset]] (in fact ${\mathbb{R}}$-vector subspace) of $\mathbb{C}$ by identifying $a$ with $a + \mathrm{i} 0$.  $\mathbb{C}$ is equipped with a $\mathbb{R}$-linear [[involution]] , called __[[complex conjugation]]__, that maps $\mathrm{i}$ to $\bar{\mathrm{i}} = -\mathrm{i}$. Concretely, $ \overline{a + \mathrm{i} b} = a - \mathrm{i} b $.

Complex conjugation is the nontrivial [[field]] [[automorphism]] of $\mathbb{C}$ which leaves ${\mathbb{R}}$ invariant. In other words, the Galois group $Gal({\mathbb{C}}/\mathbb{R})$ is cyclic of order two and generated by complex conjugation.
$\mathbb{C}$ also has an [[absolute value]]:
$$
|{a + \mathrm{i} b}| = \sqrt{a^2 + b^2} ;
$$
notice that the absolute value of a complex number is a nonnegative real number, with
$$
|z|^2 = z \bar{z} .
$$

Most concepts in analysis can be extended from $\mathbb{R}$ to $\mathbb{C}$, as long as they do not rely on the order in $\mathbb{R}$.  Sometimes $\mathbb{C}$ even works better, either because it is algebraically closed or because of Goursat\'s theorem.  Even when the order in $\mathbb{R}$ is important, often it is enough to order the absolute values of complex numbers.  See [[ground field]] for some of the concepts whose precise definition may vary with the choice of $\mathbb{R}$ or $\mathbb{C}$ (or even other possibilities).

## Properties

### Automorphisms

+-- {: .num_prop #AutomorphismsOfComplexNumbersIsZ2}
###### Proposition

The [[automorphism group]] of the [[complex numbers]], as an [[associative algebra]] over the [[real numbers]], is [[Z/2]], [[action|acting]] by [[complex conjugation]].

=--

See also at _[normed division algebra -- automorphism](normed+division+algebra#Automorphisms)_.

Over other subfields, the automorphism group may be considerably larger. Over the [[rational numbers]], for instance, $\mathbb{C}$ has transcendence degree equal to the cardinality of the continuum, i.e., there is an algebraic extension $\mathbb{Q}(X) \hookrightarrow \mathbb{C}$ with ${|X|} = \mathfrak{c} = 2^{\aleph_0}$. Any [[bijection]] $X \to X$ induces a field automorphism $\mathbb{Q}(X) \to \mathbb{Q}(X)$ which may be extended to an automorphism of $\mathbb{C}$ over $\mathbb{Q}$. Therefore the number of automorphisms of $\mathbb{C}$ is at least $2^\mathfrak{c}$ (and in fact at most this as well, since the number of functions $\mathbb{C} \to \mathbb{C}$ is also $2^\mathfrak{c}$).

See also at _[[automorphism of the complex numbers]]_.

### Geometry of complex numbers

The complex numbers form a plane, the __complex plane__. Indeed, a map $\mathbb{C} \to \mathbb{R}^2$ given by sending $\mathrm{x} + \mathrm{i}\mathrm{y}$ to the standard real-valued coordinates $(\mathrm{x},\mathrm{y})$ on this plane is a bijection. Much of [[complex analysis]] can be understood through [[differential topology]] by identifying $\mathbb{C}$ with $\mathbb{R}^2$, using either $\mathrm{x}$ and $\mathrm{y}$ or $\mathrm{z}$ and $\bar{\mathrm{z}}$.  (For example, Cauchy\'s integral theorem is Green\'s/Stokes\'s theorem.)

It is often convenient to use the [[Alexandroff compactification]] of $\mathbb{C}$, the [[Riemann sphere]] $\mathbb{C}P^1$.  One may think of $\mathbb{C}P^1$ as $\mathbb{C} \cup \{\infty\}$; functions valued in $\mathbb{C}$ but containing 'poles' may be taken to be valued in $\overline{\mathbb{C}}$, with $f(\zeta) = \infty$ whenever $\zeta$ is a pole of $f$.

## Related concepts

* [[automorphism of the complex numbers]]

* [[absolute value]], [[phase]]

* [[real part]], [[imaginary part]]

* [[complex plane]], [[complex projective space]], [[Riemann sphere]]

* [[complex vector space]]

  * [[complex line]]

* [[conjugate transpose matrix]]

* [[complex vector bundle]]

  * [[complex line bundle]]

* [[contour integration]]

* [[complex geometry]]

* [[polydisc]]

* [[p-adic complex number]]

[[!include exceptional spinors and division algebras -- table]]

## References

On the history of the notion of complex numbers:

* [[Jean-Pierre Tignol]], p. 19 of: *Galois' Theory of Algebraic Equations*, World Scientific (2001) [[doi:10.1142/4628](https://doi.org/10.1142/4628)]

* Orlando Merino, *A Short History of Complex Numbers* (2006) [[pdf](https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf), [[Merino-ComplexNumbers.pdf:file]]]

* [[Leo Corry]], *A Brief History of Numbers*, Oxford University Press (2015) [[ISBN:9780198702597](https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597)]

See also:

* Wikipedia, _[Complex numbers](https://en.wikipedia.org/wiki/Complex_number)_

* [[Tom Leinster]], _Objects of categories as complex numbers_, [arXiv:math/0212377](http://arxiv.org/abs/math/0212377)

[[!redirects complex numbers]]

[[!redirects complex number system]]
[[!redirects complex number systems]]

[[!redirects complex plane]]
[[!redirects complex planes]]


complex of groups

This is an entry on the notion of **complex of groups** introduced by [[André Haefliger]] and [[Jon Corson]] as a higher dimensional generalisation of the Bass-Serre theory of [[graphs of groups]]. (It does _not_ refer to the idea of chain complexes of groups, i.e., [[chain complexes]] in the (more or less) usual sense.)

#Contents#
* table of contents
{:toc}

##Idea

A _complex of groups_ is a [[diagram]] of [[groups]], [[homomorphisms]] and [[conjugations]], corresponding, abstractly, to the system of inclusions of the [[stabiliser]] subgroups of an [[action]] of a group on a [[simplicial cell complex]] or equivalently on a [[small category without loops]]. If the complex is 1-dimensional one obtains a [[graph of groups]] - note however, the category of 1-complexes of groups is not equivalent as a category to the category of graphs of groups (see [[A. Thomas]] Proposition 2.1).

##Definition

We will initially give the definition in its 'bare hands' form. Here $K$ is a [[simplicial complex]]

 A _complex of groups_, $G(K)$, on $K$ is specified by the data, $(\{G_\sigma\}, \{\psi_a\}, \{g_{a, b}\})$
given by

* a group, $G_{\sigma}$, for each simplex, $\sigma$, of $K$;

* an injective homomorphism,

$$\psi_a :G_{i (a)} \rightarrow G_{t(a)},$$

for each edge, $a \in E_K$, of the barycentric subdivision of $K$;

* for each pair of composable edges, $a$ and $b$, in $E_K$, an element
  $g_{a, b} \in G_{t(a)}$ is given such that

$$g^{- 1}_{a, b} \psi_{ba} (_-) g_{a, b} = \psi_a \psi_b$$

and such that the   'cocycle condition'

$$g_{a, cb} \psi_a (g_{b, c}) = g_{ab, c} g_{a, b}$$

holds.

