Hopf degree theorem

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Algebraic topology
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[[!include algebraic topology - contents]]
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#Contents#
* table of contents
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## Statement

### In ordinary homotopy theory

+-- {: .num_prop}
###### Proposition
**(Hopf degree theorem)**

Let $n \in \mathbb{N}$ be a [[natural number]] and $X \in Mfd$ be a [[connected topological space|connected]] [[orientation|orientable]] [[closed manifold]] of [[dimension]] $n$. Then the $n$th [[cohomotopy]] classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in [[bijection]] with the [[degree of a continuous function|degree]] $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

$$
  \pi^n(X)
    \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow}
   H^n(X,\mathbb{Z})
     \;\simeq\;
   \mathbb{Z}
$$

from $n$th [[cohomotopy]] to $n$th [[integral cohomology]] is a [[bijection]].

=--

([Pontrjagin 55, Sec. 9](#Pontrjagin55), review in [Kosinski 93, IX (5.8)](#Kosinski93), [Kobin 16, 7.5](#Kobin16))

### In equivariant homotopy theory
 {#InEquivariantHomotopyTheory}

The [[equivariant Hopf degree theorem]] -- Theorem \ref{EquivariantHopfDegreeTheorem} below -- is a generalization of the Hopf degree theorem to [[equivariant homotopy theory]], due to [tomDieck 79, 8.4](#tomDieck79). It implies a fairly explicit characterization of [[equivariant cohomotopy]] of [[representation spheres]] $S^V$ in [[RO(G)-degree]] $V$ (Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} below).

We need the following list of ingredients and assumptions:

Let $G$ be a [[finite group]]. For $H \subset G$ a [[subgroup]], write

\[
  \label{WeylGroup}
  W_G H \coloneqq (N_G H) / H
\]

for its [Weyl group](Weyl+group#in_equivariant_homotopy_theory).

For $X$ a [[G-space]], we write

\[
  \label{IsotropySubgroups}
  Isotr_X(G) \subset Sub(G)
\]

for the [[subset]] of the [[subgroup lattice]] on the [[isotropy groups]] of $X$, hence those [[subgroups]] which appear as [[stabilizer subgroups]] $Stab_G(x)$ of some point $x \in X$. This means  that if $H_1, H_2 \in Iso_X(G)$ and $H_1 \lt H_2$ is a strict inclusion, then the [[fixed loci]] differ $X^{H_1} \gt X^{H_2}$.

+-- {: .num_defn #MatchingGSpaces}
###### Definition
**(matching pair of $G$-spaces)**

For $G$ a [[finite group]], we say that a [[pair]] $X,Y \in G Spaces$ of [[topological G-spaces]], where $X$ is a [[G-CW-complex]], is a _matching pair_ if the following conditions are satisfied for all [[isotropy groups]] $H \in Isotr_X(G)$ (eq:IsotropySubgroups) of the $G$-action on $X$.

1. The [[fixed point space]] $X^H$ is a $W_G H = (N_G H) / H $[[G-CW-complex|-complex]] of [[finite number|finite]] [[dimension of a cell complex|dimension]] $dim\big( X^H\big) \in \mathbb{N}$;

1. $H^{dim(X^H)}\Big( X^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ ([[integral cohomology]] of the [[fixed point space]]),

   this implies that the [[action]] of $W_G(H)$ on cohomology induces a [[group homomorphism]]

   \[
     \label{OrientationBehaviour}
     e_{H,X}
     \;\colon\;
     W_G(H)
       \longrightarrow
     Aut_{Ab}(\mathbb{Z})
       \simeq
     \mathbb{Z}^\times
   \]

   to be called the _orientation behaviour_ of the action of $W_G(H)$ on $X^H$;

and

1. $Y^H$ is $(dim(X^H)-1)$-[[n-connected topological space|connected]]

   $$
     pi_{ \lt \mathrm{dim}\big( X^H \big) }
     \big(
       Y^H
     \big)
     \;=\;
     0
   $$

   (hence [[connected topological space|connected]] if $dim\left(X^H\right) = 1$, [[simply connected topological space|simply connected]] if $dim\left(X^H\right) = 2$, etc.);

1. $\pi_{dim(X^H)}\big( Y^H\big) \simeq \mathbb{Z}$ ([[homotopy groups]] of [[fixed point space]]),

   with the previous point this implies (by the [[Hurewicz theorem]]) that  $H^{dim(X^H)}\Big( Y^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ and hence orientation behaviour (eq:OrientationBehaviour) $e_{H,Y} \;\colon\; W_G(H) \to \mathbb{Z}^\times$

1. $e_{H,X} = e_{H,Y}$, the orientation behaviour (eq:OrientationBehaviour) of $X$ and $Y$ agrees at all [[isotropy groups]].

For simplicity we also demand that

* $dim(X^H) \geq 1$.

Given a matching pair of $G$-spaces, we say that a choice of generators in the [[cohomology groups]] of all the fixed strata

\[
  \label{ChoiceOfStratumWiseOrientations}
  \begin{aligned}
    or_{X,H}
    & =
    \pm 1 \in
    \mathbb{Z}
    \simeq
    H^{dim(X^H)}\big(X^H, \mathbb{Z} \big)
    \\
    or_{Y,H}
    & =
    \pm 1 \in
    \mathbb{Z}
    \simeq
    H^{dim(X^H)}\big(Y^H, \mathbb{Z} \big)
  \end{aligned}
\]

is a _choice of singularity-wise [[orientations]]_.  Given such a choice and an equivariant continuous function $f \colon X \to Y$, we have for each [[isotropy group]] $H \in Isotr_X(G)$ that the continuous function $f^H \;\colon\; X^H \to Y^H$ has a well-defined [[integer]] [[degree of a continuous function|degree]]

\[
  \label{IntegerDegreeOfEquivariantFunctionOnFixedStrata}
  deg(f^H)
  \;
    \in
  \;\mathbb{Z}
  \,.
\]

=--

([tom Dieck 79, p. 212 and p. 213](#tomDieck79))

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

Let $G$ be a [[finite group]] and $V \in RO(G)$ a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] [[linear representation]] of $G$. Then the pair

$$
  (X \coloneqq S^V,\; Y \coloneqq S^V)
$$

consisting of two copies of the [[representation sphere]] of $V$ is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}.

=--

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

First notice that an $H$-[[fixed locus]] of any $G$-representation sphere is itself a $W_G(H)$-representation sphere, since $\left(S^V\right)^H \simeq S^{\left( V^H\right)}$. That these are all [[G-CW-complex|WH-CW-complexes]] follows because generally [[G-representation spheres are G-CW-complexes]]. Moreover, since the [[topological spaces]] underlying all these fixed loci are [[n-spheres]], and since we have the same spheres for $X$ and $Y$, the connectivity and orientability conditions in Def. \ref{MatchingGSpaces} are evidently satisfied.

=--

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

Let $G$ be a [[finite group]] which arises as the [[point group]] $G \simeq S/N$ of a [[crystallographic group]] $S \subset Iso(E)$ of some [[Euclidean space]] $E$. Then the pair

$$
  \big(
    X \coloneqq E/N
    \,;
    Y \coloneqq S^E
  \big)
$$

consisting of

1. the [[torus]] $E/N$ which is the [[quotient space]] of $E$ by the given crystallographic sub-lattice $N \subset E$ and equipped with the $G$-[[action]] descending from that on $E$ ([this Prop.](crystallographic+group#InducedPointGroupActionOnTorus));

1. the [[representation sphere]] of the [[linear representation|linear action]] of the point group $G$ on $E$

is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}.

=--

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

The [[torus]] $E/N$ carries the [[structure]] of a [[smooth manifold]] for which the $G$-[[action]] is [[smooth]]. Since $G$ is [[finite group|finite]], also all its [[fixed loci]] $(E/N)^H$ are smooth manifolds ([this Prop.](equivariant+differential+topology#FixedLociOfSmoothProperActionsAreSubmanifolds)). By the [[equivariant triangulation theorem]], all these are [[G-CW-complexes|WH-CW-complexes]].

