Whitehead integral formula

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

An expression for [[Hopf invariants]] in terms of [[secondary characteristic classes]] induced from the [[intersection pairing]], also known as _[[functional cup products]]_ ([Steenrod 49](#Steenrod49)) or _[[homotopy periods]]_ ([Sinha-Walter 13](#SinhaWalter13)).

The original expression due to [Whitehead 47](#Whitehead47) (see [Bott-Tu 82, Prop. 17.22](#BottTu82)) is in terms of [[smooth functions]] to an [[n-sphere]], which implies that [[wedge product]]/[[cup product]] of the [[pullback of differential forms|pullback]] of the [[volume form]] with itself vanishes identically.

More generally the homotopy Whitehead formula applies to general [[cocycles]] in [[cohomotopy]]. Its existence was suggested in [Haefliger 78, p. 17](#Haefliger78), worked out for the case of maps from the [[3-sphere]] to the [[2-sphere]] in [Griffith-Morgan 81, Section 14.5](#GriffithMorgan13) and stated generally but without proof in [Sinha-Walter 13, Example 1.9](#SinhaWalter13). A transparent proof of the general expression via lifts in [[cohomotopy]] through [[Hopf fibrations]] is in [FSS 19](#FSS19), relating the expression to the [[Hopf-Wess-Zumino term]] of the [[M5-brane]].

## Related concepts

* [[functional cup product]]

## References

* {#Whitehead47} [[J. H. C. Whitehead]], _An expression of Hopf's invariant as an integral_, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 ([jstor:87688](https://www.jstor.org/stable/87688))

* {#Steenrod49} [[Norman Steenrod]], _Cohomology Invariants of Mappings_, Annals of Mathematics Second Series, Vol. 50, No. 4 (Oct., 1949), pp. 954-988 ([jstor:1969589 ](https://www.jstor.org/stable/1969589 ))

* [[Hassler Whitney]], Section 31 in _Geometric Integration Theory_, 1957 ([pup:3151](https://press.princeton.edu/titles/3151.html))

* {#Haefliger78} [[André Haefliger]], p. 3 of _Whitehead products and differential forms_, In: P.A. Schweitzer (ed.), _Differential Topology, Foliations and Gelfand-Fuks Cohomology_, Lecture Notes in Mathematics, vol 652. Springer 1978 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Prop. 17.22 in _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics **82**, Springer (1982) [[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)]

* Lee Rudolph, _Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number_, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 ([arXiv:0912.4974](https://arxiv.org/abs/0912.4974))

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], Section 14.5 of _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhauser (1981, 2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))

* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883  ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))

* [[Felix Wierstra]], *Hopf Invariants in Real and Rational Homotopy Theory* (2017) [[diva:146246](http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-146246), [pdf](https://www.diva-portal.org/smash/get/diva2:1136442/FULLTEXT02.pdf)]

* {#FSS19} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], pp. 18 of: _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_, Comm. Math. Phys. **384** (2021) 403-432 [[arXiv:1906.07417](https://arxiv.org/abs/1906.07417), [doi:10.1007/s00220-021-03951-0](https://doi.org/10.1007/s00220-021-03951-0)]

[[!redirects Whitehead integral formulas]]

[[!redirects Whitehead's integral formula]]
[[!redirects Whitehead's integral formulas]]

[[!redirects homotopy Whitehead integral formula]]
[[!redirects homotopy Whitehead integral formulas]]

[[!redirects homotopy Whitehead's integral formula]]
[[!redirects homotopy Whitehead's integral formulas]]

[[!redirects Whitehead integral]
[[!redirects Whitehead integrals]]

[[!redirects Whitehead's integral]]
[[!redirects Whitehead's integrals]]

[[!redirects homotopy Whitehead integral]]
[[!redirects homotopy Whitehead integrals]]

[[!redirects homotopy Whitehead's integral]]
[[!redirects homotopy Whitehead's integrals]]


Whitehead link

The __Whitehead link__ is a famous [[link]] which shows the difference between the notion of [[equivalence]].  If links are only allowed to move by [[isotopy|isotopies]], then the two components are linked. However, if a link is allowed to pass through itself, then they can be unlinked.

[[!include Whitehead link - SVG]]

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###### Note
To include this picture in a page, write
<nowiki><code>[[!include Whitehead link - SVG]]</code></nowiki>.
=--

[[!redirects Whitehead link]]
[[!redirects Whitehead links]]


Whitehead link - SVG

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category: svg


Whitehead principle of nonabelian cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _[[nonabelian cohomology]]_ is the general [[cohomology|intrinsic cohohomology]] of any [[(∞,1)-topos]] $\mathbf{H}$ with coefficients in any object $A \in \mathbf{H}$, not necessarily an [[Eilenberg-MacLane object]].

But there is a general notion of [[Postnikov tower in an (∞,1)-category]] that applies in any [[locally presentable (∞,1)-category]], in particular in [[(∞,1)-topos]]es.

This implies that every object $A\in \mathbf{H}$ has a decomposition as a sequence of objects

$$
  A \to \cdots \to A_3 \to A_2\to A_1 \to A_0 \to *
  \,,
$$

where $A_k$ is an $k$-[[truncated]] object, in fact the $k$-truncation of $A$.

This implies that every $n$-[[truncated]] [[connected]] object $A$ is given by a possibly nonabelian 0-truncated group object $G$ and a sequence of abelian extensions of the [[delooping]] $\mathbf{B}G$ in that we have [[fiber sequence]]s

$$
  \mathbf{B}^2 K_1 \to A_2 \to \mathbf{B}G = A_1
$$

etc.

(...)

It follows that any [[cocycle]] $X \to A_2$ decomposes into the [[principal bundle]] classified by $X \to \mathbf{B}G$ and an abelian $\mathbf{B}^2 K$-cocycle on its total space

(...)

## Examples

A [[string structure]] is a nonabelian cocycle with coefficients in the [[string 2-group]]. This is equivalently a $\mathbf{B}U(1)$-cocycle (a [[bundle gerbe]]) on the total space of the underlying $Spin$-[[principal bundle]]. See the section In terms of classes on the total space.

## References

The term "Whitehead principle" for nonabelian cohomology is used in

* {#Toen02} [[Bertrand Toën]], p. 8 of: _Stacks and Non-abelian cohomology_, lecture at _[Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory](https://www.msri.org/realvideo/index04.html)_, MSRI 2002 ([slides](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html), [ps](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps), [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf), [[ToenStacksAndNonabelianCohomology.pdf:file]])

[[!redirects Whitehead principle of non-abelian cohomology]]


Whitehead product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Basic definition

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

For $X$ a [[pointed space|pointed]] [[topological space]] ([[CW-complex]]), the _Whitehead products_ ([Whitehead 41, Section 3](#Whitehead41)) are the [[bilinear maps]] on the elements of the [[homotopy groups]] $\pi_\bullet(X)$ of $X$ of the form

\[
  \label{WhiteheadProductMapInSingleDegrees}
  [-,-]_{Wh}
  \;\colon\;
  \pi_{n_1}\big( X \big)
  \otimes_{\mathbb{Z}}
  \pi_{n_2}\big( X \big)
  \longrightarrow
  \pi_{n_1 + n_2 - 1}\big( X \big)
  \phantom{AA}
  \text{for all}
  \;
  n_i \in \mathbb{N}
  ;
  \;
  n_i \geq 1
  \,,
\]

given by sending any [[pair]] of [[homotopy classes]]

$$
  \big[
    S^{n_i} \overset{\phi_i}{\longrightarrow} X
  \big]
  \;\in\;
  \pi_{ n_i }
  \big(
    X
  \big)
$$

to the [[homotopy class]] of the top [[composition|composite]] in the [[diagram]]

$$
  \array{
    S^{ n_1 + n_2 -1 }
    &
    \overset{
      f_{n_1,n_2}
    }{
      \longrightarrow
    }
    &
    S^{n_1}
    \vee
    S^{n_2}
    &
    \overset{
      (\phi_1, \phi_2)
    }{\longrightarrow}
    &
    X
    \\
    \big\downarrow
    &
    (po)
    &
    \big\downarrow
    \\
    D^{ n_1 + n_2 }
    &\underset{}{\longrightarrow}&
    S^{n_1} \times S^{n_2}
  }
$$

where $f_{n_1, n_2}$ is [[generalized the|the]] [[attaching map]] exhibiting the [[product space]] $S^{n_1} \times S^{n_2}$ as the result of a [[cell attachment]] to the [[wedge sum]] $S^{n_1} \vee S^{n_2}$.