###Developability and the universal cover

(to come later)

###The universal group and fundamental group

(to come later)

##Examples

(to come later)

## Complexes of groups as pseudofunctors.

see paper by [[Tom Fiore]] et al (below)

##References

* [[M. Bridson]] and [[A. Haefliger]], 1999, _Metric Spaces of Non-Positive Curvature_, number 31 in
Grundlehren der Math. Wiss, Springer.

* [[A. Haefliger]], 1991, _Complexes of Groups and Orbihedra_, in Group Theory from a Geometric
viewpoint , 504 – 540, ICTP, Trieste, 26 March- 6 April 1990, World Scientific.

* [[J. M. Corson]], _Complexes of Groups_, Proc. London Math. Soc., 65, (1992), 199–224.

* [[A. Thomas]], 2006 _Lattices acting on right-angled buildings_, Alg. Geom. Top., 6, 1215-1238.

See also:

* [[Thomas M. Fiore]], Wolfgang Lück, and Roman Sauer, _Euler Characteristics of Categories and Homotopy Colimits_, ([arXiv: 1007.3868](http://arxiv.org/abs/1007.3868))

[[!redirects complexes of groups]]


complex orbifold

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=--

#Contents#
* table of contents
{:toc}

## Idea

A _complex orbifold_  is an [[orbifold]] with [[complex structure]], hence like a [[complex manifold]] but possibly with [[singularities]] modeled on [[fixed points]] of local [[actions]] of [[finite groups]] by [[holomorphic maps]].

## Related concepts

* [[Riemannian orbifold]]

* [[symplectic orbifold]]

* [[complex analytic stack]]

* [[Lorentzian orbifold]]

## References

* Brian Steer, Andrew Wren, _Grothendieck Toplogy and the Picard group of a complex orbifold_, In: Jean-Pierre Meyer, [[Jack Morava]] and W. Stephen Wilson (eds.) _Homotopy Invariant Algebraic Structures: A Conference in Honor of J. Michael Boardman_, AMS 1999 ([doi:10.1090/conm/239](http://dx.doi.org/10.1090/conm/239))

* [[Edward Frenkel]], Matthew Szczesny, _Chiral de Rham complex and orbifolds_, J. Algebraic Geom. 16 (2007), 599-624 ([doi:10.1090/S1056-3911-07-00466-3](https://doi.org/10.1090/S1056-3911-07-00466-3))

[[!redirects complex orbifolds]]


This page collects links related to

* [[Michael Hopkins]],

  _Complex oriented cohomology theories and the language of stacks_

  1999

  ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf), [[HopkinsComplexOrientedCohomology.pdf:file]])

on [[complex oriented cohomology]], the [[Adams spectral sequence]], indicating the modern picture of [[E-infinity geometry]] over the "[[derived stacks]]" formally dual to [[S]] and [[MU]].

A predecessor is

* [[Frank Adams]], _[[Stable homotopy and generalised homology]]_, 1974

Much of this is later discussed more comprehensively in the lectures

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, 2010

See also

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

category: reference


complex oriented cohomology theory

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* table of contents$X$
{:toc}

## Idea

A _complex oriented cohomology theory_ is a [[Whitehead-generalized cohomology theory]] which is [[orientation in generalized cohomology|oriented]] on all [[complex vector bundles]]. Examples include [[ordinary cohomology]], complex [[topological K-theory]], [[elliptic cohomology]] and [[cobordism cohomology]].

The collection of all complex oriented cohomology theories turns out to be parameterized over the [[moduli stack of formal group laws]]. The [[stratification]] of this stack by the [[height of a formal group|height of formal group]] leads to the stratification of complex oriented cohomology theory by "[[chromatic level]]", a perspective also known as _[[chromatic homotopy theory]]_.

For more detailed introduction see at _[[Introduction to Cobordism and Complex Oriented Cohomology]]_.

## Definition
 {#Definition}

### In terms of generalized first Chern classes
 {#DefInTermsOfGeneralizedFirstChernClass}

Write $\mathbb{C}P^\infty \simeq B U(1) \simeq K(\mathbb{Z},2)$ for the infinite [[complex projective space]], equivalently the [[classifying space]] for [[circle group]]-[[principal bundles]] (an [[Eilenberg-MacLane space]]); write $S^2$ for the [[2-sphere]] and write

$$
  i \;\colon\; S^2 \longrightarrow B U(1)
$$

for a representative of $1 \in \mathbb{Z} \simeq  \pi_2(B U(1))$, classifying the [[universal complex line bundle]]. Regard both $S^2$ and $B U(1)$ as [[pointed homotopy types]] and take $i$ to be a pointed morphism.

Let $E^\bullet$ be a [[multiplicative cohomology theory]], i.e. a [[functor]] $X \mapsto \pi_\bullet[X,E]$ for $E$ a [[ring spectrum]]. Write $\tilde E^\bullet$ for the corresponding [[reduced cohomology]] on [[pointed topological spaces]], such that for any pointed space $X$ there is a canonical [[direct sum]] decomposition ([this prop.](generalized+%28Eilenberg-Steenrod%29+cohomology#UnreducedCohomologyIsReducedPlusPointValue))

$$
  E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)
  \,.
$$

By the [[suspension isomorphism]] there is an identification

$$
  \tilde E^2(S^2) \simeq \tilde E^0(S^0) \simeq E^0(\ast) \simeq \pi_0(E)
$$

with the [[commutative ring]] underlying $E$. Write $1 \in \pi_0(E)$ for the multiplicative identity element in this ring.

+-- {: .num_defn #ComplexOrientedCohomologyTheory}
###### Definition
**(complex oriented cohomology theory)**

A [[multiplicative cohomology theory]] $E$ is _complex orientable_ if the following equivalent conditions hold:

1. The morphism

   $$
     i^\ast \;\colon\; E^2(B U(1)) \longrightarrow E^2(S^2)
   $$

   is [[surjection|surjective]].

1. The morphism on [[reduced cohomology]]

   $$
     \tilde i^\ast
     \;\colon\;
     \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2)
     \simeq \pi_0(E)
   $$

   is [[surjection|surjective]].

1. The [[ring unit]] $1 \in \pi_0(E)$ is in the [[image]] of the morphism $\tilde i^\ast$.

A **complex orientation** on a [[multiplicative cohomology theory]] $E^\bullet$ is an element

$$
  c_1^E \in \tilde E^2(B U(1))
$$

(the "first [[generalized Chern class]]") such that

$$
  i^\ast c^E_1 = 1 \in \pi_0(E)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Since $B U(1) \simeq K(\mathbb{Z},2)$ is the [[classifying space]] for [[complex line bundles]], it follows that a complex orientation on $E^\bullet$ induces an $E$-[[generalized Chern class|generalization]] of the [[first Chern class]] which to a [[complex line bundle]] $\mathcal{L}$ on $X$, classified by $\phi \colon X \to B U(1)$  assigns the class $c_1(\mathcal{L}) \coloneqq \phi^\ast c_1^E$. This construction extends to a general construction of $E$-[[Chern classes]] ([[Conner-Floyd Chern classes]]).