Moreover, the orientability and connectivity assumptions in Def. \ref{MatchingGSpaces} are evidently satisfied, using the fact that both $E/N$ as well as $S^E$ are modeled on the same linear $G$-representation space $E$.

=--

+-- {: .num_theorem #EquivariantHopfDegreeTheorem}
###### Theorem
**([[equivariant Hopf degree theorem]])**

Given a matching pair of $G$-spaces $X, Y$ (Def. \ref{MatchingGSpaces}) the function
(from $G$-[[equivariant homotopy theory|equivariant]] [[homotopy classes]] to [[tuples]] of [[degree of a continuous function|degrees]] labeled by [[isotropy groups]])
which sends any equivariant homotopy class $[f]$ of an equivariant continuou sfunction $f \colon X \to Y$ to its $H$-degrees (eq:IntegerDegreeOfEquivariantFunctionOnFixedStrata)

$$
  \array{
    deg
    &\colon&
    \pi_0 \mathrm{Maps}\big( X,Y \big)^G
    &
    \overset{
      \phantom{AAAA}
    }{\hookrightarrow}
    &
    \underset{
      { H \in Isotr_X(G) }
    }{\prod} \mathbb{Z}
    \\
    &&
    [f]
      &\mapsto &
    \big(
      H \mapsto \mathrm{deg}\big( f^H\big)
    \big)
  }
$$

is an [[injective function|injection]].

Moreover, for each $[f] \in \pi_0 \mathrm{Maps}\big( X,Y \big)^G$ and for each $H \in Isotropy_X(G)$

1. the $H$-degree (eq:IntegerDegreeOfEquivariantFunctionOnFixedStrata) modulo the [[order of a group|order]] of the [[Weyl group]]

   \[
     \label{DegreeConstraintsInEquivariantHopfDegreeTheorem}
     deg\left( f^H\right)
     \;mod\;
     {\vert W_G(H)\vert}
     \;\in\;
     \mathbb{Z}/{\vert W_G(H)\vert}
   \]

   is fixed by the degrees $deg\big( f^K\big)$ for all $K \gt H$;

1. there exists $f'$ with specified degrees $deg\big( (f')^K\big) = deg\big( f^K\big)$ for $K \gt H$ and realizing any of the degrees $deg\big( (f')^H\big) \in \mathbb{Z}$ allowed by the  constraint  (eq:DegreeConstraintsInEquivariantHopfDegreeTheorem).

=--

([tom Dieck 79, 8.4](#tomDieck79))

As a special case of the [[equivariant Hopf degree theorem]] (Theorem \ref{EquivariantHopfDegreeTheorem}), we obtain the following:

+-- {: .num_prop #EquivariantHomotopyOfSVInRODegreeV}
###### Proposition
**([[equivariant cohomotopy]] of [[representation sphere]] $S^V$ in [[RO(G)-degree]] $V$)**

Let $G \in \mathrm{Grp}_{\mathrm{fin}}$ and $V \in \mathrm{RO}(G)$. Then the pointed [[equivariant cohomotopy]]  of the [[representation sphere]] $S^V$  in [[RO(G)-degree]] $V$ is the [[Cartesian product]] of one copy of the [[integers]] for each _proper_ [[isotropy group|isotropy]] subgroup (eq:IsotropySubgroups) $H \underset{\neq}{\subset} G$ in $S^V$, and a copy of $\mathbb{Z}_2$ or $\mathbb{Z}$ depending on whether $V^G = 0$ or not:

$$
  \array{
      \pi^V\left( S^V\right)^{\{\infty\}/}
      &
        \overset{\simeq}{\longrightarrow}
      &
      \left\{
        \array{
          \mathbb{Z}_2 &\vert& V^G = 0
          \\
          \mathbb{Z} &\vert& \text{otherwise}
        }
      \right\}
      \times
      \underset{
        {
          {
              {H \in \mathrm{Isotr}_{S^V}(G)}
              \atop
              {H \neq G}
          }
        }
      }{\prod}
      \;\;
      {\vert W_G(H)\vert } \cdot \mathbb{Z}
      \\
      \big[
        S^V
          \overset{c}{\longrightarrow}
        S^V
      \big]
      &\mapsto&
      \Big(
        H
          \mapsto
        \mathrm{deg}
        \big(
          c^H
        \big)
        -
        \mathrm{offs}(c,H)
      \Big)
  }
$$

where on the right

$$
  \mathrm{deg}
  \Big(
    \big(
      S^V
    \big)^H
    \overset{
      c^H
    }{\longrightarrow}
    \big(
      S^V
    \big)^H
  \Big)
  \in
  \left\{
    \array{
      \mathbb{Z} &\vert& dim\left(V^H\right) \gt 0
      \\
      \mathbb{Z}_2 &\vert& dim\left( V^H\right) = 0
    }
  \right\}
$$

is the [[winding number]] of the underlying [[continuous function]] of $c$ ([[corestriction|co]])[[restriction|restricted]] to $H$-[[fixed points]], and part of the claim is  that in the cases with $dim\left( V^H\right) \gt 0$ this is an integer multiple of the order of the [[Weyl group]] $W_G(H)$ (eq:WeylGroup) up to an offset

$$
  \mathrm{offs}(f,H)
  \;\in\;
  \big\{
    0,1, \cdots,  \left\vert W_G(H)\right\vert
  \big\}
  \;\subset\;
  \mathbb{Z}
$$

which depends in a definite way on the degrees of $c^K$ for  all isotropy groups $K \gt H$.

=--

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

This follows as a special case of the equivariant Hopf degree theorem (Theorem \ref{EquivariantHopfDegreeTheorem}).

Here $(S^V, S^V)$ is a matching pair of $G$-spaces according to Example \ref{RepresentationSphereToRepresentationSphereIsMatchingPair}.

This equivariant Hopf degree theorem is stated above under the simplifying assumption that the dimension of all fixed loci is positive. But the proof from [tomDieck 79, 8.4](#tomDieck79) immediately applies to our situation where the dimension of the fixed locus at the full subgroup $H = G$ may be 0, with $\left( S^V\right)^G = S^0$. This gives a choice in $\mathbb{Z}_2$ in the first step of the inductive argument in [tomDieck 79, 8.4](#tomDieck79), and from there on the proof applies verbatim.

Alternatively, if $V^G = 0$ we may consider maps $S^{1+V} \to S^{1+V}$ which restrict on $S^1 \to S^1$ to degree zero or one.

=--

+-- {: .num_example #EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}
###### Example
**([[equivariant cohomotopy]] of $S^{\mathbb{R}_{sgn}}$ in [[RO(G)-degree]] the [[sign representation]] $\mathbb{R}_{sgn}$)

Let $G = \mathbb{Z}_2$ the [[cyclic group of order 2]] and $\mathbb{R}_{sgn} \in RO(\mathbb{Z}_2)$ its 1-dimensional [[sign representation]].

Under equivariant [[stereographic projection]] ([here](representation\+sphere#Construction)) the corresponding [[representation sphere]] $S^{\mathbb{R}_{sgn}}$ is equivalently the [[unit circle]]

$$
  S^1
  \simeq
  S(\mathbb{R}^2)
$$

equipped with the $\mathbb{Z}_2$-[[action]] whose [[involution]] element $\sigma$ [[reflection|reflects]] one of the two [[coordinate functions|coordinates]] of the ambient [[Cartesian space]]

$$
  \sigma \;\colon\; (x_1,x_2) \mapsto (x_1, -x_2)
  \,.
$$

Equivalently, if we identify

\[
  \label{CircleAsQuaotientOfRByZ}
  S^1
  \;\simeq\;
  \mathbb{R}/\mathbb{Z}
\]

then the involution action is

$$
  \begin{aligned}
    \sigma
    \;\colon\;
    t
    \mapsto
    &
    \phantom{\sim}
    1 - t
    \\
    & \sim
    \phantom{1} - t
  \end{aligned}
  \,.
$$

This means that the [[fixed point space]] is the [[0-sphere]]

$$
  \big( S^1\big)^{\mathbb{Z}_2}
  \;\simeq\;
  S^0
$$

being two antipodal points on the circle, which in the presentation (eq:CircleAsQuaotientOfRByZ) are labeled $\{0,1/2\} \simeq S^0$.