=--

In this form this appears for instance in [Félix-Halperin-Thomas, p. 176 with  p. 177](#FelixHalperinThomas00).

### Generalized version

There is also a **generalized Whitehead product** where we can take more general homotopy classes ([[continuous maps]] up to [[homotopy]]) $\alpha\in [S^\cdot Y,X]$ and $\beta\in [S^\cdot Z,X]$ to produce a class $[\alpha,\beta]_{Wh}\in[Y\star Z,X]$. Here $S^\cdot$ denotes the [[reduced suspension]] operation on pointed spaces and $\star$ denotes the [[join of spaces|join]] of CW-complexes. Notice that $pt\star Z = C^\cdot(Z)$ and the reduced [[cone]] of a point is $C^\cdot(pt)=S^1$. Thus for $Y=Z=pt$ the generalized Whitehead product reduced to the usual Whitehead product.

## Properties

### Super Lie algebra structure
  {#SuperLieAlgebraStructure}

If one assigns degree $n-1$ to the $n$th [[homotopy group]] $\pi_n$,
then the degree-wise Whitehead products (eq:WhiteheadProductMapInSingleDegrees)
organize into a single degree-0 bilinear pairing on the [[graded abelian group]] which is the [[direct sum]] of all the [[homotopy groups]]:

\[
  \label{DegreeShiftedHomotopyGroups}
  \array{
    \pi_{\bullet + 1}(X)
    =
    &
    \pi_1(X)
    &\oplus&
    \pi_2(X)
    &\oplus&
    \pi_3(X)
    &\oplus&
    \cdots
    \\
    deg =
    &
    0
    &&
    1
    &&
    2
  }
\]

This unified Whitehead product is _graded_ skew-symmetric in that for $\phi_i \in \pi_{n_i}\big( X \big)$ it satisfies

$$
  \big[
    \phi_1,
    \,
    \phi_2
  \big]_{Wh}
  \;=\;
  (-1)^{ n_1 n_2 }
  \big[
    \phi_2,
    \,
    \phi_1
  \big]_{Wh}

$$

and it satisfies the corresponding graded [[Jacobi identity]] ([Hilton 55, Theorem B](#Hilton55)).

This makes the Whitehead bracket the [[Lie bracket]] of a [[super Lie algebra]] [[structure]] on $\pi_{\bullet-1}(X)$ (eq:DegreeShiftedHomotopyGroups), over the [[ring]] of [[integers]] (sometimes called, in this context, a _graded quasi-Lie algebra_, see [below](#OfAnElementWithItself)).

### Of elements with themselves
 {#OfAnElementWithItself}

Beware that the skew-symmetry of [[Lie algebras]] over the [[integers]], as opposed to over a [[field]] of [[characteristic zero]], implies for any element $\phi$ of [[even number|even]] homogeneous degree --
hence here for elements of [[homotopy groups]] in [[odd number|odd]] degree -- only that the bracket with itself vanishes after multiplication by 2

$$
  [\phi,\phi]_{Wh} = - [\phi,\phi]_{Wh}
  \phantom{AA}
  \text{hence equivalently}
  \phantom{AA}
  2 \cdot [\phi,\phi]_{Wh} = 0
$$

but not necessarily that $[\phi,\phi]_{Wh} = 0$ by itself -- since [[multiplication]] by 2 is not an [[isomorphism]] over the [[integers]].

But this means that the Whitehead bracket of any even-degree [[element]] with itself -- hence of any [[element]] of a [[homotopy group]] in [[odd number|odd]] degree -- has [[order of a group element|order]] at most 2, hence is in the 2-[[torsion subgroup]] of the respective [[homotopy group]].

### As primary homotopy operations

The Whitehead products form one of the [[primary homotopy operations]].

In fact, together with [[composition operations]] and [[fundamental group]]-[[actions]] they [[generators and relations|generate]] all such operations.

This is related to the definition of [[Pi-algebras]].

### Relation to Pontrjagin product
 {#RelationToPontrjaginProduct}

Under the [[Hurewicz homomorphism]], the Whitehead product on homotopy groups is the [[commutator]] of the [[Pontrjagin product]] on integral [[ordinary homology|homology]] groups of a [[based loop space]].

This is due to [Samelson (1953)](#Samelson53) and for higher Whitehead brackets due to [Arkowitz (1971)](#Arkowitz71).

In fact, in [[characteristic zero]] the [[Pontrjagin ring]] is the [[universal enveloping algebra]] of the Whitehead bracket [[Lie algebra]] [[Milnor & Moore (1965) Appendix](#MilnorMoore65)].

A textbook account is in [Whitehead (1978) Thm. X.7.10](#Whitehead78).

### In terms of Samelson products

In the context of [[simplicial groups]], representing [[connected]] [[homotopy types]], there is a formula for the Whitehead product in terms of a [[Samelson product]], which in turn is derived from a [[shuffle]] product which is a sort of non-commutative version of the [[Eilenberg-Zilber map]].  These simplicial formulae come from an analysis of the structure of the [[product of simplices]].

(This formula for the Whitehead product is due to [[Dan Kan]] and can be found in the old survey article of [[Ed Curtis]].  The proof that it works was never published. For more pointers see [MO:q/296479/381](https://mathoverflow.net/q/296479/381))

### Relation to the Sullivan models
 {#RelationToSullivanModels}

We discuss (Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} below) how the [[rationalization]] of the [[Whitehead product]] is the co-binary part of the [[Sullivan model|Sullivan differential]] in [[rational homotopy theory]]. First we make explicit some notation and normalization conventions that enter this statement:

In the following, for $W$ a $\mathbb{Z}$-[[graded module]], we write

$$
  W \wedge W
  \;\coloneqq\;
  Sym^2(W)
  \;\coloneqq\;
  \big(
    W \otimes W
  \big)
  /
  \big(
   \alpha \otimes \beta
     \sim
   (-1)^{ n_\alpha n_\beta }
   \beta \otimes \alpha
  \big)
  \,,
$$

where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that "$(-)\wedge(-)$" is not to imply here a degree shift of the generators (as it typically does in the usual notation for [[Grassmann algebras]]).

Let $X$ be a [[simply connected topological space]] with [[Sullivan model]]

\[
  \label{SullivanModelX}
  CE( \mathfrak{l} X )
  \;=\;
  \big(
    Sym^\bullet\big(V^\ast\big), d_X
  \big)
\]

for $V^\ast$ the [[graded vector space]] of generators, which is the $\mathbb{Q}$-linear [[dual space|dual]] [[graded vector space]] of the [[graded object|graded]] $\mathbb{Z}$-[[module]] (=[[graded abelian group]]) of [[homotopy groups]] of $X$:

$$
  V^\ast
  \;\coloneqq\;
  Hom_{Ab}\big(
    \pi_\bullet(X),
    \mathbb{Q}
  \big)
  \,.
$$

Declare the [[wedge product]] pairing to be given by

\[
  \label{WedgeProductNormalization}
  \array{
    V^\ast
    \wedge
    V^\ast
    &\overset{\Phi}{\longrightarrow}&
    Hom_{Ab}
    \big(
      \pi_\bullet(X) \wedge \pi_\bullet(X)
      ,
      \mathbb{Q}
    \big)
    \\
    (\alpha, \beta)
    &\mapsto&
    \Big(
      v \wedge w
      \;\mapsto\;
      (-1)^{ n_\alpha \cdot n_\beta }
      \alpha(v)\cdot \beta(w)
      +
      \beta(v)\cdot \alpha(w)
    \Big)
  }
\]

where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.

(Notice that the usual normalization factor of $1/2$ is _not_ included on the right. This normalization follows [Andrews & Arkowitz 1978, above Thm. 6.1](#AndrewsArkowitz78).)

Finally, write

\[
  \label{PojectionToBinary}
  [-]_2
  \;\colon\;
  Sym^\bullet\big(V^\ast\big)
  \longrightarrow
  V^\ast \wedge V^\ast
\]

for the linear projection on quadratic polynomials in the graded [[symmetric algebra]].

Then:

+-- {: .num_prop #CoBinarySullivanDifferentialIsWhiteheadProduct}
###### Proposition
**([[co-binary Sullivan differential is Whitehead product]])**

Let $X$ be a [[simply connected topological space]] of rational [[finite type]], so that it has a [[Sullivan model]] with Sullivan differential $d_X$ (eq:SullivanModelX).