=--

+-- {: .num_remark #ComplexOrientationByExtensionsAndTheirObstructions}
###### Remark
**(complex $E$-orientation by [[extensions]] and their [[obstructions]])**

In terms of [[Brown representability theorem|classifying maps]], Def. \ref{ComplexOrientedCohomologyTheory} means that a complex orientation $c_1^E$ in $E$-cohomology theory is equivalently an [[extension]] (in the [[classical homotopy category]]) of the map $\Sigma^2 1 \,\colon\, \mathbb{C}P^1 \longrightarrow \Omega^{\infty-2} E$ (which classifies the suspended identity in the cohomology ring) along the canonical inclusion of [[complex projective spaces]]

\[
  \label{ComplexOrientationAsExtension}
  \array{
     \mathbb{C}P^1
     &
     \overset{
       \Sigma^2 1_E
     }{
       \longrightarrow
     }
     &
     \Omega^{\infty - 2} E
     \\
     \big\downarrow
     &
     \nearrow \mathrlap{ {}_{c_1^E} }
     \\
     \mathbb{C}P^\infty
  }
\]

Notice that the [[complex projective spaces]] form a [[cotower]]

$$
  \ast
  \,=\,
  \mathbb{C}P^0
  \hookrightarrow
  \mathbb{C}P^1
  \hookrightarrow
  \mathbb{C}P^2
  \hookrightarrow
  \mathbb{C}P^3
  \hookrightarrow
  \cdots
  \hookrightarrow
  \mathbb{C}P^\infty
  \,=\,
  \underset{\longrightarrow}{\lim}
  \mathbb{C}P^\bullet
$$

where each inclusion stage is (by [this Prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace), see at _[[cell structure of projective spaces]]_) the coprojection of a [[pushout]] [of topological spaces](Top#UniversalConstructions) (or rather: of [[pointed topological spaces]]) of the form

$$
  \array{
    D^{2n+2}
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{C}P^{n+1}
    \\
    \big\uparrow
    &\mathclap{^{_{(po)}}}&
    \big\uparrow
    \\
    S^{2n+1}
    &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}&
    \mathbb{C}P^n
  }
$$

(where $h^{2n+1}_{\mathbb{C}}$ is the [[complex Hopf fibration]] in dimension $2n+1$) hence of a [[homotopy pushout]] of [[classical model structure on topological spaces|underlying homotopy types]] (rather: [[classical model structure on pointed topological spaces|of pointed homotopy types]]) of this form:

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{C}P^{n+1}
    \\
    \big\uparrow
    &\mathclap{^{_{(hpo)}}}&
    \big\uparrow
    \\
    S^{2n+1}
    &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}&
    \mathbb{C}P^n
  }
$$

Therefore, a complex orientation by extension (eq:ComplexOrientationAsExtension) is equivalently the homotopy colimiting map of a sequence

$$
  \big(
    \Sigma^2 1
      \,=\,
    c_1^{E,0}
    ,\,
    c_1^{E,1}
    ,\,
    c_1^{E,2}
    ,\,
    \cdots
  \big)
$$

of finite-stage extensions

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{C}P^{n+1}
    &
    \overset{ c_1^{E,n+1} }{\longrightarrow}
    &
    \Omega^{\infty -2} E
    \\
    \big\uparrow
    &\mathclap{^{_{(hpo)}}}&
    \big\uparrow
    &
    \nearrow \mathrlap{ {}_{c_1^{E,n}} }
    \\
    S^{2n+1}
    &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}&
    \mathbb{C}P^n
    \,.
  }
$$

Moreover, by the defining [[universal property]] of the [[homotopy pushout]], the extension $c_1^{E,n+1}$ of $c_1^{E,n}$ is equivalently a choice of [[homotopy]] which trivializes the pullback of $c_1^{E,n}$ to the [[n-sphere|2n+1-sphere]]:

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \Omega^{\infty - 2} E
    \\
    \big\uparrow
    &
    {}_{  c_1^{E,n+1}  }
    \seArrow
    &
    \big\uparrow \mathrlap{ ^{_{ c_1^{E,n} }} }
    \\
    S^{2n+1}
    &\underset{ h^{2n+1}_{\mathbb{C}} }{\longrightarrow}&
    \mathbb{C}P^n
    \,.
  }
$$

This means, first of all, that the non-triviality of the pullback class

$$
  \big(
    h^{2n+1}_{\mathbb{C}}
  \big)^\ast
  ( c_1^{E,n} )
  \;\in\;
  \widetilde E^2
  \big(
    S^{2n+1}
  \big)
  \;\simeq\;
  E_{2n-1}
$$

is the [[obstruction]] to the existence of the extension/orientation at this stage.

It follows that if these obstructions all vanish, then a complex $E$-orientation does exist.

{#ComplexOrientationOfEvenCohomologyTheories} A sufficient condition for this is, evidently, that the reduced $E$-cohomology of all odd-dimensional spheres vanishes, hence, that the [[graded-commutative ring|graded]] $E$-[[cohomology ring]] $E_\bullet$ is trivial in odd degrees (an [[even cohomology theory]]).

=--

(see also [Lurie, Lecture 6, Remark 4](#LurieLecture))

### In terms of orientations of fibers of complex vector bundles
 {#InTermsOfOrientations}

\begin{remark}
  \label{ComplexEOrientationsFromFirstEChernClass}
  **(complex $E$-orientations from first $E$-Chern class)**

To see that a generalized Chern class $c^E_1$ in the [above sense](#DefInTermsOfGeneralizedFirstChernClass) indeed has to do with _complex orientations_, namely of, first of all, [[fibers]] of [[complex line bundles]], notice that any [[complex projective space]] is equivalently the [[Thom space]] of the [[dual vector bundle|dual]] [[tautological line bundle]] (see [there](tautological+line+bundle#DualTautologicalLineBundle)) on the [[complex projective space]] of one complex dimension lower:

\begin{xymatrix@R=20pt@C=30pt}
    \mathllap{
      S^{2}
      =
      \;\;
    }
    \mathbb{C}P^1
    \ar@{^{(}->}[r]
    &
    \mathbb{C}P^2
    \ar@{=}[d]
    \ar@{^{(}->}[r]
    &
    \;\;
    \cdots
    \;\;
    \ar@{^{(}->}[r]
    &
    \mathbb{C}P^{n+1}
    \ar@{=}[d]
    \;
    \ar@{^{(}->}[rr]
      |-{
        \;
        \scalebox{.76}{
        \raisebox{-5pt}{$
          \cdots
        $}
        }
      \;}
    &
    &
    \mathbb{C}P^{\infty}
    \ar@{=}[d]
    \\
    &
    \mathrm{Th}
    \big(
      \mathcal{L}^\ast_{\mathbb{C}P^1}
    \big)
    \ar@{^{(}->}[r]
    &
    \;\;
    \cdots
    \;\;
    \ar@{^{(}->}[r]
    &
    \mathrm{Th}
    \big(
      \mathcal{L}^\ast_{\mathbb{C}P^n}
    \big)
    \;
    \ar@{^{(}->}[rr]
      |-{
        \;
        \scalebox{.76}{
        \raisebox{-5pt}{$
          \cdots
        $}
        }
      \;}
    &
    &
    \mathrm{Th}
    \big(
      \mathcal{L}^\ast_{\mathbb{C}P^\infty}
    \big)
    \ar@{=}[r]
    &
    \mathrm{Th}
    \big(
      E \mathrm{U}(1)
        \!\!
        \underset{\scalebox{.63}{$\mathrm{U}(1)$}}{\times}
        \!\!
      \mathbb{C}
    \big)
    \\
    \\
    &
    \mathllap{
      S^{2}
      =
      \;\;
    }
    \mathbb{C}P^1
    \ar@{^{(}->}[r]
    \ar[uu]
     |-{
       0_{{}_{
         \mathrm{Th}
         (
           \mathcal{L}^\ast_{\mathbb{C}P^1}
         )
       }}
       \mathclap{\phantom{\vert_{\vert_{\vert_{\vert}}}}}
     }
    &
    \;\;\cdots\;\;
    \ar@{^{(}->}[r]
    &
    \mathbb{C}P^{n}
    \ar[uu]
     |-{
       0_{{}_{
         \mathrm{Th}
         (
           \mathcal{L}^\ast_{\mathbb{C}P^n}
         )
       }}
       \mathclap{\phantom{\vert_{\vert_{\vert_{\vert}}}}}
     }
    \;
    \ar@{^{(}->}[rr]
      |-{
        \;
        \scalebox{.76}{
        \raisebox{-5pt}{$
          \cdots
        $}
        }
      \;}
    &
    &
    \mathbb{C}P^{\infty}
    \ar[uu]
     |-{
       0_{{}_{
         \mathrm{Th}
         (
           \mathcal{L}^\ast_{\mathbb{C}P^\infty}
         )
       }}
       \mathclap{\phantom{\vert_{\vert_{\vert_{\vert}}}}}
     }
     \ar@{=}[r]
     &
     B \mathrm{U}(1)
     \ar[uu]
     |-{
       \hspace{3pt}
       \scalebox{.7}{
         \rotatebox{-90}{$
           \;\simeq\;
         $}
       }
     }
\end{xymatrix}