Notice that the map

\[
  \label{ConstantParameterFunctionFromSignRepresentationSphereToItself}
  \array{
    S^1 &\overset{n}{\longrightarrow}& S^1
    \\
    t &\mapsto& n\cdot t
  }
\]

of constant parameter speed and [[winding number]] $n \in \mathbb{N}$ is equivariant for this $\mathbb{Z}_2$-[[action]] on both sides:

\begin{center}
\begin{xymatrix}
  t
    \ar@{|->}[r]
    \ar@{|->}[d]_{\sigma}
    &
  n\cdot t
    \ar@{|->}[d]^{\sigma}
  \\
  -t
    \ar@{|->}[r]
    &
  -n \cdot t
\end{xymatrix}
\end{center}

Now the restriction of the map $ n \cdot(-)\in \mathbb{Z}$ from (eq:ConstantParameterFunctionFromSignRepresentationSphereToItself) to the [[fixed points]]

$$
  \array{
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right)
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
    \\
    {}^{ \mathllap{  \left( \cdot n\right)^{\mathbb{Z}_2} } }
    \big\downarrow
      &&
    \big\downarrow^{\mathrlap{\cdot n}}
    \\
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right)
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
  }
$$

sends (0 to 0 and) $1/2$ to either $1/2$ or to $0$, depending on whether the [[winding number]] is [[odd number|odd]] or [[even number|even]]:

$$
  \array{
     S^0
       &\overset{ \left(\cdot n\right)^{\mathbb{Z}_2} }{\longrightarrow}&
     S^0
     \\
     1/2 &\mapsto&
     \left\{
       \array{
         1/2 &\vert& n \;\text{is odd}
         \\
         0 &\vert& n \text{is even}
       }
     \right.
  }
$$

Hence if the restriction to the [[fixed locus]] is taken to be the [[identity function|identity]] (bipointed [[equivariant cohomotopy]]) then, in accord with Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} there remains the [[integers]] worth of equivariant [[homotopy classes]], where each integer $k \in \mathbb{Z}$ corresponds to the odd winding integer $1 + 2k$

$$
  \array{
    \pi^{\mathbb{R}_{sgn}}
    \left(
      S^{\mathbb{R}_{sgn}}
    \right)^{\{0,\infty\}/}
    &\simeq&
    2 \cdot \mathbb{Z} + 1
    &\simeq&
   \mathbb{Z}
    \\
    \left[
      \mathbb{R}/\mathbb{Z} \overset{c}{\to} \mathbb{R}/\mathbb{Z}
    \right]_{{0 \mapsto 0} \atop {1/2 \mapsto 1/2}}
    &\mapsto&
    deg(c)
    &\mapsto&
    \big( deg(c) - 1\big)/2
  }
$$

=--

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

This result

$$
  \pi^{ \mathbb{R}_{sgn} }\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z}_2 \times \mathbb{Z}
$$

from Example \ref{EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}

becomes, after [[stabilization]] to [[equivariant stable homotopy theory]], the stable homotopy groups of the [[equivariant sphere spectrum]] in [[RO(G)-grading]] given by

$$
  \pi^{ \mathbb{R}_{sgn} }_{stab}\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z} \times \mathbb{Z}
$$

see [there](equivariant+sphere+spectrum#Z2equivariance).

=--

+-- {: .num_example #EquivariantCohomotopyOfRepresentationSphereOfQuaternionsInThatDegree}
###### Example
**([[equivariant cohomotopy]] of $S^{\mathbb{H}}$ in [[RO(G)-degree]] the [[quaternions]] $\mathbb{H}$)**

Let $G \subset SU(2) \simeq S(\mathbb{H})$ be a non-[[trivial group|trivial]] [[finite subgroup of SU(2)]] and let $\mathbb{H} \in RO(G)$ be the [[real vector space]] of [[quaternions]] regarded as a [[linear representation]] of $G$ by left multiplication with unit [[quaternions]].

Then the bi-pointed [[equivariant cohomotopy]] of the [[representation sphere]] $S^{\mathbb{H}}$ in [[RO(G)-degree]] $\mathbb{H}$ is

$$
  \array{
    \pi^{\mathbb{H}}
    \left(
      S^{\mathbb{H}}
    \right)^{\{0,\infty\}/}
    &\simeq&
    {\left\vert G\right\vert} \cdot \mathbb{Z} + 1
    &\simeq&
    {\left\vert G\right\vert} \cdot \mathbb{Z}
    &\simeq&
    \mathbb{Z}
    \\
    \left[
      S^{\mathbb{H}}
      \overset{c}{\longrightarrow}
       S^{\mathbb{H}}
    \right]
    &\mapsto&
    deg\left( c^{ \{e\}  }\right)
    &\mapsto&
    deg\left( c^{ \{e\}  }\right) - 1
    &\mapsto&
    \big(
      deg\left( c^{ \{e\}  }\right) - 1
    \big)/ {\left\vert G\right\vert}
  }
$$

=--

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

The only [[isotropy groups|isotropy]] [[subgroups]] of the left action of $G$ on $\mathbb{H}$ are the two extreme cases $Isotr_{\mathbb{H}}(G) = \{1, G\} \in Sub(G)$. Hence the only multiplicity that appears in Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} is

$$
  \left\vert W_G(1)\right\vert
  \;=\;
  \left\vert G \right\vert
  \,.
$$

and all degrees  must differ from that of the class of the [[identity function]] by a multiple of this multiplicity. Finally, the offset of the identity function is clearly $offs\left(id_{S^{\mathbb{H}}},1\right) = deg\left( id_{S^{\mathbb{H}}}\right) = 1$.

=--

## Applications

* [[Burnside ring is equivariant stable cohomotopy of the point]]

## Related statements

* [[Hurewicz theorem]]

* [[Boardman homomorphism]]

* [[Hopf invariant]]

* [[Poincaré–Hopf theorem]]

* [[d-invariant]]

## References

Due to [[Heinz Hopf]].

* {#Pontrjagin55} [[Lev Pontrjagin]], Section 9 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))

Textbook accounts:

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Chapter 11 of  _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* {#DubrovinNovikovFomenko85} B. A. Dubrovin, [[S. P. Novikov]], A. T. Fomenko, section 13.3 of: _Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds_, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

* {#Kosinski93} [[Antoni Kosinski]], chapter IX, Cor. 5.8 of: _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C))

* [[John Milnor]], p. 62 of: _Topology from the differential viewpoint_, Princeton University Press, 1997. ([ISBN:9780691048338](https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf))

* {#Kobin16} Andrew Kobin, Section 7.5 of: _Algebraic Topology_, 2016 ([[KobinAT2016.pdf:file]])

Generalization to [[equivariant cohomotopy]] and [[equivariant cohomology]]

* {#tomDieck79} [[Tammo tom Dieck]], section 8.4 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766 Springer 1979

[[!redirects Hopf theorem]]


Hopf E-∞ algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The pairing of the concepts of [[commutative Hopf algebra]] and [[E-∞ algebra]]: the [[formal dual]] of a [[spectral group scheme]], that is, a [[group scheme]] in [[E-∞ geometry]].

## References

* {#LurieI} [[Jacob Lurie]], def. 3.9.7 of: *[[Elliptic Cohomology I]]* [[pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf)]

On Hopf (semi-)ring structure on the [[symmetric algebra]] of the [[sphere spectrum]]:

* {#StricklandTurner97} [[Neil Strickland]], [[Paul Turner]], _Rational Morava $E$-theory and $D S^0$_, Topology **36** 1 (1997) 137-151 [doi:10.1016/0040-9383(95)00073-9[pdf](http://hopf.math.purdue.edu/Strickland-PTurner/rme.pdf)]

[[!redirects Hopf E-infinity-algebra]]
[[!redirects Hopf E-infinity-algebra]]
[[!redirects Hopf E-infinity algebra]]
[[!redirects Hopf E-infinity algebras]]
[[!redirects Hopf E-∞ algebras]]

[[!redirects Hopf E-infinity ring]]
[[!redirects Hopf E-infinity rings]]
[[!redirects Hopf E-infinity-ring]]
[[!redirects Hopf E-infinity-rings]]

[[!redirects Hopf ring spectrum]]
[[!redirects Hopf ring spectra]]


Hopf envelope

For any family $\{ C_i \}_{i\in I}$ of [[coalgebra]]s,
$T(\coprod_i C_i) = \coprod_i T(C_i)$ where
the coproduct on the right-hand side is the coproduct in the category of [[bialgebra]]s, i.e. the [[free product]] of algebras with natural induced coalgebra structure. If the index set consists of
nonnegative integers and $C_{i+1} = C_i^{\rm cop}$,
the left hand side specializes to an intermediate
stage in building Takuechi's [[free Hopf algebra]] $H(C)$ on the coalgebra $C$.