Then the co-binary component (eq:PojectionToBinary) of the Sullivan differential equals the $\mathbb{Q}$-[[linear dual map]] of the [[Whitehead product]] $[-,-]_X$ on the [[homotopy groups]] of $X$:

$$
  [d_X \alpha]_2
  \;=\;
  [-,-]_X^\ast
  \,.
$$

More explicitly, the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    V^\ast
      &\overset{ [-]_2\circ d_X }{\longrightarrow}&
    V^\ast \wedge V^\ast
    \\
    \big\downarrow\mathrlap{^=}
    &&
    \big\downarrow\mathrlap{^\Phi}
    \\
    Hom_{Ab}
    \big(
      \pi_\bullet(X),
      \mathbb{Q}
    \big)
    &
      \underset{
        Hom_{Ab}\big(
          [-,-]_X
          ,
          \;
          \mathbb{Q}
        \big)
      }{
        \longrightarrow
      }
    &
    Hom_{Ab}
    \big(
      \pi_\bullet(X)
        \wedge
      \pi_\bullet(X),
      \;
      \mathbb{Q}
    \big)
  }
  \,,
$$

where the wedge product on the right is normalized as in (eq:WedgeProductNormalization).

=--

([Andrews & Arkowitz 1978, Thm. 6.1](#AndrewsArkowitz78), following [Deligne, Griffiths, Morgan & Sullivan 1975](#DeligneGriffithsMorganSullivan75))

\begin{remark}
\label{AbsenceOfJacobiatorInWhiteheadBracket}
Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} says in particular that the binary bracket of the [[L-infinity-algebra|$L_\infty$-algebra]] dual to a [[Sullivan model]] is always an actual [[super Lie bracket]] in that it satisfies its super-[[Jacobi identity]], even if there happens to also be a nontrivial trinary bracket which would serve as a "[[Jacobiator]]".

This is due to the [[minimal model|minimality]] of Sullivan models, which implies that the co-unary part of their [[differential]] vanishes, and hence that that the unary bracket of the corresponding $L_\infty$-algebra vanishes: Since the failure of the Jacobi identity on binary brackets in an $L_\infty$-algebra is measured not by the trinary bracket itself but by its composition with the unary bracket, this vanishes in the above case.
\end{remark}

### Relation to Goodwillie Calculus:

On the relation to [[Goodwillie calculus]] see e.g. [Scherer & Chorny 2011, Sec. 1](#SchererChorny11), which also gives an application of the relationship between the Whitehead and [[Samelson products]].

## Examples

+-- {: .num_example #WhiteheadProductCorrespondingToComplexHopfFibration}
###### Example
**([[Whitehead product]] corresponding to [[complex Hopf fibration]])**

For $X = S^2$ the [[2-sphere]], consider the following two [[elements]] of its [[homotopy groups]] ([[homotopy groups of spheres|of spheres]], as it were):

1. $id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the [[identity function]] $S^2 \to S^2$)

1. $h_{\mathbb{C}} \in \pi_3\big(  S^2 \big)$ (represented by the [[complex Hopf fibration]])

Then the Whitehead product satisfies

$$
  \big[
    id_{S^2},
    \;
    id_{S^2}
  \big]
  \;=\;
  2 \cdot h_{\mathbb{C}}
  \,.
$$

=--

\linebreak

## Related concepts

* [[Pontrjagin product]]

* [[Samelson product]]

## References

### General

The concept is originally due to

* {#Whitehead41} [[J. H. C. Whitehead]], Section 3 of _On Adding Relations to Homotopy Groups_, Annals of Mathematics Second Series, Vol. 42, No. 2 (Apr., 1941), pp. 409-428 ([jstor:1968907](https://www.jstor.org/stable/1968907))

with early discussion in

* {#HiltonWhitehead53} [[Peter Hilton]], [[J. H. C. Whitehead]], _Note on the Whitehead Product_, Annals of Mathematics Second Series, Vol. 58, No. 3 (Nov., 1953), pp. 429-442  ([jstor:1969746](https://www.jstor.org/stable/1969746))

* {#Hilton55} [[Peter Hilton]], _On the homotopy groups of unions of spheres_, J. London Math. Soc., 1955, 30, 154–172 ([[Hilton55.pdf:file]], [doi:doi.org/10.1112/jlms/s1-30.2.154]( https://doi.org/10.1112/jlms/s1-30.2.154))

Proof that the Whitehead product is the [[commutator]] of the [[Pontrjagin product]]:

* {#Samelson53} [[Hans Samelson]], *A Connection Between the Whitehead and the Pontryagin Product*, American Journal of Mathematics **75** 4 (1953) 744–752  [[doi:10.2307/2372549](https://doi.org/10.2307/2372549)]

and in [[characteristic zero]]:

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], Appendix of: _On the structure of Hopf algebras_, Annals of Math. __81__ (1965) 211-264 [[doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf)]

and for higher Whitehead brackets:

* {#Arkowitz71} [[Martin Arkowitz]], *Whitehead Products as Images of Pontrjagin Products*, Transactions of the American Mathematical Society, **158** 2 (1971) 453-463 [[doi:10.2307/1995917](https://doi.org/10.2307/1995917)]

Textbook account:

* {#Whitehead78} [[George W. Whitehead]], §X.7 in: *Elements of Homotopy Theory*, Springer (1978) [[doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0)]

See also

* Wikipedia _[Whitehead product](http://en.wikipedia.org/wiki/Whitehead_product)_

Discussion of Whitehead products specifically of [[homotopy groups of spheres]]:

* {#James56} [[Ioan Mackenzie James]], _On the Suspension Triad_, Annals of Mathematics Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 191-247 ([arXiv:1969607](https://www.jstor.org/stable/1969607))

* {#James57} [[Ioan Mackenzie James]], _On the Suspension Sequence_, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 ([arXiv:1969666](https://www.jstor.org/stable/1969666))

As [[n-excisive functors]]:

* {#SchererChorny11} [[Jerome Scherer]], [[Boris Chorny]], _Goodwillie calculus and Whitehead products_, Forum Math. 27 (2015), no. 1, 119 - 130 ([arXiv:1109.2691](https://arxiv.org/abs/1109.2691))

Discussion of Whitehead products in [[homotopy type theory]]:

* [[Guillaume Brunerie]], section 3.3 of _On the homotopy groups of spheres in homotopy type theory_ ([arXiv:1606.05916](https://arxiv.org/abs/1606.05916))

### In rational homotopy theory
  {#ReferencesInRationalHomotopyTheory}

Discussion of Whitehead products in [[rational homotopy theory]] ([[the co-binary Sullivan differential is the dual Whitehead product]]):

* {#Quillen69} [[Daniel Quillen]], section I.5 of _Rational Homotopy Theory_, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](https://www.jstor.org/stable/1970725))

* Christopher Allday, _Rational Whitehead products and a spectral sequence of Quillen_, Pacific J. Math. Volume 46, Number 2 (1973), 313-323 ([euclid:1102946308](https://projecteuclid.org/euclid.pjm/1102946308))

* Christopher Allday, _Rational Whitehead product and a spectral sequence of Quillen, II_, Houston Journal of Mathematics, Volume 3, No. 3, 1977 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.434.8821&rep=rep1&type=pdf))

* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

* {#AndrewsArkowitz78} Peter Andrews, [[Martin Arkowitz]], *Sullivan's Minimal Models and Higher Order Whitehead Products*, Canadian Journal of Mathematics, **30** 5 (1978) 961-982 [[doi:10.4153/CJM-1978-083-6](https://doi.org/10.4153/CJM-1978-083-6)]

* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]], J. C. Thomas, Prop. 13.16 in: _Rational Homotopy Theory_, Graduate Texts in Mathematics **205** Springer (2000)

* Francisco Belchí, [[Urtzi Buijs]], José M. Moreno-Fernández, Aniceto Murillo, _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 ([arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008))

* Takahito Naito, _A model for the Whitehead product in rational mapping spaces_ ([arXiv:1106.4080](https://arxiv.org/abs/1106.4080))

The Whitehead product of $\mathbb{C}P^1 \vee \mathbb{C}P^1 \to \mathbb{C}P^\infty \times \mathbb{C}P^\infty$ in relation to the [[Dolbeault complex]]:

* Shamuel Auyeung, Jin-Cheng Guu, Jiahao Hu, pp. 4 of: *On the algebra generated by $\overline{\mu}$, $\overline{\partial}$, $\partial$, $\mu$*, Complex Manifolds **10** 1 (2023) [[arXiv:2208.04890](https://arxiv.org/abs/2208.04890), [doi:10.1515/coma-2022-0149](https://doi.org/10.1515/coma-2022-0149)]

[[!redirects Whitehead products]]

[[!redirects Whitehead bracket]]
[[!redirects Whitehead brackets]]

[[!redirects Whitehead Lie algebra]]
[[!redirects Whitehead Lie algebras]]

[[!redirects Whitehead super Lie algebra]]
[[!redirects Whitehead super Lie algebras]]

[[!redirects Whitehead bracket super Lie algebra]]
[[!redirects Whitehead bracket super Lie algebras]]

[[!redirects Whitehead algebra]]
[[!redirects Whitehead algebras]]

[[!redirects Whitehead L-infinity algebra]]
[[!redirects Whitehead L-infinity algebras]]

[[!redirects Whitehead bracket L-infinity algebra]]
[[!redirects Whitehead bracket L-infinity algebras]]


Whitehead theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### (∞,1)-Topos theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Classical case

The classical **Whitehead theorem** asserts that

+-- {: .num_theorem }
###### Theorem (Whitehead)

Every [[weak homotopy equivalence]] between [[CW-complexes]] is a [[homotopy equivalence]].

=--

(See also the discussion at [[m-cofibrant space]]).

Using the [[homotopy hypothesis]]-theorem this may be reformulated:

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

In the [[(∞,1)-category]] [[∞Grpd]] every [[weak homotopy equivalence]] is a [[homotopy equivalence]].

=--

## Simplicial version

\begin{theorem}
A [[simplicial map]] $f\colon X\to Y$ between [[Kan complexes]]
is a [[simplicial homotopy equivalence]] if and only if
for any $a$ and $b$ that make the following square commute
$$\array{
\partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^\iota&{}^{\exists d}\nearrow&\downarrow^f\\
\Delta^n &\stackrel{b}{\to}&Y\\
}
$$
there is a diagonal arrow $d$ that makes the upper triangle
commutative and the lower triangle commutative up to a homotopy
$h\colon \Delta^1\times\Delta^n\to Y$
that is constant on the boundary $\Delta^1\times\partial\Delta^n$.
\end{theorem}

Of course, this statement can be reformulated using
homotopy groups like the version for topological spaces,
but the above statement is more practical.

\begin{remark}
In the above criterion, the boundary inclusion
$$\partial\Delta^n\to\Delta^n$$
can be replaced by any weakly equivalent [[cofibration]].
\end{remark}

\begin{remark}
If $X$ or $Y$ is not a [[Kan complex]],
one can formulate a similar criterion
using [[barycentric subdivisions]] of $\partial\Delta^n$ and $\Delta^n$.
A [[simplicial map]] $f\colon X\to Y$ between [[simplicial sets]]
is a [[weak homotopy equivalence]] if and only if
for any $k\ge0$ and for any $a$ and $b$ that make the following square commute
$$\array{
Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^k \iota}&{}^{\exists d}\nearrow&\downarrow^f\\
Sd^k \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
there is $l\ge k$ such that in the outer rectangle in the diagram
$$\array{
Sd^l \partial\Delta^n&\to&Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^l \iota}&&\downarrow^{Sd^k \iota}&&\downarrow^f\\
Sd^l \Delta^n &\to&Sd^k \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
we can find a diagonal arrow
$$d\colon Sd^l \Delta^n \to X$$
that makes the upper triangle in the diagram
$$\array{
Sd^l \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^l \iota}&{}^{\exists d}\nearrow&\downarrow^f\\
Sd^l \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
commutative and the lower triangle commutative up to a homotopy
$$h\colon Sd^l(\Delta^1\times\Delta^n)\to Y$$
that is constant on the boundary $Sd^l(\Delta^1\times\partial\Delta^n)$.
\end{remark}

## Equivariant version

In $G$-[[equivariant homotopy theory]] the statement is that $G$-homotopy equivalences between [[G-CW complexes]] are equivalent to maps that are weak homotopy equivalences on [[fixed point]] spaces $H^H$ for all [[closed subset|closed]] [[subgroups]] $H \subset G$ (e.g. [Greenlees-May 95, theorem 2.4](#GreenleesMay95)). See at _[[equivariant Whitehead theorem]]_.

## In general $(\infty,1)$-toposes

There is a notion of [[homotopy groups]] for objects in every [[∞-stack]] [[(∞,1)-topos]], as described at [[homotopy group (of an ∞-stack)]]. Accordingly, there is a notion of [[weak homotopy equivalence]] in every [[∞-stack]] [[(∞,1)-topos]] and hence an analog of the statement of Whiteheads theorem. One finds that

**Warning** Whitehead's theorem **fails** for general [[(∞,1)-toposes]] and non-[[truncated object in an (infinity,1)-category|truncated]] objects.

The [[∞-stack]] [[(∞,1)-topos]]es in which the Whitehead theorem does hold are the [[hypercomplete (∞,1)-topos]]es. These are precisely the ones that are [[presentable (∞,1)-category|presented]] by a local [[model structure on simplicial presheaves]].

For instance the hypercomplete $(\infty,1)$-topos [[Top]] is presented by the model structure on simplicial presheaves on the point, namely the [[model structure on simplicial sets]].

## In homotopy type theory

Since [[homotopy type theory]] admits models in [[(∞,1)-toposes]] (and in particular in non-hypercomplete ones), Whitehead's theorem is not provable when regarded as a statement about types in homotopy type theory.  From this perspective, the truth of Whitehead's theorem is a [[foundational axiom]] that may be regarded as a "classicality" property, akin to [[excluded middle]] or the [[axiom of choice]] --- we call it **Whitehead's principle** (not to be confused with [[Whitehead's problem]], another statement that is independent of the usual axioms of set theory).

Whitehead's principle does hold, however, for maps between [[homotopy n-types]] for any finite $n$; this is provable in homotopy type theory by [[induction]] on $n$.

## Related concepts

* [[mod p Whitehead theorem]]

* [[equivariant Whitehead theorem]]

* [equivariant stable Whitehead theorem](G-spectrum#EquivariantWhitehead)

* [[CW-approximation theorem]]

* [[cellular approximation theorem]]

## References

Originally proved by [[J. H. C. Whitehead]]:

* [[J. H. C. Whitehead]], _Combinatorial homotopy.  I_, Bulletin of the American Mathematical Society 55:3 (1949), 213–246.  [doi](http://dx.doi.org/10.1090/s0002-9904-1949-09175-9).

The simplicial version is due to [[Daniel M. Kan]], see Theorem 7.2 in

* [[Daniel M. Kan]], _On c.s.s. categories_, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94.  [PDF](https://dmitripavlov.org/scans/kan-on-css-categories.pdf).

Review:

* {#Bredon93} [[Glen Bredon]], Cor. 11.14 in: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.),  _[[Handbook of Algebraic Topology]]_,  1995 Amsterdam: North-Holland, pp. 213–253,   ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], Thm. 6.3.31 in: *Algebraic topology from a homotopical viewpoint*, Springer (2008) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))

Discussion of the [[equivariant Whitehead theorem]]:

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

The [[(infinity,1)-topos|$(\infty,1)$-topos]] theoretic version:

* [[Jacob Lurie]], section 6.5 of [[Higher Topos Theory]]

The analogous formulation in [[homotopy type theory]]:

* {#UFP13} [[Univalent Foundations Project]], §8.8 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

Corresponding formalization in [[Agda]]:

* [[Dan Licata]], _[Whitehead.agda](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Whitehead.agda)_

category: foundational axiom

[[!redirects Whitehead's theorem]]
[[!redirects Whitehead's principle]]
[[!redirects Whitehead principle]]

[[!redirects simplicial Whitehead theorem]]


Whitehead torsion

#Contents#
* table of contents
{:toc}

## Idea

(...)