The morphism of [E-cohomology Milnor sequences](lim1+and+Milnor+sequences#MilnorSequenceForGeneralizedCohomology) that this transformation induces shows (see the proof of Prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory} for details) that [[pullback in cohomology|pullback]] along the [[zero section]] identifies the $E$-cohomology of the [[Thom space]] of the [[universal complex line bundle]] with that of the base [[classifying space]]:

\begin{xymatrix@R=26pt@C=30pt}
    \widetilde E {}^\bullet
    \big(
      S^2
    \big)
    \ar@{=}[d]
    \ar@{<-}[rr]^-{
      \mbox{
        \tiny
        \color{blue}
        \bf
        restriction to fiber
      }
    }
    &&
    \widetilde E {}^\bullet
    \Big(
      \mathrm{Th}
      \big(
        E \mathrm{U}(1)
          \underset{\scalebox{.6}{$\mathrm{U}(1)$}}{\times}
        \mathbb{C}
      \big)
    \Big)
    \ar[d]
     |-{
       \hspace{3pt}
       \scalebox{.7}{
         \rotatebox{-90}{$
           \;\simeq\;
         $}
       }
     }
    &
    \mathrm{th}^E_1
    \ar@{|->}[d]
    \\
    \widetilde E^{\bullet}
    \big(
      S^2
    \big)
    \ar@{<-}[rr]_-{
      \mbox{
        \tiny
        \color{blue}
        \bf
        restriction to $\mathbb{C}P^1 \subset \mathbb{C}P^\infty$
      }
    }
    &&
    \widetilde E {}^{\bullet}
    \big(
      B\mathrm{U}(1)
    \big)
    &
    c^E_1
\end{xymatrix}

(In fact the [[zero-section]] of the [[Thom space]] of the [[universal complex line bundle]] is itself a [[weak homotopy equivalence]], see [this Lemma](universal+complex+orientation+on+MU#UniversalComplexLineBundleThomSpace).)

But this means that the $E$-Chern class $c^E_1$ has a unique pre-image which is an $E$-[[Thom class]] $th^E_1$ on the [[universal complex line bundle]], and hence [[pullback in cohomology|pulls back]] to an $E$-[[Thom class]] on any [[complex line bundle]].

In fact more is true: The choice of $c^E_1$ induces a sequence of [[universal characteristic class|universal]] [[Conner-Floyd Chern classes]] $c^E_n$ for all $n \in \mathbb{N}$, and by an elaboration of the previous argument these are [[pullback in cohomology|pullbacks]] along the [[zero-section]] of universal $E$-[[Thom classes]] on the [[universal complex vector bundle]] of any [[rank]] $n$ (see [this Lemma](universal+complex+orientation+on+MU#UniversalComplexVectorBundleThomSpace)).

\end{remark}

### In terms of genera
 {#InTermsOfGenera}

Complex orientation in the [above](#Definition) sense is indeed universal [[MU]]-[[orientation in generalized cohomology]]:

+-- {: .num_prop #ComplexOrientationIsMUAlgebraStructure}
###### Proposition

For $E$ a  [[homotopy commutative ring spectrum]], there is a [[bijection]] between complex orientations of $E$-cohomology and  [[homotopy commutative ring spectrum]]-[[homomorphisms]] $MU \longrightarrow E$ out of [[MU]].

=--

([Hopkins 99, section 4](#Hopkins99), [Lurie, lecture 6, theorem 8](#LurieLecture))

See at _[[universal complex orientation on MU]]_.

+-- {: .num_remark}
###### Remark

One might hope that the above universal property can be refined to say that that if $E$ is an $E_\infty$-ring spectrum, then complex orientations of $E$ are in bijection with $E_\infty$ ring maps $MU \to E$. This is not the case; Ando classified $H_\infty$ ring maps $MU \to E$ in [his thesis](#Ando92) and in particular showed that not every complex orientation gives rise to an $H_\infty$ ring map out of $MU$. More recently, [Hopkins and Lawson](#HopkinsLawson16) have classified the further structure constituting an $E_\infty$ ring map out of $MU$.

=--

## Examples

Examples of complex orientable cohomology theories:

+-- {: .num_example }
###### Example
**([[ordinary cohomology]])**

For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], the ordinary [[first Chern class]]

$$
  c_1 \in H^2(B U(1), \mathbb{Z})
$$

defines a complex orientation of $H\mathbb{Z}$.
=--

+-- {: .num_example }
###### Example
**([[topological K-theory]])**

For $E = KU$ complex [[topological K-theory]], then the class of the image of the [[universal complex line bundle]] $\mathcal{O}(1)$ in [[reduced K-theory]] is a complex orientation.

The induced [[formal group law]] (by prop. \ref{ComplexOrientedCohomologyTheoryFormalGroupLaw}) is the [[multiplicative formal group law]].

For details see at _[[topological K-theory]]_ the section _[Complex orientation and Formal group law](topological+K-theory#ComplexOrientationAndFormalGroupLaw)_.

=--

+-- {: .num_example #ComplexCobordism}
###### Example
**(complex cobordism)**

For $E = MU$ [[complex cobordism cohomology theory]], the canonical map

$$
  B U(1) \stackrel{\simeq}{\to} MU(1) \to MU
$$

defines a complex orientation.

=--

+-- {: .num_example #BrownPeterson}
###### Example

[[Brown-Peterson cohomology]] $E = B P^\bullet$.