Manin generalized the RHS.
He replaces $T(C_i)$ with any bialgebra $B_i$
with $B_{i+1} = B_i^{\rm cop, op}$.
Notice that the algebra structure is also opposite between
even and odd cases (a superfluous/iunvisible condition in the case of the tensor algebra $T(C_i)$ appearing in [[free Hopf algebra|Takeuchi's construction]]).
Let $\mathcal{B} = \coprod_i B_i$ and $S : \mathcal{B} \rightarrow \mathcal{B}$
be again defined by a shift in index by $+1$. Then the 2-sided
[[ideal]] $I_S \subset \mathcal{B}$ generated by relations
$\sum b_{(1)} S(b_{(2)}) - \epsilon(b) 1$ and
$\sum S(b_{(1)}) b_{(2)} - \epsilon(b) 1$, for all $b \in B_i$, is $S$-stable ideal and
the quotient $H(B) = \mathcal{B}/I_S$ is a [[Hopf algebra]],
the **Hopf envelope** of the bialgebra $B$.

It satisfies the following universal property:
for any Hopf algebra $H'$ and a bialgebra map
$\phi : B_0 \rightarrow H'$ there is a unique Hopf algebra
map $H(\phi) : H(B) \rightarrow H'$ such that
$H(\phi) \circ i = \phi$ where $i : B_0 \rightarrow H(B)$ is the
composition of the inclusion into $\mathcal{B}$ and the canonical projection $\mathcal{B} \rightarrow H(B)$.

Manin has introduced this construction in

* Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988,

and applied it mainly to matrix Hopf algebras (e.g. quantum linear groups). The Hopf envelope of the matrix Hopf algebra with basis $t^i_j$ whose underlying bialgebra is the free bialgebra on $n^2$ generators $t^i_j$ is sometimes called the free matrix Hopf algebra, cf. section 13 of

* Z. Škoda, Localizations for construction of quantum coset spaces, [math.QA/0301090](http://arxiv.org/abs/math/0301090), "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.

for more details.


Hopf fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _complex Hopf fibration_ [[Hopf (1931)](#Hopf31)] is a canonical nontrivial  [[circle principal bundle]] over the [[2-sphere]] whose total space is the [[3-sphere]].

$$
  S^1 \hookrightarrow S^3 \to S^2
  \,.
$$

Its canonically [[associated bundle|associated]] [[complex line bundle]] is the [[basic line bundle on the 2-sphere]].

This we discuss below in

* _[On the 3-sphere](#DefinitionFor3Sphere)_

More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in

* _[On the 1-sphere, 3-sphere, 7-sphere and 15-sphere](#OnAllFourSpheres)_.

## On the 3-sphere
 {#DefinitionFor3Sphere}

### Homotopy-theoretic characterization
 {#HomotopyTheoreticCharacterization}

The [[Eilenberg-MacLane space]] $K(\mathbb{Z},2) \simeq B S^1$ is the [[classifying space]] for [[circle group]] [[principal bundles]]. By its very nature, it has a single nontrivial [[homotopy group]], the second, and this is isomorphic to the group of [[integers]]

$$
  \pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}
  \,.
$$

This means that there is, up to [[homotopy]], a canonical (up to sign), continuous map from the 2-sphere

$$
  \phi : S^2 \to K(\mathbb{Z},2)
  \,,
$$

such that $[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}$.

As any map into $K(\mathbb{Z},2)$ this classifies a [[circle group]] [[principal bundle]] over its [[domain]]. This is the Hopf fibration, fitting into the long [[fiber sequence]]

$$
  \array{
    S^1 &\hookrightarrow& S^3
    \\
    && \downarrow
    \\
    && S^2
    &\stackrel{\phi}{\to}&
    B S^1 \simeq K(\mathbb{Z},2)
  }
  \,.
$$

In other words, the Hopf fibration is the $U(1)$-bundle with unit first [[Chern class]] on $S^2$.

### Realization via the complex numbers

An explicit [[topological space]] presenting the Hopf fibration may be obtained as follows. Identify

$$
  S^3 \simeq \{(z_0, z_1) \in \mathbb{C}\times \mathbb{C} \,|\, {|z_0|}^2 + {|z_1|}^2 = 1\} = S(\mathbb{C}^2)
$$

and

$$S^2 \simeq \mathbb{C P}^1 \simeq \mathbb{C} \sqcup \{\infty\}$$

Then the [[continuous function]] $S^3 \to S^2$ defined by

$$(z_0, z_1) \mapsto \frac{z_0}{z_1}$$

gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical [[line bundle]].) Alternatively, if we use

$$
  S^2 \simeq \{(z, x) \in \mathbb{C} \times \mathbb{R}
  \,|\,
  {|z|}^2 + x^2 = 1\} = S(\mathbb{C} \times \mathbb{R})
  \,.
$$

and identify this presentation of the 2-sphere with the complex projective line via [[stereographic projection]], the Hopf fibration is identified with the map  $S^3 \to S^2$ given by sending

$$
  (z_0, z_1) \mapsto (2 z_0 \overline{z_1} , {|z_0|}^2 - {|z_1|}^2).
$$

### Realization via quaternions
 {#RealizationViaQuaternions}

Alternatively, we may regard $S^3 \simeq S(\mathbb{H})$ as the [[unit sphere]] in the [[quaternions]] and $S^2 \simeq S\left( \mathbb{H}_{\mathrm{im}}\right)$ as the unit sphere in the [[imaginary part|imaginary]] [[quaternions]]. Under this identification, the complex Hopf fibration is equivalently represented by

$$
  \array{
    S(\mathbb{H})
    &\longrightarrow&
    S\left( \mathbb{H}_{\mathrm{im}}\right)
    \\
    q
      &\mapsto&
    q \cdot \mathbf{i} \cdot \overline{q}
  }
$$

where $\mathbf{i} \in S\left( \mathbb{H}_{\mathrm{im}}\right)$ is any unit imaginary quaternion.

### Realization via the Hopf construction

Regard $S^1 = U(1)$ as equipped with its [[circle group]] structure. This makes $S^1$ in particular an [[H-space]]. The Hopf fibration $S^1 \to S^3 \to S^2$  is the [[Hopf construction]] applied to this H-space.

### [[Spin(3)]]-[[action|equivariance]]

Consider

1. the [[Spin(3)]]-[[action]] on the [[2-sphere]] $S^2$ which is induced by the defining action on $\mathbb{R}^3$ under the identification $S^2 \simeq S(\mathbb{R}^3)$;

1. the [[Spin(3)]]-action on the [[3-sphere]] $S^3$ which is induced under the exceptional [[isomorphism]] $Spin(3) \simeq Sp(1) = U(1,\mathbb{H})$ by the canonical left action of $U(1,\mathbb{H})$ on $\mathbb{H}$ via $S^3 \simeq S(\mathbb{H})$.

Then the complex Hopf fibration $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$ is [[equivariant]] with respect to these [[actions]].