## References

* {#Milnor66} [[John Milnor]], *Whitehead torsion*, Bull. Amer. Math. Soc. **72** (1966) 358–426 [[doi:10:1090/S0002-9904-1966-11484-2](https://www.ams.org/journals/bull/1966-72-03/S0002-9904-1966-11484-2), [maths.ed.ac.uk](https://web.archive.org/web/20160529051526/http://www.maths.ed.ac.uk/~s1122173/surgerygroup12/milnorwh.pdf), [jstor:bams/1183527946](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-72/issue-3/Whitehead-torsion/bams/1183527946.full)]

* Wikipedia, _[Whitehead torsion](https://en.wikipedia.org/wiki/Whitehead_torsion)_

category: topology


Whitehead tower

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Whitehead tower_ of a [[pointed object|pointed]] [[homotopy type]] $X$ is an interpolation of the point inclusion $* \to X$ by a sequence of homotopy types

$$
  * \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)}
  \simeq X
$$

that are obtained from right to left by _removing [[homotopy groups]] from below_, hence such that

* each $X^{(n)}$ is $(n-1)$-[[n-simply connected space|connected]]

* and each [[morphism]] $X^{(n+1)} \to X^{(n)}$ induces an [[isomorphism]] on all [[homotopy groups]] in degree $k \geq (n+1)$ (and the inclusion $1 \to \pi_n(X^{(n)})$ in degree $n$ as well as the identity $1 = 1$ in degree $k \lt n$).

The notion of Whitehead tower is [[duality|dual]] to the notion of _[[Postnikov tower]]_, which instead is a factorization of the terminal morphism $X \to *$ into a tower, where homotopy groups are _added_ from right to left.

In fact, the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X$ to be the _[[homotopy fiber]]_ of the corresponding map into the $(n+1)$st stage of the [[Postnikov tower]].

## Definition

The construction of Whitehead towers is traditionally done for [[topological spaces]] regarded up to [[weak homotopy equivalence]], hence as objects of the [[(∞,1)-category]] [[Top]]. The discussion directly generalizes to any [[(∞,1)-topos]].

The **Whitehead tower** of a [[pointed object|pointed]] [[homotopy type]] $X$ is a sequence of [[homotopy types]]

$$* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X$$

where [[generalized the|the]] space $X^{(n)}$ is [[generalized the|the]] [[homotopy fiber]] of [[generalized the|the]] map $X \to X_{(n+1)}$ into the item $X_{(n+1)}$ in the [[Postnikov tower]] of $X$.

Here each [[homotopy pullback]]

$$
  \array{
    X^{(n)} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\to& X_{(n+1)}
  }
$$

in the [[(∞,1)-category]] [[Top]] may be computed (as described at [[homotopy pullback]]) as an ordinary [[pullback]] in the 1-[[category]] [[Top]] of a fibrantly replaced diagram, for instance with the point $*$ replaced by the path fibration $P X_{(n+1)} \simeq *$, which is a [[Hurewicz fibration]] $P X_{(n+1)} \to X_{(n+1)}$. In this case also the ordinary [[pullback]] $X^{(n)}\to X$

$$
  \array{
    X^{(n)} &\to& P X_{(n+1)}
    \\
    \downarrow && \downarrow
    \\
    X &\to& X_{(n+1)}
  }
$$

is a fibration, and this is often taken as part of the definition of the Whitehead tower.

From this perspective the _Whitehead tower_ of a [[pointed space]] $(X,x)$ is a sequence of fibrations

$$
\ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X
$$

where each $X\langle n\rangle \to X\langle n-1 \rangle$ induces isomorphisms on [[homotopy group]]s $\pi_i$ for $i\gt n$ and such that $X\langle n\rangle$ is $n$-[[n-connected|connected]] (has trivial [[homotopy group]]s $\pi_i$ for $i \leq n$).
The homotopy long exact sequence then shows that the fiber of $X\langle n\rangle \to X\langle n-1 \rangle$ is a $K(\pi_{n-1}(X,x),n-1)$ [[Eilenberg-Mac Lane space]]. One has a model for $K(\pi_{n-1}(X,x),n-1)$ which is an abelian topological group; this has a remarkable consequence when $(X,x)=(G,e)$ is a [[topological group]]. Indeed, in this case one sees inductively that $G\langle n\rangle$ has a model which is a topological group, which is an abelian group extension:

$$
1\to K(\pi_{n-1}(X,x),n-1) \to G\langle n\rangle \to G\langle n-1 \rangle \to 1
$$

For instance, the [[string group]] can be realized as a topological group as a $K(\mathbb{Z},2)$-extension of the [[spin group]].

For $n=0$ we require that $X\langle 0 \rangle \hookrightarrow X$ is the inclusion of the path-component of $x$. Really this is defined up to [[homotopy (as an operation)|homotopy]], but we have a canonical model. If $X$ is locally connected and semilocally path-connected, then $X\langle 1\rangle$ can be chosen as the [[universal covering space]].

In traditional models this construction is highly non-[[functor]]ial, except for nice spaces in low dimensions as remarked above.

## Constructions

### Whitehead's construction

[Whitehead 1952](#Whitehead) answered the question, posed by [[Witold Hurewicz]], of the existence of what we would now call $n$-connected \'covers\' of a given space $X$, taking this to mean a fibration $X\langle n\rangle \to X$ with $X\langle n\rangle$ $n$-connected and otherwise inducing isomorphisms on homotopy groups.

The construction proceeds as follows (using modern terminology). Given a pointed space $(X,x)$,

* Choose a representative for the [[Postnikov section]] $X_n$ such that $X \hookrightarrow X_n$ is a closed subspace (I would be tempted to make it a closed cofibration, but I don't know any reason for this to be necessary -DMR).

* Form the $\infty$-connected cover of $X_n$, i.e. the [[path fibration]] $P X_n$. This is a [[Hurewicz fibration]].

* Pull this back to $X$, to get $p\colon X\langle n\rangle \to X$, which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that $p$ has the desired properties.

This gives us a single $n$-connected cover, but by considering the [[Postnikov tower]]
$$
X \to (\ldots \to X_n \to X_{n-1} \to \ldots \to X_1 \to X_0)
$$
of $X$, where each map $X \to X_n$ is the inclusion of a closed subspace, it is simple to see there are induced maps $X\langle n\rangle \to X\langle n-1\rangle$ over $X$ for all $n$.

One way of obtaining a Postnikov section as above is to choose representatives $\phi_g\colon S^{n+1} \to X$ of generators $g$ of $\pi_{n+1}(X,x)$ and attaching cells: $X(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X$. We then choose representatives for the generators of $\pi_{n+2}(X(1),x)$ and attach cells and so on. The colimit $\lim_{\to n} X(n)$ is then a Postnikov section with the properties we require.

Understandably, this process is unbelievably non-canonical, and so we are generally reduced to existence theorems using this method -- unless there is a functorial way to construct Postnikov sections. Strictly speaking we can only say _an_ $n$-connected cover (except in special cases, like when $n=1$ and $X$ is a [[well-connected space]]).

### Functorial constructions

The $n$th stage of the Whitehead tower of $X$ is the [[homotopy fiber]] of the map from $X$ to the $n$th (or so) stage of its [[Postnikov tower]], so one can use a functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the [[path object|path space]] of the target).

The $n$th stage of the Whitehead tower of $X$ is also the cofibrant replacement for $X$ in the right [[Bousfield localization]] of [[Top]] with respect to the object $S^n$ (or so). Since [[Top]] is right proper and cellular this localization exists by the result of chapter 5 of Hirschhorn's book on [[localization]]s of [[model category|model categories]].

## Examples

### Whitehead tower of the orthogonal group
 {#OfTheOrthogonalGroup}

The Whitehead tower of the [[classifying space]]/[[delooping]] of the [[orthogonal group]] $O(n)$ starts out as

$$
  \array{
    & \mathbf{\text{Whitehead tower}}
    \\
    &\vdots
    \\
    & B Fivebrane &\to& \cdots &\to& *
    \\
    & \downarrow && && \downarrow
    \\
    \mathbf{\text{second frac Pontr. class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}&  B^8 \mathbb{Z}
    &\to& *
    \\
    & \downarrow && && \downarrow && \downarrow
    \\
    \mathbf{\text{first frac Pontr. class}} & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & *
    \\
    & \downarrow && && \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\text{second SW class}} & B S O
    &\to&
    \cdots
    &\to&
    &\to&
    & \stackrel{w_2}{\to} &
    \mathbf{B}^2 \mathbb{Z}_2
    &\to&
    *
    \\
    & \downarrow && && \downarrow && \downarrow &&  \downarrow && \downarrow
    \\
    \mathbf{\text{first SW class}} & B O
    &\to&
    \cdots
    &\to&
    \tau_{\leq 8 } B O
    &\to&
    \tau_{\leq 4 } B O
    &\to&
    \tau_{\leq 2 } B O
    &\stackrel{w_1}{\to}&
    \tau_{\leq 1 } B O
    \simeq
    B \mathbb{Z}_2
    &
    \mathbf{\text{Postnikov tower}}
  }
$$

where each square and each composite rectangle is a [[homotopy pullback]] square (all controled by the [[pasting law]]),

where the stages are the deloopings of

...
$\to$
[[fivebrane group]]
$\to$
[[string group]]
$\to$
[[spin group]]
$\to$
[[special orthogonal group]]
$\to$
[[orthogonal group]],

where lifts through the stages correspond to

* [[orthogonal structure]]

* [[orientation]]

* [[spin structure]]

* [[string structure]]

* [[fivebrane structure]]

and where the [[obstruction]] classes are the [[universal characteristic classes]]

* [[first Stiefel-Whitney class]] $w_1$

* [[second Stiefel-Whitney class]] $w_2$

* [[Pontryagin class|first fractional Pontryagin class]] $\tfrac{1}{2}p_1$

* [[Pontryagin class|second fractional Pontryagin class]] $\tfrac{1}{6}p_2$

and where every possible square in the above is a [[homotopy pullback]] square (using the [[pasting law]]).