=--

## Properties

### Cohomology ring of $B U(1)$
 {#CohomologyRingOfBU1}

+-- {: .num_prop #CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}
###### Proposition

Given a complex oriented cohomology theory $(E^\bullet, c^E_1)$ according to def. \ref{ComplexOrientedCohomologyTheory}, then there are [[isomorphisms]] of [[graded rings]]

1. $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$

   (between the $E$-[[cohomology ring]] of $B U(1)$ and the [[formal power series]] (but see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}) in one generator of even degree over the $E$-[[cohomology ring]] of the point);

1. $E^\bullet(B U(1) \times B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E \otimes 1 , 1 \otimes c_1^E ] ]$.

=--

+-- {: .proof}
###### Proof

We may realize the [[classifying space]] $B U(1)$ as the infinite [[complex projective space]] $\mathbb{C}P^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n$ ([exmpl.](classifying+space#RealComplexProjectiveSpaceAsGrassmannian)).
There is a standard [[CW-complex]]-structure on the [[classifying space]] $\mathbb{C}P^\infty$, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$ ([prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace)). With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.

Then consider the [[Atiyah-Hirzebruch spectral sequence]] for the $E$-cohomology of $\mathbb{C}P^n$.

$$
  H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \;\Rightarrow\;
  E^\bullet(\mathbb{C}P^n)
  \,.
$$

Since ([prop.](complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace)) the [[ordinary cohomology]] with [[integer]] [[coefficients]] of projective space is

$$
  H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n+1})
  \,,
$$

where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the [[ordinary homology]] of $\mathbb{C}P^n$ is a [[free abelian group]] ([prop.](complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace)), hence a [[projective object]] in abelian groups ([prop.](projective+object#ProjectiveObjectsInAbAreFreeGroups)), the [[Ext]]-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the [[universal coefficient theorem]] ([prop.](universal+coefficient+theorem#OrdinaryStatementInCohomology)) gives that the second page of the spectral sequence is

$$
  H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \simeq
  E^\bullet(\ast)[ c_1 ] / (c_1^{n+1})
  \,.
$$

By the standard construction of the [[Atiyah-Hirzebruch spectral sequence]] ([here](Atiyah–Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace)) in this identification the element $c_1$ is identified with a generator of the [[relative cohomology]]

$$
  E^2((\mathbb{C}P^n)_2, (\mathbb{C}P^n)_1)
  \simeq
  \tilde E^2(S^2)
$$

(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).

This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument, see ([Pedrotti 16](#Pedrotti16))).

The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials ([prop.](multiplicative+cohomology+theory#LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory)), implies that all these elements are permanent cocycles.

Since the AHSS of a multiplicative cohomology theory is a [[multiplicative spectral sequence]] ([prop.](multiplicative+cohomology+theory#AHSSForMultiplicativeCohomologyTheoryIsMultiplicative)) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give

$$
  \mathcal{E}_\infty^{\bullet,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1^{E} ] / ((c_1^E)^{n+1})
$$

or in more detail:

$$
  \mathcal{E}_\infty^{p,\bullet}
  \simeq
  \left\{
    \array{
      E^\bullet(\ast) & \text{if}\; p \leq  2n \; and\; even
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are [[free modules]] over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-[[module]] homomorphisms ([prop.](multiplicative+cohomology+theory#RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory)) the [extension problem](spectral+sequence#ExtensionProblem) trivializes, in that all the [[short exact sequences]]

$$
  0 \to F^{p+1}E^{p+\bullet}(X)
  \longrightarrow F^{p}E^{p+\bullet}(X) \longrightarrow \mathcal{E}_\infty^{p,\bullet} \to 0
$$

[[split exact sequence|split]] (since the [[Ext]]-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the [[free module]], hence [[projective module]] $\mathcal{E}_\infty^{p,\bullet}$).

In conclusion, this gives an isomorphism of graded rings

$$
  E^\bullet(\mathbb{C}P^n)
  \simeq
  \underset{p}{\oplus} \mathcal{E}_\infty^{p,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1 ] / ((c_1^{E})^{n+1})
  \,.
$$

A first consequence is that the projection maps

$$
  E^\bullet((\mathbb{C}P^\infty)_{2n+2})
  =
  E^\bullet(\mathbb{C}P^{n+1})
  \to
  E^\bullet(\mathbb{C}P^{n})
  =
  E^\bullet((\mathbb{C}P^\infty)_{2n})
$$

are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] ([def.](lim1#MittagLefflerCondition), [exmpl.](lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections)) and therefore the [[Milnor exact sequence]] for generalized cohomology ([prop.](lim1#MilorSequenceForReducedCohomologyOnCWComplex)) finally implies the claim:

$$
  \begin{aligned}
    E^\bullet(B U(1))
     & \simeq
    E^\bullet(\mathbb{C}P^\infty)
    \\
    & \simeq
    E^\bullet( \underset{\longrightarrow}{\lim}_n \mathbb{C}P^n )
   \\
    &\simeq \underset{\longleftarrow}{\lim}_n E^\bullet(\mathbb{C}P^n)
   \\
   &\simeq \underset{\longleftarrow}{\lim}_n ( E^\bullet(\ast) [c_1^E] / ((c_1^E)^{n+1})  )
   \\
   & \simeq E^\bullet(\ast)[ [ c_1^E ] ]
   \,,
  \end{aligned}
$$

where the last step is [this prop.](formal+scheme#FormalPowerSeries).

=--

+-- {: .num_remark #ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}
###### Remark

There is in general a choice to be made in interpreting the [[cohomology groups]] of a [[multiplicative cohomology theory]] $E$ as a [[ring]]:

a priori $E^\bullet(X)$ is a sequence

$$
  \{E^n(X)\}_{n \in \mathbb{Z}}
$$

of [[abelian groups]], together with a system of group homomorphisms

$$
  E^{n_1}(X) \otimes E^{n_2}(X)
    \longrightarrow
  E^{n_1 + n_2}(X)
  \,,
$$

one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.

In turning this into a single [[ring]] by forming [[formal sums]] of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.

In the former case the ring obtained is the [[direct sum]]

$$
  \oplus_{n \in \mathbb{N}} E^n(X)
$$

while in the latter case it is the [[Cartesian product]]

$$
  \prod_{n \in \mathbb{N}} E^n (X)
  \,.
$$

These differ in general. For instance if $E$ is [[ordinary cohomology]] with [[integer]] [[coefficients]] and $X$ is infinite [[complex projective space]] $\mathbb{C}P^\infty$, then ([prop.](complex+projective+space+OrdinaryCohomologyOfComplexProjectiveSpace))

$$
  E^n(X)
  =
  \left\{
    \array{
      \mathbb{Z} & n \; even
      \\
      0 & otherwise
    }
  \right.
$$

and the product operation is given by

$$
  E^{2{n_1}}(X)\otimes E^{2 n_2}(X)
    \longrightarrow
  E^{2(n_1 + n_2)}(X)
$$

for all $n_1, n_2$ (and zero in odd degrees, necessarily).  Now taking the [[direct sum]] of these, this is the [[polynomial ring]] on one generator (in degree 2)

$$
  \oplus_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z}[c_1]
  \,.
$$

But taking the [[Cartesian product]], then this is the [[formal power series ring]]

$$
  \prod_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z} [ [ c_1 ] ]
  \,.
$$

A priori both of these are sensible choices. The former is the usual choice in traditional [[algebraic topology]]. However, from the point of view of regarding [[ordinary cohomology]] theory as a [[multiplicative cohomology theory]] right away, then the second perspective tends to be more natural;

The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the [[inverse limit]] of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the [[category]] of [[commutative rings]] right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the [[formal power series ring]] (see [here](formal+scheme#FormalPowerSeries)).