A way to make the $Spin(3)$-equivariance of the complex Hopf fibration fully explicit is to observe that it is equivalently the following map of [[coset spaces]]:

$$
  \array{
    S^1
      &\overset{fib(h_{\mathbb{C}})}{\longrightarrow}&
    S^{3}
      &\overset{h_{\mathbb{C}}}{\longrightarrow}&
    S^2
    \\
     = && = && =
    \\
    \frac{Spin(2)}{Spin(1)}
    &\longrightarrow&
    \frac{Spin(3)}{Spin(1)}
    &\longrightarrow&
    \frac{Spin(3)}{Spin(2)}
  }
$$

## On the 1-sphere, 3-sphere, 7-sphere and 15-sphere
 {#OnAllFourSpheres}

### Via norms and projections

For each of the [[normed division algebra|normed division algebras]] over $\mathbb{R}$, the [[real numbers]], [[complex numbers]], [[quaternions]], [[octonions]]

$$A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O},$$

there is a corresponding Hopf fibration of [[Hopf invariant one]].

The total space of the fibration is the space of pairs $(\alpha, \beta) \in A^2$ of unit norm: ${|\alpha|}^2 + {|\beta|}^2 = 1$. This gives [[spheres]] of dimension 1, [[3-sphere|3]], [[7-sphere|7]], and 15 respectively. The base space of the fibration is [[projective space|projective]] 1-space $\mathbb{P}^1(A)$, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map

$$S^{2^n - 1} \to S^{2^{n-1}}$$

($n = 1, 2, 3, 4$) which sends the pair $(\alpha, \beta)$ to $\alpha/\beta$.

### Via the Hopf construction

When $X$ is a [[sphere]] that is an $H$-space, namely, one of the [[groups]] $S^0 = 1$ the [[trivial group]], $S^1 = \mathbb{Z}/2$ the [[group of order 2]], the [[3-sphere]] [[special unitary group]] $S^3 = SU(2)$;  or the [[7-sphere]] $S^7$ with its [[Moufang loop]] structure, then the Hopf construction produces the above four Hopf fibrations:

1. $S^0 \hookrightarrow S^1 \to S^1 $ -- [[real Hopf fibration]]
1. $ S^1 \hookrightarrow S^3 \to S^2 $ -- [[complex Hopf fibration]]
1. $ S^3 \hookrightarrow S^7 \to S^4 $ -- [[quaternionic Hopf fibration]]
1. $ S^7 \hookrightarrow S^{15} \to S^8 $ -- [[octonionic Hopf fibration]]

## Properties

### Relation to stable homotopy groups of spheres
 {#RelationToStableHomotopyGroupsOfSpheres}

Let

$H_{\mathbb{C}} \in \pi_3(S^2)$,

$H_{\mathbb{H}} \in \pi_7(S^4)$

$H_{\mathbb{O}} \in \pi_{15}(S^8)$

be the [[homotopy classes]] of the [[complex Hopf fibration]], the [[quaternionic Hopf fibration]] and the [[octonionic Hopf fibration]], respectively. Then their [[suspensions]] are the generators of the corresponding [[stable homotopy groups of spheres]]:

$$
  \begin{aligned}
  \Sigma H_{\mathbb{C}}
  & =
  \pm 1
    \in
  \mathbb{Z}_2
  \simeq
  \pi_1^{st}
  \\
  \Sigma H_{\mathbb{H}}
  & =
  \pm 1
    \in
  \mathbb{Z}_{24}
    \simeq
  \pi_3^{st}
  \\
  \Sigma H_{\mathbb{O}}
  & =
  \pm 1
    \in
  \mathbb{Z}_{240}
    \simeq
  \pi_7^{st}
  \end{aligned}
$$

see also at

* [[first stable homotopy group of spheres]]

* [[third stable homotopy group of spheres]]

and [this MO comment](https://mathoverflow.net/a/224082/381)

## Applications

### Magnetic monopoles

When [[line bundles]] are regarded as models for the topological structure underlying the [[electromagnetic field]] the Hopf fibration is often called "the [[magnetic monopole]]". We may think of the $S^2$ homotopically as being the 3-dimensional [[Cartesian space]] with origin removed $\mathbb{R}^3 - \{0\}$ and think of this as being 3-dimensional physical space with a unit point [[magnetic charge]] at the origin removed. The corresponding [[electromagnetic field]] away from the origin is given by a [[connection on a bundle|connection]] on the corresponding Hopf fibration bundle.

### K-theory

In complex [[K-theory]], the Hopf fibration represents a class $H$ which generates the [[cohomology ring]] $K_U(S^2)$, and satisfying the relation $H^2 = 2 \cdot H - 1$, or $(H-1)^2 = 0$. (So in particular $H$ has an [[inverse]] $H^{-1} = 2- H $, see at [[Bott generator]].)

A succinct formulation of [[Bott periodicity]] for [[complex K-theory]] is that for a space $X$ whose [[homotopy type]] is that of a [[CW-complex]], we have

$$K(S^2 \times X) \cong K(S^2) \otimes K(X)$$

(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for $K(S^2)$ to the topos of sheaves over $X$.)

The Hopf fibrations over other [[normed division algebras]] also figure in the more complicated case of [[real K-theory]] $K_O$: they can be used to provide generators for the non-zero [[homotopy groups]] $\pi_n(B O)$ for the [[classifying space]] of the [[stable orthogonal group]], which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra $\mathbb{O}$). [To be followed up on.]

## Related concepts

* [[Hopf construction]]

* [[Hopf invariant one]]

* [[real Hopf fibration]], [[quaternionic Hopf fibration]], [[octonionic Hopf fibration]]

* [[complex projective space]]

* [[Hopf construction in homotopy type theory]]

## References
  {#References}

Original articles:

* {#Hopf31} [[Heinz Hopf]], *Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche*, Mathematische Annalen **104**  (1931) 637–665 [[doi:10.1007/BF01457962](https://doi.org/10.1007/BF01457962)]

* {#Hopf35} [[Heinz Hopf]], *Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension*, Fundamenta Mathematicae **25** 1 (1935) 427-440 [[eudml:212801](https://eudml.org/doc/212801)]

Exposition:

* Saifuddin Syed, _Group structure on spheres and the Hopf fibration_ ([[SyedHopfFibration.pdf:file]])

Review:

* [[Herman Gluck]], [[Frank Warner]], [[Chung Tao Yang]], Section 6 of: _Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics_, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 ([euclid:dmj/1077303489](https://projecteuclid.org/euclid.dmj/1077303489))

* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, **32** (1986), 173-198 [[ResearchGate](https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations), [[GluckWarnerZiller-HopfFibrations.pdf:file]]]

* _[Third homotopy group of the sphere and Hopf fibration](https://chiasme.wordpress.com/2014/01/04/third-homotopy-group-of-the-sphere-and-hopf-fibration/)_

Formulation in [[homotopy type theory]]:

* [HoTT Wiki](https://ncatlab.org/homotopytypetheory/show/HomePage), _[[homotopytypetheory:Hopf fibration]]_

Relation to [[skyrmions]]:

* [[Sven Bjarke Gudnason]], [[Muneto Nitta]], _Linking number of vortices as baryon number_ ([arXiv:2002.01762](https://arxiv.org/abs/2002.01762))

Discussion of [[supersymmetry|supersymmetric]] Hopf fibrations:

* [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of _Gauge Symmetries and Fibre Bundles_, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 ([arXiv:1702.08910](https://arxiv.org/abs/1702.08910))

* Simon Davis, section 3 of _Supersymmetry and the Hopf fibration_ ([doi:10.4995/agt.2012.1623](https://doi.org/10.4995/agt.2012.1623))

[[!redirects Hopf fibrations]]

[[!redirects complex Hopf fibration]]


Hopf invariant

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

#Contents#
* table of contents
{:toc}

## Definition

### In ordinary cohomology
  {#InOrdinaryCohomology}

For $n \in \mathbb{N}$ with $n \gt 1$, consider [[continuous functions]] between [[spheres]] of the form

\[
  \label{TheMap}
  f \;\colon\; S^{2n-1} \longrightarrow S^n
  \,.
\]

The [[homotopy cofiber]] of $f$ (the [[attaching space]] induced by $f$)

$$
  C_f \;\coloneqq\; S^n \underset{S^{2n-1}}{\cup} D^{2n}
$$

has [[ordinary cohomology]]

$$
  H^k(C_f, \mathbb{Z})
  \simeq
  \left\{
    \array{
      \mathbb{Z} & for\; k = n, 2n;
      \\
      0 &  otherwise
    }
  \right.
  \,.
$$

Hence for $\alpha_n, \beta_{2n}$ generators of the [[cohomology groups]] in degree $n$ and $2n$ (unique up to choice of sign), respectively, there exists an [[integer]] $HI(f)$ which expresses the [[cup product]] square of $\alpha_n$ as a multiple of $\beta_{2n}$:

\[
  \label{HopfInvariantFormula}
  \alpha_n \cup \alpha_n \;=\; HI(f) \cdot \beta_{2n}
  \,.
\]

This [[integer]] $HI(f) \in \mathbb{Z}$ is called the _Hopf invariant_ of $f$ (e.g. [Mosher-Tangora 86, p. 33](#MosherTangora86)).