For instance $w_2$ can be identified as such by representing $B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}$ by a [[Kan fibration]] (see at [[Postnikov tower]]) between [[Kan complexes]] so that then the [[homotopy pullback]] (as discussed there) is given by an ordinary pullback. Since $sSet$ is a [[simplicial model category]], $sSet(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $ B O \to \tau_{\leq 2} B O$ to an isomorphism on connected components. This identifies  $B SO \to B^2 \mathbb{Z}$ as being an [[isomorphism]] on the second [[homotopy group]]. Therefore, by the [[Hurewicz theorem]], it is also an isomorphism on the [[cohomology group]] $H^2(-,\mathbb{Z}_2)$. Analogously for the other characteristic maps.

In summary, more concisely, the tower is

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    B Fivebrane
    \\
    \downarrow
    \\
    B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z}
    \\
    \downarrow
    \\
    B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z}
    \\
    \downarrow
    \\
    B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2
    \\
    \downarrow
    \\
    B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    B GL
  }
  \,,
$$

where each "hook" is a [[fiber sequence]].

Via the [[J-homomorphism]] this corresponds to the [[stable homotopy groups of spheres]]:

[[!include image of J -- table]]

## Whitehead tower in general $(\infty,1)$-toposes

While a notion of [[Postnikov tower in an (∞,1)-category]] depends on the _categorical_ [[homotopy groups in an (∞,1)-topos|homotopy groups in an (∞,1)-category]], the notion of Whitehead tower makes good sense with respect to the _geometric_ homotopy groups.

A good notion of geometric [[homotopy groups in an (∞,1)-topos]] exist in a [[locally contractible (∞,1)-topos]]. The notion of Whitehead tower in this context is discussed at

* [[Whitehead tower in an (∞,1)-topos]].

## Related concepts

* Applying the [[Hurewicz theorem]] stagewise to a [[Whitehead tower]] yields an method for computing the [[homotopy groups]] of the original [[space]]. This process, or rather the refinement thereof for Whitehead towers generalized to [[Adams resolutions]], is formalized by the _[[Adams spectral sequence]]_, see there for more.

* _[symmetric group -- Whitehead tower](https://ncatlab.org/nlab/show/permutation#WhietheadTowerAndSupersymmetry)_

[[!include Lurie spectral sequences -- table]]

## References

The original reference is

*  [[George Whitehead]], _Fiber Spaces and the Eilenberg Homology Groups_, PNAS **38**, No. 5 (1952)
 {#Whitehead}

A textbook account is around example 4.20 in

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

A more detailed useful discussion happens to be in section 2.A, starting on p. 11 of

* [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_,  Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false))

[[!redirects Whitehead tower]]
[[!redirects Whitehead towers]]


Whitehead tower in an (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If an [[(∞,1)-topos]] $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] it has a good notion of  [[homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of its objects. In terms of these, there is an analog in $\mathbf{H}$ of the notion of the classical notion of [[Whitehead tower]] in the archetypical $(\infty,1)$-topos [[Top]]:

the Whitehead tower of an object $X \in \mathbf{H}$ is the tower of $n$-[[connected]] [[homotopy fiber]]s of the canonical morphism into the [[Postnikov tower in an (∞,1)-category|(∞,1)-topos-theoretic Postnikov tower]] $\cdots \to \mathbf{\Pi}_{n+1}(X) \to \mathbf{\Pi}_n(X) \to \cdots$ of the [[schreiber:path ∞-groupoid|structured path ∞-groupoid]] $\mathbf{\Pi}(X)$ of $X$.

See also the section Universal coverings and geometric Whitehead towers at [[cohesive (∞,1)-topos]].

## Definition

Let $\mathbf{H}$ be a [[locally ∞-connected (∞,1)-topos]] $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd$. Write

$$
  \mathbf{\Pi} := LConst \circ \Pi : \mathbf{H} \to \mathbf{H}
$$

for the _internal_ [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor. From the adjunction relation this comes with the canonical natural morphism

$$
  X \to \mathbf{\Pi}(X)
  \,.
$$

For $n \in \mathbb{N}$ write

$$
  \mathbf{H}_{\leq n} \stackrel{\overset{\tau_{\geq n}}{\leftarrow}}{\overset{}{\hookrightarrow}} \mathbf{H}
$$

for the [[reflective (∞,1)-subcategory]] of [[n-truncated object of an (∞,1)-category|n-truncated objects]] of $\mathbf{H}$ and write $\mathbf{\tau}_{\leq n}$ for the [[localization of an (∞,1)-category|localization]]

$$
  \mathbf{\tau}_{\leq n} : \mathbf{H} \stackrel{\tau_{\leq n}}{\to}
  \mathbf{H}_{\leq n} \hookrightarrow \mathbf{H}
  \,.
$$

Write

$$
  \mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\Pi}}{\to} \mathbf{H} \stackrel{\mathbf{\tau}_{\leq n}}{\to} \mathbf{H}
$$

for the internal **homotopy $n$-groupoid**. For $X \in \mathbf{H}$ we have the [[Postnikov tower in an (∞,1)-category|(∞,1)-Postnikov tower]]

$$
  \cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)
$$

of $\mathbf{\Pi}(X)$.

+-- {: .un_def}
###### Definition
**(Whitehead tower)**

For $X \in \mathbf{H}$, its **Whitehead tower** is the sequence of objects

$$
  * \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X
$$

in $\mathbf{H}$, where for each $n \in \mathbb{N}$ the object $X^{(n+1)}$ is the [[homotopy fiber]] of the canonical morphism $X \to \mathbf{\Pi}_{n+1}$, i.e. the object defined by the [[pullback]] diagram

$$
  \array{
    X^{(n+1)} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_{n+1}(X)
  }
  \,.
$$

=--

Here the morphisms $X^{(n+1)} \to X^{(n)}$ are induced from the universality of the pullback:

$$
  \array{
    X^{(n+1)}&\to&X^{(n)}&& \mathbf{\Pi}_{(n+1)}(X)
    \\
    &\searrow &\downarrow&\nearrow& \downarrow
    \\
    &&X &\to& \mathbf{\Pi}_n(X)
  }
$$

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

We have that $\mathbf{\Pi}_n(X) \simeq LConst \tau_{\leq n} \Pi(X)$.

A homotopy-commuting diagram

$$
  \array{
    X^{(n)} &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_n(X)
  }
$$

in $\mathbf{H}$ corresponds by the [[adjoint (∞,1)-functor|adjunction relation]] to diagram

$$
  \array{
    \Pi(X^{(n)}) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    \Pi(X) &\to& {\Pi}_n(X)
  }
$$

in [[∞Grpd]]. This being universal means that $\Pi(X^{(n)})$ is $n$-connected, and universal with that property as an object over $\Pi(X)$.

=--

## Properties {#Properties}

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

For $* \to X \in \mathbf{H}$ a [[pointed object]] and $n \in \mathbb{N}$, $n \geq 1 $, define the object $\mathbf{B}^n \mathbf{\pi}_n(X)$ to be the [[homotopy fiber]] of $\mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{n-1}(X)$, so that we have a [[fibration sequence]]

$$
  \mathbf{B}^n \mathbf{\pi}_n(X) \to \mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{n-1}(X)
  \,.
$$

=--

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

With $\Pi(X) \in \infty Grpd \simeq Top$ the underlying topological space of $X \in \mathbf{H}$ (its [geometric realization](http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#GeomReal)) we have that

$$
  \mathbf{B} \mathbf{\pi}_n(X)
  \simeq
  LConst \mathcal{B}^n \pi_n(X)
  \,,
$$

where $\mathcal{B}^n \pi_n(X)$ denotes the [[homotopy fiber]] of
$\Pi_n(X) \to \Pi_{(n-1)}(X)$ in [[∞Grpd]].