See also for instance remark 1.1. in [[Jacob Lurie]]: _[[A Survey of Elliptic Cohomology]]_.

=--

\begin{example}\label{ComplexKTheoryOfClassifyingSpaceOfCircleGroup}
**([[topological K-theory]] of [[classifying space]] of [[circle group]])**
  For the case of [[topological K-theory]] $E = $ [[KU]], the cohomology ring $KU^\bullet(B U(1)) \;\simeq\; \mathbb{Z}[ [ c_1^{KU}] ]$ (Prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}) may also be computed via the [[Atiyah-Segal completion theorem]], see the [Example there](Atiyah-Segal+completion+theorem#ComplexTopologicalKTheoryOfBS1).
\end{example}

### Formal group law
 {#FormalGroupLaw}

Let again $B U(1)$ be the [[classifying space]] for [[complex line bundles]], modeled, in particular, by infinite [[complex projective space]] $\mathbb{C}P^\infty)$.

+-- {: .num_lemma #BU1HomotopyGroupStructure}
###### Lemma

There is a [[continuous function]]

$$
  \mu
    \;\colon\;
  \mathbb{C}P^\infty \times \mathbb{C}P
    \longrightarrow
  \mathbb{C}P^\infty
$$

which represents the [[tensor product of line bundles]] in that under the defining equivalence, and for $X$ any [[paracompact Hausdorff space]] (notably a [[CW-complex]], since all [[CW-complexes are paracompact Hausdorff spaces]]), then

$$
  \array{
     [X, \mathbb{C}P^\infty \times \mathbb{C}P^\infty]
     &\simeq&
     \mathbb{C}LineBund(X)_{/\sim} \times \mathbb{C}LineBund(X)_{/\sim}
     \\
     {}^{\mathllap{[X,\mu]}}\downarrow
       &&
     \downarrow^{\mathrlap{\otimes}}
     \\
     [X,\mathbb{C}P^\infty]
       &\simeq&
     \mathbb{C}LineBund(X)_{/\sim}
  }
  \,,
$$

where $[-,-]$ denotes the [[hom-sets]] in the (Serre-Quillen-)[[classical homotopy category]] and $\mathbb{C}LineBund(X)_{/\sim}$ denotes the set of [[isomorphism classes]] of [[complex line bundles]] on $X$.

Together with the canonical point inclusion $\ast \to \mathbb{C}P^\infty$, this makes $\mathbb{C}P^\infty$ an [[abelian group|abelian]] [[group object]] in the [[classical homotopy category]] (an abelian [[H-group]]).

=--

+-- {: .proof}
###### Proof

By the [[Yoneda lemma]] (the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]]) there exists such a morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \longrightarrow \mathbb{C}P^\infty$ in the [[classical homotopy category]]. But since $\mathbb{C}P^\infty$ admits the structure of a [[CW-complex]] ([prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace)) it is cofibrant in the [[standard model structure on topological spaces]], as is its [[Cartesian product]] with itself ([prop.](CW+complex#ClosureOfCWComplexesUnderCartesianProduct)). Since moreover all spaces are fibrant in the [[classical model structure on topological spaces]], it follows (by [this lemma](Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)) that there is an actual [[continuous function]] representing that morphism in the homotopy category.

That this gives the structure of an [[abelian group|abelian]] [[group object]] now follows via the [[Yoneda lemma]] from the fact that each $\mathbb{C}LineBund(X)_{/\sim}$ has the structure of an [[abelian group]] under [[tensor product of line bundles]], with the [[trivial bundle|trivial]] line bundle (wich is classified by maps factoring through $\ast \to \mathbb{C}P^\infty$) being the neutral element, and that this group structure is [[natural transformation|natural]] in $X$.

=--

+-- {: .num_remark}
###### Remark

The space $B U(1) \simeq \mathbb{C}P^\infty$ has in fact more structure than that of an [[H-group]] from lemma \ref{BU1HomotopyGroupStructure}. As an object of the [[homotopy theory]] represented by the [[classical model structure on topological spaces]], it is a _[[2-group]]_, a [[truncated object in an (infinity,1)-category|1-truncated]] [[infinity-group]].

=--

+-- {: .num_prop #ComplexOrientedCohomologyTheoryFormalGroupLaw}
###### Proposition

Let $(E, c_1^E)$ be a [[complex oriented cohomology theory]]. Under the identification

$$
  E^\bullet(\mathbb{C}P^\infty)
   \simeq
  \pi_\bullet(E)[ [ c^E_1 ] ]
  \;\;\;\,,
  \;\;\;
  E^\bullet(\mathbb{C}P^\infty \times \mathbb{C}P^\infty)
   \simeq
  \pi_\bullet(E)[ [ c^E_1 \otimes 1 , \, 1 \otimes c^E_1 ] ]
$$

from prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}, the operation

$$
  \pi_\bullet(E)
  [ [ c^E_1 ] ]
    \simeq
  E^\bullet(\mathbb{C}P^\infty)
    \longrightarrow
  E^\bullet( \mathbb{C}P^\infty \times \mathbb{C}P^\infty )
    \simeq
  \pi_\bullet(E)[ [ c_1^E \otimes 1, 1 \otimes c_1^E ] ]
$$

of pullback in $E$-cohomology along the maps from lemma \ref{BU1HomotopyGroupStructure} constitutes a 1-dimensional
graded-commutative [[formal group law]] ([exmpl.](formal+group#Commutative1DimFormalGroupLaw)) over the [[graded commutative ring]] $\pi_\bullet(E)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)). If we consider $c_1^E$ to be in degree 2, then this formal group law is compatibly graded.

=--

+-- {: .proof}
###### Proof

The associativity and commutativity conditions follow directly from the respective properties of the map $\mu$ in lemma \ref{BU1HomotopyGroupStructure}. The grading follows from the nature of the identifications in prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}.

=--

+-- {: .num_remark}
###### Remark

That the grading of $c_1^E$ in prop. \ref{ComplexOrientedCohomologyTheoryFormalGroupLaw} is in negative degree is because by definition

$$
  \pi_\bullet(E) = E_\bullet = E^{-\bullet}
$$

([rmk.](Introduction+to+Stable+homotopy+theory+--+1-2#EMHomology)).

Under different choices of orientation, one obtains different but isomorphic formal group laws.

=--

+-- {: .num_example}
###### Example

The [[formal group law]] of [[complex cobordism cohomology theory]], example \ref{ComplexCobordism} is [[universal property|universal]] in that for every [[commutative ring]] $R$ there is a [[natural bijection]]

$$
  CRing(MU^\bullet, R)
  \simeq
  FormalGroupLaws_{/R}
  \,.
$$

$MU^\bullet$ is the _[[Lazard ring]]_.

This is _[[Milnor-Quillen's theorem on MU]]_ (involving [[Lazard's theorem]]).

=--

+-- {: .num_example}
###### Example

The [[formal group law]] of [[Brown-Peterson cohomology theory]], example \ref{BrownPeterson} is [[universal property|universal]] for $p$-local cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, [[p-typical formal group laws]].