This depends on the choices made only up to sign. In particular it has a well-defined mod-2 reduction image $[HI(f)] \in \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$ (in [[cyclic group of order 2|Z/2]]), and as such it is the [[Steenrod square]]

$$
  [HI(f)] \cdot (-)
    \;\colon\;
  \mathbb{F}_2
  \;\simeq\;
  H^n
  \big(
    C_f;
    \,
    \mathbb{F}_2
  \big)
  \stackrel{Sq^n}{\longrightarrow}
  H^{2n}
  \big(
    C_f;
    \,
    \mathbb{F}_2
  \big)
   \;\simeq\;
  \mathbb{F}_2
  \,.
$$

### In generalized cohomology
 {#InGeneralizedCohomology}

Here is a more abstract picture of the Hopf invariant in abstract [[homotopy theory]] (following [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)):

Let $E$ be a [[multiplicative cohomology theory]], assumed to vanish in degree $2n - 1$

\[
  \label{ECohomologyVanishingInDegree2nMinus1}
  \widetilde E
  \big(
    S^{2n-1}
  \big)
  \,\simeq\,
  \pi_{2n-1}(E)
  \,\simeq\,
  0
  \,.
\]

We write
$
  E_n \;\coloneqq\; \Omega^\infty \Sigma^n E
$
for its classifying spaces, hence for the component spaces of its [[Brown representability theorem|representing]] [[spectrum]].

(For the case of [[ordinary cohomology]] $E = H \mathbb{Z}$ we have $E_n \,\simeq\, K(\mathbb{Z},n)$ an [[Eilenberg-MacLane space]].)

Now let

$$
  S^{2n-1} \overset{f}{\longrightarrow} S^n
$$

be a map (eq:TheMap). Then its $E$-Hopf invariant "is" the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] of [[pointed homotopy types]] (see also at _[[e-invariant is Todd class of cobounding (U,fr)-manifold]]_):

\begin{imagefromfile}
        "file_name": "HopfInvariantPastingDiagram.jpg",
        "web": "nlab",
        "width": 480,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting the Hopf invariant",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

* the top square is defined to be a [[homotopy pushout]], exhibiting the [[attaching space]] $C_f$;

* the total rectangle is defined to be a [[homotopy pushout]], exhibiting the [[suspension]] of $S^{2n-1}$ to $S^{2n}$;

* the bottom square is hence a [[homotopy pushout]] by the [[pasting law]];

* by assumption (eq:ECohomologyVanishingInDegree2nMinus1) the restriction of $\Sigma^{2n} 1$ to $S^{2n-1}$ trivializes, exhibited by a choice of [[homotopy]] filling the full top part of the diagram; and by the [[universal property]] of the top homotopy-pushout this corresponds equivalently to the dashed morphism $c$;

* {#RestrictionOfCupSquareTrivializes} the restriction of the [cup square cohomology operation](cohomology+operation#CupPowersInMultiplicativeCohomology) on $c$ to $S^n$ trivializes: $c$ factors through the $n$ component space of the [[connective cover]] $E\langle 0\rangle$, whence its [cup square](cohomology+operation#CupPowersInMultiplicativeCohomology) (by the discussion at _[connective cover -- For ring spectra](connective+cover#ForRingSpectra)_) factors through the $2n$-connective $2n$th component space of the same connective cover (as in a standard argument in [[complex oriented cohomology theory]], e.g. [Lurie, Lec. 4. Exmpl. 8](complex+oriented+cohomology+theory#LurieLecture), or see [[The Relation of Cobordism to K-Theories|Conner-Floyd 66, Part I, Cor. 7.2]]):

\begin{xymatrix}
  S^n
  \ar[r]
    _>>>{\ }="s"
  \ar[d]
    ^>>>{\ }="t"
  &
  \Omega^{\infty-n}
  \big(
    E\langle0\rangle
  \big)
  \ar[d]
    ^-{ (-)^{2_{\cup}} }
  \ar[r]
    ^-{ \;\
      \Omega^{\infty-n}(
        \epsilon_E
      )
    \; }
  &
  \Omega^{\infty-n}
  \big(
    E
  \big)
  \ar[d]
    ^-{ (-)^{2_{\cup}} }
  \\
  \ast
  \ar[r]
  &
  \Omega^{\infty-2n}
  \big(
    E\langle0\rangle
  \big)
  \ar[r]
    _-{ \;
      \Omega^{\infty-2n}
      ( \epsilon_E )
    \;}
  &
  \Omega^{\infty-2n}
  \big(
    E
  \big)
  %
  \ar@{=>}
    ^-{ \exists ! }
    "s"; "t"
\end{xymatrix}

* This yields the [[homotopy]]  filling the full bottom part of the diagram above; and by the [[universal property]] of the bottom homotopy-pushout this corresponds equivalently to a dashed morphism $S^{2n} \to E_{2n}$, labeled by  some class

  $$
    \kappa
    \,\in\,
    \pi_0(E)
    \,\simeq\,
    \widetilde E(S^0)
    \overset{\Sigma^{2n}}{\longrightarrow}
    \widetilde E(S^{2n})
    \,
    =
    \,
    \big[
      S^{2n} \longrightarrow E_{2n}
    \big]
    \,;
  $$

* finally, by the homotopy-pushout property of the total rectangle, this class $\kappa$ also labels the total homotopy filling the full diagram.

We see that:

* for $E = H \mathbb{Z}$ being [[ordinary cohomology]], $\kappa \,\in\, \mathbb{Z}$ is the traditional Hopf invariant as discussed [above](#InOrdinaryCohomology);

* for $E = KU$ being [[complex topological K-theory]], $\kappa \,\in\, \mathbb{Z}$ is the Hopf invariant in K-theory as used in [Adams-Atiyah 66](Hopf+invariant+one#AdamsAtiyah66).

In the case that the map $f$ is one the classical [[Hopf fibrations]], the [[attaching space]] above is a [[projective space]] (by the discussion at _[[cell structure of projective spaces]]_) and the choice of homotopy $c$ is the choice of an _orientation_ in $E$-cohomology theory to second stage. Specifically:

\begin{imagefromfile}
        "file_name": "OrientationInECohomologyAndHopfInvariant.jpg",
        "web": "nlab",
        "width": 600,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting orientation and the Hopf invariant",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

* for the [[complex Hopf fibration]] $f = h_{\mathbb{C}}$ the attaching space is [[complex projective space]] $\mathbb{C}P^2$ and the choice of homotopy $c$ is a choice of [[complex oriented cohomology theory|complex orientation]] to second stage;

* for the [[quaternionic Hopf fibration]] $f = h_{\mathbb{H}}$ the attaching space is [[quaternionic projective space]] $\mathbb{H}P^2$ and the choice of homotopy $c$ is a choice of [[quaternionic oriented cohomology theory|quaternionic orientation]] to second stage.

{#PastingDiagramForHomotopyWhiteheadIntegral}  Moreover, the de-composition of this [[pasting diagram]] exhibits the [[homotopy Whitehead integral]]/[[functional cup product]]-formula for the Hopf invariant:

\begin{imagefromfile}
        "file_name": "EWhiteheadIntegralPastingDiagram.jpg",
        "web": "nlab",
        "width": 700,
        "unit": "px",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting the homotopy Whitehead integral",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

## Properties

### Generic values

For $n$ odd, the Hopf invariant necessarily vanishes. For $n$ even however, then there is a homomorphism

$$
  \pi_{2n-1}(S^n) \longrightarrow  \mathbb{Z}
$$

whose [[image]] contains at least the even integers.