=--

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

> check

This follows from $\mathbf{\tau}_{\leq n} LConst \Pi(X) \simeq LConst \tau_{\leq n} \Pi(X)$. This, in turn, can for instance be checked in terms of the [[model structure on simplicial presheaves]], using that $\tau$ is objectwise the [[coskeleton]] operation. More on that at [[Postnikov tower in an (∞,1)-category]].

=--

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

For each $n \geq 1$ we have a [[fibration sequence]]

$$
  X^{(n)} \to X^{(n-1)} \to \mathbf{B}^n \mathbf{\pi}_n(X)
  \,.
$$

=--

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

Regard the diagram

$$
  \array{
    X^{(n)} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X^{(n-1)} & \to & \mathbf{B}^n \mathbf{\pi}_n(X) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_n(X) &\to& \mathbf{\Pi}_{(n-1)}(X)
  }
  \,.
$$

Here the right square is the defining $(\infty,1)$-pullback diagram of $\mathbf{B}^n \mathbf{\pi}_n(X)$ from above. Take also the left bottom square to be a homotopy pullback. Then from the pasting rules of pullbacks it follows that the composite bottom rectangle is also a pullback, which identifies the object $X^{(n-1)}$ on the left as indicated.

Similarly, form now the top square as a pullback. Then by the composition law of pullbacks we find that the composite vertical rectangle is a pullback, which identifies the top left object as $X^{(n)}$.

=--

## Examples

### For $\infty$-groupoids

In the archetypical $(\infty,1)$-topos $\mathbf{H} = $ [[∞Grpd]] the functors $\Pi$ and $\mathbf{\Pi}$ are the identity and so
$\cdots \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)$ is just the standard [[Postnikov tower]] $\cdots \to X_1 \to  X_0 $.

If we use the [[model structure on simplicial sets]] in order to [[presentable (infinity,1)-category|present]] $\infty Grpd$, then the Postnikov tower may be realized by the simplicial [[coskeleton]] operation

$$
  \cdots \to \mathbf{cosk}_2(X) \to \mathbf{cosk}_1(X)
  \,.
$$

If $X$ is [[connected]], the [[homotopy pullback]]

$$
  \array{
    X^{(n+1)} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\to& X_{(n)}
  }
$$

defining the Whitehead tower of an $\infty$-groupoid incarnated as a [[Kan complex]] $X$ may be computed as an ordinary [[pullback]] using a fibrant replacement diagram, such as thatr replacing the [[point]] by the [[decalage]] simplicial set, hence as the ordinary pullback

$$
  \array{
    X^{(n+1)} &\to& Dec \mathbf{cosk}_n X
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{cosk}_n X
  }
$$

in [[sSet]].

If $X$ is a pointed $\infty$-groupoid then $\mathbf{B}^n \mathbf{\pi}_n(X)$ is the [[Eilenberg-Mac Lane space]] $K(\pi_n(X),n)$.

### For topological $\infty$-groupoids {#TopInfGrpds}

It is often useful to think of [[topological space]]s as embedded into the [[(∞,1)-topos]] of [[topological ∞-groupoid]]s, i.e. the [[(∞,1)-category of (∞,1)-sheaves]]/[[∞-stack]]s on (a [[small category|small]] version of) the [[site]] of "all" topological spaces.

$$
  \mathbf{H} = Sh_{(\infty,1)}(Top)
  \,.
$$

#### The universal covering spaces

For $X$ a topological space regarded as a representable object in $Sh_{(\infty,1)}(Top)$, we have that $\mathbf{\Pi}_1(X)$ is the [[fundamental groupoid]] of $X$, regarded as a [[topological ∞-groupoid]] in the obvious way.

Then the [[homotopy fiber]] $X^{(1)}$ of the constant path inclusion $X \to \mathbf{\Pi}_1(X)$ is the [[universal covering space]] of $X$, as described in detail there.

#### The higher covering spaces

The analog of this statement for the higher items $X^{(n)}$ with $n \gt 1$ in the Whitehead tower of a topological spaces regarded in the $(\infty,1)$-topos of topological groupoids  has been studied in

* [[David Roberts]], _[[Fundamental Bigroupoids and 2-Covering Spaces]]_

The following reviews some central ideas of this.

##### Non-traditional approach in $Top$

**The following is some semblance of current research, except the stuff about the string 2-group. All errors and stupid ideas are mine - [[David Roberts]]**

For locally nice spaces (say locally contractible), it is desirable to have a functorial construction of the Whitehead tower. A shadow of a low-dimensional case of this can be seen in the construction of the [[string 2-group]] given by Baez-Crans-Schreiber-Stevenson. Since finite-dimensional Lie groups have non-trivial third homotopy group, it is not possible to form the 3-connected cover in the [[category]] $fin.dim.LieGrp$, like it is possible to take the 0-, and 1-connected covers. While most people give up the smoothness and make do with the [[topological group]] $String_G$, The BCSS construction leaps out of that [[category]] into that of strict ([[Frechet space|Frechet]]) Lie [[2-group]]s. The construction is also functorial.

+--{: .query}
[[David Roberts]]: As an aside, I know of two classes of spaces with explicitly  constructed (i.e. not via co-killing homotopy groups) 2-connected covers: the 2-sphere and its quotients by $\mathbf{Z}/n$ (lens spaces), and the [[loop group]] $\Omega G$ of a compact, simple, simply-connected Lie group $G$ (the latter is the level-1 central extension by $U(1)$, the first needs no introduction). Does anybody know of any other $n$-connected covers (other than, say, the Hopf fibrations) that are canonically given?
=--

However, without one could demand a conceptually similar approach to the $n$-connected cover of a general (locally nice) space or smooth manifold. For $n=2$, and taking only [[topological space]]s for consideration, this is contained in [[David Roberts|my]] thesis work.
The rough result is that one gets a 2-connected topological groupoid $X^{(2)}$ equipped with a map to $X$ that factors through the universal covering space $X^{(1)}$. This is functorial, and generalises to higher connected covers (at least heuristically - I don't have $n$-categorical superpowers).

But the general idea is that one would get an $(n-1)$-groupoid $X^{(n)}$ over $X$ which is $n$-connected and such that the map to $X$ factors through the $(n-2)$-groupoid $X^{(n-1)}$. The map to $X$ should induce [[isomorphism]]s on homotopy groups $\pi_i$ for $i\gt n$, as in the usual Whitehead tower.

If we want to consider arbitrary $n$ and retain some sort of local triviality on our connected covers we cannot get away from the assumption that the space in question is locally contractible. An assumption of this sort appears in

*  [[Bertrand Toen]], Vers une interpretation Galoisienne de la theorie de l'homotopie, Cahiers de Top. et Geom. Diff. Cat., Vol. XLIII-4 (2002), 257-312.

in the context of locally constant $\infty$-stacks and their monodromy (I haven't got this article at the moment, and I suspect they may be $(\infty,1)$-stacks, but I'm not 100 percent certain -DMR). Technically speaking, we don't need local contractibility but the existence of a basis of open sets $U \hookrightarrow X$ such that this inclusion map is null-homotopic. But I will continue to call this local contractibilty, for lack of a better term (if there is such a term, I'd like to know -DMR).

##### Construction using topological $n$-groupoids

Consider the fundamental $n$-groupoid $\Pi_n(X)$ of the locally contractible space $X$ (as a [[Trimble n-category|Trimble n-groupoid]], say), or at least for now its underlying globular set. We can take the compact open-topology on the set of $k$-morphisms for $k \lt n$. As the space is locally contracible, in particular semi-locally $n$-connected, the space $Hom(S^{n-1},X)$ is semi-locally simply-connected (I have a fragment of a paper saying this is true for the \'absolute case\' - that is, _locally_ $n$-connected implies the mapping space locally simply connected, but I expect it to be true for the relative case -DMR). In particular, we can take the fundamental groupoid $\Pi_1(Hom(S^{n-1},X))$, which has a topology given in the [[topological fundamental groupoid|usual way]]. The arrow space of this fundamental groupoid is then non other than the space of $n$-arrows of $\Pi_n(X)$. It needs to be checked that the $n$ compositions $\#_k$, $k=0,\ldots,n-1$ are continuous, as well as a bunch of other stuff, but I think this should follow from (unique) lifting theorems for covering spaces.