=--

### Cohomology ring of $B U(n)$
 {#TheCohomologyRingOfBUn}

+-- {: .num_prop #CohomologyRingOfBUn}
###### Proposition

For $E$ a complex oriented cohomology theory and $n \in \mathbb{N}$, restriction along the canonical map

$$
  (B U(1))^n  \longrightarrow B U(n)
$$

induces an [[isomorphism]]

$$
  E^\bullet(B U(n))
    \stackrel{\simeq}{\longrightarrow}
  (\pi_\bullet E)[ [ c^E_1, \cdots, c^E_n ] ]
  \simeq
  E^\bullet((B U(1))^n)^{\Sigma_n}
  \hookrightarrow
  E^\bullet((B U(1))^n)
  \simeq
  (\pi_\bullet E)[ [(c_1^E)_1, \cdots (c_1^E)_n ] ]
  \,,
$$

of $E^\bullet(B U(n))$ with the [[cyclic group]]-[[invariants]] in $E^\bullet((B U(1))^n)$, hence with the [[power series]] ring in the [[elementary symmetric polynomials]] $c_i^E$ (the [[generalized Chern classes]]) in the $c_1^E$-s (the generalized first Chern classes of prop. \ref{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}).

=--

Use [this proposition](ordinary+homology+spectra+split#WhenGeneralizedHomologySpectraSplit) to reduce to the situation for ordinary [[Chern classes]]. (e.g. [Lurie 10, lecture 4](#LurieLecture))

### Canonical orientation on complex vector bundles

The follows says that complex oriented cohomology theories in the sense of def. \ref{ComplexOrientedCohomologyTheory}, indeed canonically have an [[orientation in generalized cohomology]] for the ([[spherical fibration]] of) any [[complex vector bundle]].

For more details see at _[[universal complex orientation on MU]]_.

+-- {: .num_prop #ThomSpaceOfZetan}
###### Proposition

For $E$ any [[cohomology theory]] and $n \in \mathbb{N}$, $n \geq 1$, there is a canonical [[isomorphism]] of [[relative cohomology]]

$$
  E^\bullet(B U(n), B U(n-1))
  \simeq
  E^\bullet( B( \zeta_n), S( \zeta_n) )
  \,,
$$

where $\zeta_n \coloneqq E U(n) \underset{U(n)}{\times} \mathbb{R}^{2n}$ is the [[universal complex vector bundle]].

=--

+-- {: .proof}
###### Proof

Observe that the [[sphere bundle]] $S(\zeta_n) \to B U(n)$ of the [[universal complex vector bundle]] is equivalently the canonical map $B U(n-1) \to B U(n)$.

This follows form the fact that $S^{2n-1} \simeq U(n)/U(n-1)$ and that hence the [[unit sphere]] bundle is equivalently the quotient of the $U(n)$-[[universal principal bundle]] by $U(n-1)$

$$
  \array{
    U(n) &\longrightarrow& \ast
    \\
    && \downarrow
    \\
    && B U(n)
  }
  \;\;\;\; \stackrel{}{\mapsto}\;\;\;\;
  \array{
     S^{2n-1} \simeq & U(n)/U(n-1) &\longrightarrow& B U(n-1)
     \\
     & && \downarrow
     \\
     & &&  B U(n)
  }
  \,.
$$

The unit ball bundle $B(\zeta_n)$ is weakly equivalent to $B U(n)$, and under this identification the map $S(\zeta_n) \to B(\zeta_n)$ is equivalent to $B U(n-1) \to B U(n)$.

=--

+-- {: .num_prop }
###### Proposition

For $E$ a complex oriented cohomology theory,
its $n$th [[generalized Chern class]] $c^E_n$, prop. \ref{CohomologyRingOfBUn}, identified as an element of
$E^\bullet(B(\zeta_n), S(\zeta_n))$ via prop. \ref{ThomSpaceOfZetan},
is a [[Thom class]].

=--

(e.g. [Lurie 10, lecture 5, prop. 6](#LurieLecture))

## Related concepts

* [[real oriented cohomology theory]]

* [[quaternionic oriented cohomology theory]]

* [[equivariant complex oriented cohomology theory]]

* [[generalized (Eilenberg-Steenrod) cohomology theory]]

* [[multiplicative cohomology theory]]

* [[orientation in generalized cohomology]]

* [[complex cobordism cohomology theory]]

* [[Landweber exact functor theorem]]

* [[chromatic homotopy theory]]

[[!include chromatic tower examples - table]]

## References

### General

Textbook accounts:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], around Theorem 7.6 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* [[Frank Adams]], part II, section 2 of: _[[Stable homotopy and generalised homology]]_, 1974

* [[John Michael Boardman]], Section 5 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#Kochmann96} [[Stanley Kochmann]], section 4.3 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Doug Ravenel]], Section 4.1 of: _[[Complex cobordism and stable homotopy groups of spheres]]_

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Section 3.2 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Introduction includes

* _[[Introduction to Cobordism and Complex Oriented Cohomology]]_

* {#Pedrotti16} [[Riccardo Pedrotti]], _Complex oriented cohomology, generalized orientation and Thom isomorphism_, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])

The perspective of [[chromatic homotopy theory]] originates in

* {#Hopkins99} [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, 1999 course notes ([[HopkinsComplexOrientedCohomology.pdf:file]])

and is further developed in

* {#LurieLecture} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture notes, 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html))

  Lecture 4 _Complex-oriented cohomology theories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf))

  Lecture 6 _MU and complex orientations_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf))

See also the references at _[equivariant cohomology -- References -- Complex oriented cohomology](equivariant+cohomology#InComplexOrientedGeneralizedCohomologyTheory)_-.

A comparison between complex orientations and $H_\infty$ ring maps out of $MU$ was given in

* {#Ando92} [[Matthew Ando]], _Operations in complex-oriented cohomology theories related to subgroups of formal groups_ ([PhD Thesis](https://dspace.mit.edu/bitstream/handle/1721.1/13230/26903638-MIT.pdf?sequence=2))

More recent developments include

* {#HopkinsLawson16} [[Michael Hopkins]], [[Tyler Lawson]], _Strictly commutative complex orientation theory_ ([arXiv:1603.00047](http://arxiv.org/abs/1603.00047))

Refinement to [[equivariant complex oriented cohomology theory]]:

* [[John Greenlees]], _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 ([euclid:hha/1139840255](https://projecteuclid.org/euclid.hha/1139840255))

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _The universality of equivariant complex bordism_, Math Z 239, 455–475 (2002) ([doi:10.1007/s002090100315](https://doi.org/10.1007/s002090100315))

[[!include finite-rank complex orientation and MΩSUn -- references]]

[[!redirects complex oriented cohomology theories]]

[[!redirects complex oriented cohomology]]
[[!redirects complex-oriented cohomology]]

[[!redirects complex-oriented cohomology theory]]
[[!redirects complex-oriented cohomology theories]]

[[!redirects complex orientable cohomology theory]]
[[!redirects complex orientable cohomology theories]]


complex path integral

The action functional is real valued. Path integral is defined for such real action functionals. The Wiener integral is an analogue which is well defined in statistics and which is like an analogue for imaginary values of action functional.

Recently, some considerations were given to general complex valued functional arising from the analytic continuation of the real case. This research is related to the study of the [[wall crossing]], [[Stokes phenomenon]] for path integrals, to the [[complex Chern-Simons theory]] and so on.