### Hopf invariant one

A famous open question in the 1950s was which maps $f$ (eq:TheMap) have _Hopf invariant one_, namely $[HI(f)] = 1$ (eq:HopfInvariantFormula).

The _[[Hopf invariant one theorem]]_ ([Adams60](Hopf+invariant+one#Adams60)) states that the only maps of Hopf invariant one, $[HI(f)] = 1$, are the [[Hopf constructions]] on the four real [[normed division algebras]]:

* the [[real Hopf fibration]];

* the [[complex Hopf fibration]];

* the [[quaternionic Hopf fibration]];

* the [[octonionic Hopf fibration]].

### Via Sullivan models

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

By standard results in [[rational homotopy theory]], every [[continuous function]]

$$
  S^{4k-1} \overset{f}{\longrightarrow} S^{2k}
$$

corresponds to a unique [[dgc-algebra]] [[homomorphism]]

$$
  CE
  \big(
    \mathfrak{l}S^{4k-1}
  \big)
  \overset{ CE(\mathfrak{l}f) }{\longleftarrow}
  CE
  \big(
    \mathfrak{l}S^{2k}
  \big)
$$

between [[Sullivan models]] [[rational n-sphere|of n-spheres]].

The unique free [[coefficient]] of this homomorphism $CE(\mathfrak{l}f)$ is the Hopf invariant $HI(f)$ of $f$:

\begin{center}
\begin{imagefromfile}
  "file_name": "HopfInvariantFromSullivanModels.jpg",
  "width": 530
\end{imagefromfile}
\end{center}

=--

### Whitehead integral formula

See at _[[Whitehead integral formula]]_ and see the references [below](#ReferencesWhiteheadIntegralFormula)

## Related concepts

* [[Hopf invariant one]]

* [[Whitehead integral formula]]

  * [[functional cup product]]

  * [[Hopf-Wess-Zumino term]]

* [[e-invariant]]

* [[Hopf degree theorem]]

* [[EHP spectral sequence]]

## References

### General

* {#Husemoller66} [[Dale Husemöller]], chapter 15 of: _Fibre Bundles_, Graduate Texts in Mathematics 20, Springer New York (1966) ([doi:10.1007/978-1-4757-2261-1](https://link.springer.com/book/10.1007/978-1-4757-2261-1))

* {#MosherTangora86} [[Robert Mosher]], [[Martin Tangora]], p. 33 of _Cohomology operations and applications in homotopy theory_, Harper & Row 1986 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf))

* [[Michael Crabb]], [[Andrew Ranicki]], *The Geometric Hopf Invariant and Surgery Theory*,  Springer Monographs in Mathematics, Springer (2017) $[$[doi:10.1007/978-3-319-71306-9](https://doi.org/10.1007/978-3-319-71306-9), [slides pdf](https://www.maths.ed.ac.uk/~v1ranick/slides/hopfbeam.pdf), [[CrabbRanicki-GeometricHopf.pdf:file]]$]$

  > (in relation to [[surgery theory]])

* {#Wirthmuller12} [[Klaus Wirthmüller]], section 12 of: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* [[Gereon Quick]], _The Hopf invariant one problem via K-theory_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([[QuickHopfInvariant.pdf:file]])

See also:

* Wikipedia, _[Hopf invariant](http://en.wikipedia.org/wiki/Hopf_invariant)_

And see the references at _[[Hopf invariant one]]_ for more, in particular for the formulation via [[topological K-theory]].

### Whitehead's integral formula
  {#ReferencesWhiteheadIntegralFormula}

Discussion via [[differential forms]]/[[real homotopy theory]]/[[rational homotopy theory]] (see also at [[functional cup product]]):

* {#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))

* [[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/BFb0063500](https://doi.org/10.1007/BFb0063500))

* 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]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_, Comm. Math. Phys. 2020 ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

[[!redirects Hopf invariants]]


Hopf invariant one

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic topology]], _Hopf invariant one_ refers to [[homotopy classes]] of [[continuous functions]] between [[spheres]] of the form

$$
  \phi \;\colon\; S^{2n-1} \longrightarrow S^n
  \,,
$$

hence to elements in the [[homotopy groups of spheres]]

$$
  [\phi] \in \pi_{2n-1}(S^n)
  \,,
$$

whose [[Hopf invariant]] is equal to one:

$$
  h(\phi) = 1
  \,.
$$

Often this is regarded not in the [[integers]] but in its [[quotient]] to the [[cyclic group of order 2]] $\mathbb{Z}/2$, where it hence says that the actual integer value is an [[odd number]].

The _Hopf invariant one problem_, once a famous open problem, solved in [Adams 60](#Adams60), is the classification of maps for which Hopf invariant is indeed equal to one.

## Properties

### Relation to H-space structure and normed division algebras
 {#RelationToHSpaceStructureAndNormedDivisionAlgebras}

The existence of an element of Hopf invariant one in $\pi_{2n-1}(S^n)$
is equivalent to the existence of an [[H-space]] structure on $S^{n-1}$.

A celebrated theorem due to ([Adams 60](#Adams60), introducing and using the [[Adams spectral sequence]])  states that maps of Hopf invariant one correspond precisely to the the [[Hopf constructions]] on the four [[normed division algebras]] (see also at [[Hurwitz theorem]]): the [[real Hopf fibration]], the [[complex Hopf fibration]], the [[quaternionic Hopf fibration]] and the [[octonionic Hopf fibration]].

([Adams 60](#Adams60))

## History

> [[stable homotopy theory|Stable homotopy theory]] emerged as a distinct branch of [[algebraic topology]] with [[Frank Adams|Adams]]' introduction of [[Adams spectral sequence|his eponymous spectral sequence]] and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem.

> ([Elmendorf-Kriz-May 95](#ElmendorfKrizMay95))

## Related concepts

* [[e-invariant]]

* [[division algebra and supersymmetry]]

* [[cross product]]

* [[parallelizable manifold|parallelizable]] [[n-sphere]]

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

* [[Arf-Kervaire invariant problem]]

## References

The original proof that the only maps of Hopf invariant one are the [[Hopf constructions]] on the four [[normed division algebras]] is due to

* {#Adams60} [[John F. Adams]], *On the non-existence of elements of Hopf invariant one*, Ann. Math. **72** 1 (1960) 20-104 [[jstor:1970147](http://www.jstor.org/stable/1970147)]

* {#Adams66} [[John F. Adams]], Section 8 of: *On the groups $J(X)$ IV*, Topology **5** 21 (1966) [[pdf](http://math.unice.fr/~cazanave/Gdt/ImJ/J-IV.pdf), doi:10.1016/0040-9383(66)90004-8]

and made use of the [[classical Adams spectral sequence]].

Review:

* [[Doug Ravenel]], chapter 1, section 2 in: _Methods of computing $\pi_\bullet(S^n)$_ in: _[[Complex cobordism and stable homotopy groups of spheres]]_

* {#Rognes12} [[John Rognes]], around lemma 4.14, theorem 4.15 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

* Joseph Victor, Section 2.4 of _Stable Homotopy Groups of Spheres and The Hopf Invariant One Problem_, 2013 ([pdf](http://mathematics.stanford.edu/wp-content/uploads/2013/08/Victor-Honors-Thesis-2013.pdf))

Comments on the impact of this proof on the development of [[stable homotopy theory]]:

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

Another proof that instead uses [[topological K-theory]], [[Adams operations]] and the [[Atiyah-Hirzebruch spectral sequence]] was given in

* {#AdamsAtiyah66} [[John F. Adams]], [[Michael Atiyah]], *K-theory and the Hopf invariant*, Quart. J. Math. Oxford **17** 1 (1966) 31-38 [[pdf](http://www.maths.ed.ac.uk/~aar/papers/adamatiy.pdf), [doi:10.1093/qmath/17.1.31](https://doi.org/10.1093/qmath/17.1.31)]

see also [Adams 66, Sections 7, 8](#Adams66).