The object space of $\Pi_n(X)$ is just $X$, and so there is an inclusion $X \hookrightarrow \Pi_n(X)$, and it is this that replaces the Postnikov section in the Whitehead construction outlined above. The topological fundamental $n$-groupoid, even though it contains apparently more homotopical information than the untopologised fundamental $n$-groupoid $\Pi_n(X)^\delta$ ($\delta=$discrete topology), I posit that under the assumptions on $X$, the inclusion $\Pi_n(X)^\delta \to \Pi_n(X)$ has an ana-n-functor pseudoinverse (taking the [[Grothendieck pretopology]] of open covers should be enough). On passing to the homotopy colimit this span should become a span weak homotopy equivalences, and so we can consider the topologised and the untopologised to be different representatives of the $n$-type of $X$.

Given a basepoint $x\in X$, we can form the tangent $n$-groupoid $T_x \Pi_n(X)$, which is equivalent to the trivial $n$-groupoid $*$ (even as a _topological_ $n$-groupoid), and gives us what should be in any sensible definition a fibration $T_x\Pi_n(X) \to \Pi_n(X)$.
Pull back this fibration to $X$, and call the resulting thing $X^{(n)}$. It is fairly easy to see that $X^{(n)}$ is a topological $(n-1)$-groupoid over $X$. This then should be the $n$-connected cover of $X$. For $n=1$ this is precisely the classical construction of the universal covering space of a pointed space. For $n=2$ this is treated in

*  D.M. Roberts, Fundamental bigroupoids and 2-covering spaces, PhD thesis, available [[davidroberts:HomePage|here]]

and the two-dimensional homotopical tools developed there can be used to show that $X^{(2)}$ is 2-connected.

A word is probably in order about the notion of $k$-connectedness for topological $n$-groupoids. This has its usual meaning, once homotopy groups $\pi_i$ have been defined. The reader should be warned that these have nothing to do with the groups $\pi_0 Eq(1_{._{._{1_x}}})/\sim$ obtained from considering the autoequivalence $n-i+1$-group of the identity $(i-1)$-arrow on the identity $(i-2)$-arrow on ... on the object $x$. These should be defined in such a way as to agree with the homotopy colimit $hocolim X$ of $X$ considered as a truncated simplicial space. In particular, $\pi_1$ of a topological groupoid is _not_ the group of automorphisms of the basepoint, but a quotient of the set of _generalised paths_, an idea going back to Haefliger (for an intro see the early parts of chapter 2 of the above thesis, available at [[davidroberts:HomePage]], or the preprint H. Colman, _On the 1-homotopy type of Lie groupoids_, arXiv:math/0612257).

There are other interpretations of $X^{(n)}$:

 * The (0-)source fibre of $\Pi_n(X)$, which is:
 * The pullback of $(s_0,t_0):Hom_{\Pi_n(X)} \to X\times X$ along the inclusion $\{x\}\times X \to X\times X$, where $Hom_{\Pi_n(X)}$ is the (internalised) hom-$(n-1)$-groupoid familiar from the definition of a Trimble $n$-groupoid.
 * The \'vertical fundamental $(n-1)$-groupoid\' of $PX \to X$, the path fibration.

The last can be thought of as a families version of the usual fundamental $(n-1)$-groupoid: take vertical paths, vertical homotopies between paths etc.

[[!redirects Whitehead tower in an (∞,1)-topos]]


Whitney embedding theorem

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## Idea

The (strong) _Whitney embedding theorem_ states that every [[smooth manifold]] ([[Hausdorff space|Hausdorff]] and [[sigma-compact]]) of [[dimension]] $n$ has an [[embedding of smooth manifolds]] in the [[Euclidean space]] of dimension $2n$.

Notice that it is easy to see that every smooth manifold embeds into the Euclidean space of _some_ dimension ([this prop.](embedding+of+smooth+manifolds#ManifoldEmbedsIntoLargeDimensionalEuclideanSpace)). The force of Whitney's strong embedding theorem is to find the lowest dimension that still works in general.

## Related concepts

* The analogous statement for [[Riemannian manifolds]] and [[isometry|isometric]] embeddings is the _[[Nash embedding theorem]]_.

## References

Named after [[Hassler Whitney]].

* {#Pontrjagin55} [[Lev Pontrjagin]], Section 2.2 of: _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))

See also

* Wikipedia, _[Whitney embedding theorem](https://en.wikipedia.org/wiki/Whitney_embedding_theorem)_

* Paul Rapoport, _Introduction to Immersion, Embeddingand the Whitney Embedding Theorem_, 2015 ([pdf](http://schapos.people.uic.edu/MATH549_Fall2015_files/Survey%20Paul.pdf))

[[!redirects Whitney's embedding theorem]]
[[!redirects Whitney imbedding theorem]]
[[!redirects Whitney's imbedding theorem]]

[[!redirects Whitney's strong embedding theorem]]
[[!redirects Whitney's weak embedding theorem]]


Whitney extension theorem

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## Idea

A statement about sufficient data for extensions of a [[smooth function]] from a [[compact subset]] to an [[open neighbourhood]].

## Related concepts

[[!include extension theorems -- table]]

* [[Hadamard lemma]],

* [[Borel's theorem]]

* [[Tietze extension theorem]]

* [[Taimanov theorem]]

* [[Steenrod-Wockel approximation theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

## References

The original article is

* [[Hassler Whitney]], _Analytic extensions of differentiable functions defined in closed sets_, 1933 ([pdf](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-9947-1934-1501735-3.pdf))

Textbook accounts include

* {#Hoermander90} [[Lars Hörmander]], theore 2.3.6 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

Enhancement to a linear splitting of [[restriction]] maps on Fréchet [[spaces of sections]] with compact support of [[vector bundles]]:

* {#RobertsSchmeding18} [[David Roberts]], [[Alexander Schmeding]], _Extending Whitney's extension theorem: nonlinear function spaces_ ([arXiv:1801.04126](https://arxiv.org/abs/1801.04126))

This is then used to show the restriction map to (suitable) regular closed subsets is a submersion of mapping spaces (with maps valued in an arbitrary manifold).

See also

* Wikipedia, _[Whitney extension theorem](https://en.wikipedia.org/wiki/Whitney_extension_theorem)_

* [[eom]], _[Whitney extension theorem](https://www.encyclopediaofmath.org/index.php/Whitney_extension_theorem)_

* Armin Rainer, *Ultradifferentiable extension theorems: a survey*, Expositiones Mathematicae (2021) [[arXiv:2107.01061](https://arxiv.org/abs/2107.01061), [doi:10.1016/j.exmath.2021.12.001](https://doi.org/10.1016/j.exmath.2021.12.001)]


Whittaker function

## Related concepts

* [[geometric Langlands duality]]

## References

Name after [[Edmund T. Whittaker]].

See also:

* Wikipedia, *[Whittake function](https://en.wikipedia.org/wiki/Whittaker_function)*

Relation of Whittaker functions to [[equivariance|equivariant]] [[Gromov-Witten theory]] on [[flag varieties]] is discussed in

* [[Alexander Givental]], _Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture_, Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 vol. 180, Amer.Math.Soc.,Providence, Rhode Island, 1997, pp. 103–115; ([arXiv:alg-geom/9612001](http://arxiv.org/abs/alg-geom/9612001))

*  [[Anton Gerasimov]], Dimitri Lebedev, [[Sergey Oblezin]], _New Integral Representations of Whittaker Functions for Classical Lie Groups_ ([arXiv:0705.2886](http://arxiv.org/abs/0705.2886))

* [[Sergey Oblezin]], _On parabolic Whittaker functions_ ([arXiv:1011.4250](http://arxiv.org/abs/1011.4250))

For relation to [[geometric Langlands duality]] see [Gerasimov-Lebedev-Oblezin 09](geometric+Langlands+correspondence#GerasimovLebedevOblezin09)

[[!redirects Whittaker functions]]


why (infinity,1)-categories?

see at _[[(infinity,1)-category]]_


WhyNotZoidgebra?

![woop woop woop woop](http://i.imgur.com/ld72ZU0.jpg)