* [[Edward Witten]], _Analytic continuation of Chern–Simons theory_, [arXiv:1001.2933](http://arxiv.org/abs/1001.2933); _A new look at the path integral of quantum mechanics_, [arXiv:1009.6032](); D. Harlow, J. Maltz, E. Witten, _Analytic continuation of Liouville theory, [arxiv/1108.4417](http://arxiv.org/abs/1108.4417)
* D. D. Ferrante, G. S. Guralnik, Z. Guralnik, C. Pehlevan, _Complex path integrals and the space of theories_, [arxiv/1301.4233](http://arxiv.org/abs/1301.4233) (note a very good bibliography there)


complex phase

## Idea

The _complex [[phase]]_ of a [[complex number]] $c$ is the real number $\phi$ in the decomposition $c = {\vert c \vert} e^{i \phi}$ -- well defined modulo $2 \pi \mathbb{Z}$ if the [[modulus]]/[[absolute value]] ${|c|}$ is [[positive number|positive]], and modulo $\pi \mathbb{Z}$ if the modulus is merely [[real number|real]] (and nonzero).  If $c$ (equivalently ${|c|}$) is [[zero]], then the phase is entirely unspecified.

[[!redirects complex phase]]
[[!redirects complex phases]]


complex projective 3-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Complex projective 3-space $\mathbb{C}P^3$ is the [[complex projective space]] $\mathbb{C}P^n$ for $n = 3$.

## Properties

### Coset space realization

Complex projective space has the following [[coset space]]-realizations:

* [[special orthogonal group]] in dimension 5 quotiented by [[unitary group]] in complex dimension 2:

  $$
    \mathbb{C}P^3
    \;\simeq\;
    SO(5)/U(2)
  $$

* [[Sp(2)]] quotiented by [[Sp(1)]] [[direct product group|times]] [[circle group]]:

  $$
    \mathbb{C}P^3
    \;\simeq\;
    Sp(2)/
    \big(
      Sp(1)_L \times \mathrm{U}(1)_R
    \big)
  $$

  (e.g. [Onishchik 60, Таблица 1](#Onishchik60), [Zandi 88, 7](#Zandi88), [Iriye 90, (3)](#Iriye90), [Gorbatsevich-Onishchik 93, Table 3](#GorbatsevichOnishchik93) [Butruille 06, p. 2](#Butruille06))

### Calabi-Penrose fibration

... _[[twistor space]]_ ... _[[Calabi-Penrose fibration]]_

## Related concepts

* [[complex projective space]]

  * [[complex projective line]]

  * [[complex projective plane]]

## References

Complex projective 3-space is conceived in the guise as the [[twistor space]] of 4d [[Minkowski spacetime]] in

* {#Penrose67} [[Roger Penrose]], Section VI of: _Twistor algebra_,  Journal of Mathematical Physics 8 (2): 345–366, (1967) ([doi:10.1063/1.1705200](https://doi.org/10.1063/1.1705200))

Discussion of $\mathbb{C}P^3$ as the [[domain]] of the [[twistor fibration]] and with an eye towards [[Yang-Mills theory]]:

* [[Michael Atiyah]], Section III.1 of: _Geometry of Yang-Mills fields_, Pisa, Italy: Sc. Norm. Sup. (1979) 98 ([spire:150867](https://inspirehep.net/literature/150867))

See also

* {#Onishchik60} A. L. Onishchik, _On compact Lie groups transitive on certain manifolds_, Dokl. Akad. Nauk SSSR, 135:3 (1960), 531–534 ([mathnet:dan24279](http://mi.mathnet.ru/eng/dan24279))

* {#Zandi88} Ahmad Zandi, _Minimal immersion of surfaces in quaternionic projective space_, Tsukuba Journal of Mathematics Vol. 12, No. 2 (1988), pp. 423-440 (18 pages) ([jstor:43686661](https://www.jstor.org/stable/43686661))

* {#Iriye90} [[Kouyemon Iriye]], _Manifolds which have two projective space bundle structures from the homotopical point of view_, J. Math. Soc. Japan Volume 42, Number 4 (1990), 639-658 ([euclid:jmsj/1227108441](https://projecteuclid.org/euclid.jmsj/1227108441))

* {#GorbatsevichOnishchik93} V. Gorbatsevich, A. L. Onishchik, _Compact homogeneous spaces_, Chapter 5 in: _Lie Groups and Lie Algebras II: Lie Transformation groups_, Encyclopaedia of Mathematical Sciences, vol 20. Springer, Spinger 1993 ([doi:10.1007/978-3-642-57999-8_11](https://doi.org/10.1007/978-3-642-57999-8_11))

* {#Butruille06} Jean-Baptiste Butruille, _Homogeneous nearly Kähler manifolds_, in: Vicente Cortés (ed) _Handbook of Pseudo-Riemannian Geometry and Supersymmetry_, pp 399–423 ([arXiv:math/0612655](https://arxiv.org/abs/math/0612655), [doi:10.4171/079-1/11](https://doi.org/10.4171/079-1/11))

On the [[KO-theory]] of $\mathbb{C}P^3$:

* [[Michael Atiyah]], E. Rees, _Vector bundles on projective 3-space_, Invent. Math. 35, 131–153 (1976) ([[AtiyahReesVectorBundlesOnCP3.pdf:file]])

On symmetries of manifolds of the [[homotopy type]] of $\mathbb{C}P^3$:

* Mark Hughes, _Symmetries of Homotopy Complex Projective Three Spaces_, Transactions of the American Mathematical Society
Vol. 337, No. 1 (May, 1993), pp. 291-304 ([doi:10.2307/2154323](https://doi.org/10.2307/2154323))

On toric symmetries of $\mathbb{C}P^3$:

* Dagan Karp, Dhruv Ranganathan, Paul Riggins, Ursula Whitcher,
_Toric symmetry of $\mathbb{C}P^3$_, Advances in Theoretical and Mathematical Physics, Vol. 16, No. 4, 2012 ([arXiv:1109.5157](https://arxiv.org/abs/1109.5157))

[[!redirects CP3]]


complex projective plane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _complex projective plane_ $\mathbb{C}P^2$ is the [[complex projective space]] of [[complex numbers|complex]] [[dimension]] 2, hence the [[projective plane]] over the [[complex numbers]].

## Properties

### Rational homotopy type

A [[Sullivan model]] is given by

$$
  \array{
    d\, \alpha_2 & = 0
    \\
    d\, \alpha_5 & = \alpha_2 \wedge \alpha_2 \wedge \alpha_2
  }
$$

* Luc Menichi, Section 5.3 of: _Rational homotopy – Sullivan models_

### Quotient by complex conjugation is 4-sphere

+-- {: .num_prop}
###### Proposition
**([[Arnold-Kuiper-Massey theorem]])**

The [[4-sphere]] is the [[quotient space]] of the [[complex projective plane]] by the [[action]] on homogeneous coordinates of [[complex conjugation]]:

$$
  \mathbb{C}P^2 / (-)^*
  \simeq
  S^4
$$

=--

### Blowup to del Pezzo surfaces

The [[blowup]] at [[generic points]] is a [[del Pezzo surface]].

## Related concepts

* [[complex projective space]]

  * [[complex projective line]]

  * [[complex projective 3-space]]

## Referenced

See also

* Wikipedia, _[Complex projective plane](https://en.m.wikipedia.org/wiki/Complex_projective_plane)_

[[!redirects complex projective planes]]