Review of the [[Hopf invariant one problem]] via [[Adams operations]] in [[topological K-theory]]:

* {#Husemoller66} [[Dale Husemöller]], chapter 15 of _Fibre Bundles_, Graduate Texts in Mathematics 20, Springer New York (1966)

* {#Wirthmuller12} [[Klaus Wirthmüller]], section 12 of: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* {#Hatcher} [[Allen Hatcher]], section 2.3 of _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* {#AguilarGitlerPrieto02} [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], section 10.6 of; _Algebraic topology from a homotopical viewpoint_, Springer (2002) [[toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf)]

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 9 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Akhil Mathew]], _[K-Theory and the Hopf invariant](https://amathew.wordpress.com/2012/01/08/k-theory-and-the-hopf-invariant/)_, 2012

* [[Gereon Quick]], _The Hopf invariant one problem via K-theory_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([pdf](https://ncatlab.org/nlab/files/QuickHopfInvariant.pdf))

* [[Ulrich Pennig]], _[K-theory and the Hopf invariant](http://wwwmath.uni-muenster.de/reine/u/topos/lehre/SS2015/KTheorie-Hopf/Hopf_eng.html)_

* [[Akhil Mathew]], _[K-Theory and the Hopf invariant](https://amathew.wordpress.com/2012/01/08/k-theory-and-the-hopf-invariant/)_, 2012

* Ishan Banerjee, _The Hopf invariant one problem_, 2016 ([pdf](http://math.uchicago.edu/~may/REU2016/REUPapers/Banerjee.pdf))

Review using the [[BP]]-[[Adams-Novikov spectral sequence]] includes

* [[Doug Ravenel]], chapter 5, section 2 _[[Complex cobordism and stable homotopy groups of spheres]]_

[[!redirects Hopf invariant 1]]

[[!redirects Hopf invariant one theorem]]

[[!redirects Hopf invariant one problem]]


Hopf link

The __Hopf link__ is a famous [[link]].  It is a [[Brunnian link]] with $2$ components.

[[!include Hopf link - SVG]]

+-- {: .un_remark}
###### Note
To include this picture in a page, write
<nowiki><code>[[!include Hopf link - SVG]]</code></nowiki>.
=--

[[!redirects Hopf link]]
[[!redirects Hopf links]]


Hopf link - SVG

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category: svg


Hopf module

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The Hopf modules over bimonoids are modules in the category of comodules
or viceversa. This notion has many generalizations and variants. Relative Hopf modules
are an algebraic and possibly noncommutative analogue of a notion of an equivariant
sheaf.

## Definition

Given a $k$-[[bialgebra|bialgebra]] $(H,m_H,\eta,\Delta,\epsilon)$, a left-right **Hopf module** of $H$ is a $k$-[[module]] $M$ with the structure of left $H$-module and right $H$-[[comodule]], where the action $\nu: H\otimes M\to M$ and right $H$-coaction $\rho : M\to M\otimes H$ are compatible in the sense that the coaction is a morphism of left modules. In this, the structure of left module on $M\otimes H$ is the standard tensor product of modules over Hopf algebras, with the action given by $(\nu\otimes m_H)\circ(H\otimes \tau\otimes H)\circ(\Delta \otimes M\otimes H)$ as $k$-linear map $H\otimes(M\otimes H)\to M\otimes H$ where $\tau=\tau_{H,M}:H\otimes M\to M\otimes H$ is the standard flip of tensor factors in the [[symmetric monoidal category]] of $k$-modules.

An immediate generalization of Hopf modules is for the case where $(E,\eho_E)$ is a right $H$-comodule algebra (a monoid in the category of $H$-comodules); then one can define the category ${}_E\mathcal{M}^H$ of left $E$- right $H$- __relative Hopf modules__ (less precisely, $(E,H)$-relative Hopf modules, or simply (relative) Hopf modules), which are left $E$-modules that are right $H$-comodules with a natural compatibility condition. In [[Sweedler notation]] for comodules. where $\rho(m) = \sum m_{(0)}\otimes m_{(1)}$, $\rho_E(e) = \sum e_{(0)}\otimes e_{(1)}$, the compatibility condition for the left-right relative Hopf modules is
$\rho (e m) = \sum e_{(0)} m_{(0)} \otimes e_{(1)} m_{(1)}$ for all $m\in M$ and $e\in E$.

There are further generalizations where instead of a bialgebra $H$ and a $H$-comodule algebra $E$ one replaces $E$ by an arbitrary algebra $A$, and $H$ by a coalgebra $C$ and introduces a compatibility in the sense of a mixed [[distributive law]] or [[entwining]] (structure). Then the relative Hopf modules become a special case of so-called [[entwined module]]s, see the monograph \[BW 2003\].

Geometrically, relative Hopf modules are instances of [[equivariant object]]s (equivariant quasicoherent sheaves) in [[noncommutative algebraic geometry]], the statement of which can be made precise, cf. \[Škoda 2008\].

Furthermore, in the context of relative Hopf modules there is an analogue of the faithfully flat descent along [[torsor]]s from commutative algebraic geometry, and the Galois descent theorems in algebra. Its main instance is [[Schneider's theorem]], asserting that if $H$ is a Hopf algebra and $U\hookrightarrow E$ a faithfully flat $H$-[[Hopf-Galois extension]] then the natural adjunction between the categories of relative $(E,H)$-Hopf modules and left $U$-modules is an [[equivalence of categories]]. This corresponds to the classical theorem saying that the category of equivariant quasicoherent sheaves over the total space of a torsor is equivalent to the category of the quasicoherent sheaves over the base of the torsor.

## Hopf bimodules

One can also consider Hopf bimodules, and similar categories. A Hopf $H$-bimodule is
left and right $H$-comodule
and left and right $H$-bimodule, where all four
structure are compatible in standard way.
The category of Hopf bimodules, ${}_H^H\mathcal{M}^H_H$ is monoidally equivalent to the category of [[Yetter-Drinfeld module]]s.

## Fundamental theorem on Hopf modules

If $H$ is a Hopf algebra over a field $k$, then the category of the ordinary Hopf modules ${}_H^H\mathcal{M}$ is equivalent to the category of $k$-vector spaces. See Section 1 of [Montgomery 1993](#Mont93) for more.

The equivalence may be seen as follows. Any vector space $V$ can be endowed with a (left-) Hopf module structure, for $H$ a Hopf algebra, simply by tensoring with $H$. The action of $H$ is given as

$$
\rho: h'\otimes h\otimes v\mapsto h'h\otimes v
$$

and the coaction as

$$
\sigma: h\otimes v\mapsto \Delta(h)\otimes v
$$

for $\Delta:H\to H\otimes H$ the comultiplication. This is known as a **trivial Hopf module**.

The fundamental theorem of Hopf modules states that *any* Hopf module $M$ arises precisely in this way, as one shows that

$$
M\cong H \otimes M^{\text{co} H}
$$

where $M^{\text{co} H}:= \{m\in M \vert \sigma(m)= 1_H \otimes m\}$ is the space of coinvariant of $M$ under the coaction $\sigma$ of $H$. In fact, the operations

$$
V \mapsto H\otimes V
$$

$$
M\mapsto M^{\text{co} H}
$$

come as functors realizing an equivalence of categories between vector spaces, and $H$-Hopf modules (see [Vercruysse 2012](#Vercruysse12) for more on this).

## References

Related entries include [[comodule algebra]], [[Schneider's descent theorem]], [[Yetter-Drinfeld module]], [[entwined module]]

* BW2003: [[T. Brzeziński]], R. Wisbauer, __Corings and comodules__, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
* Škoda 2008: [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal __16__ (2009), No. 1, 183–202, [arXiv:0811.4770](https://arxiv.org/abs/0811.4770) [MR2011b:14004](https://www.ams.org/mathscinet-getitem?mr=2527623)
* {#Mont93} [[Susan Montgomery]], _Hopf algebras and their actions on rings_, CBMS Lecture Notes __82__, AMS 1993, 240p.
* Peter Schauenburg, _Hopf modules and Yetter - Drinfel′d modules_, J. Algebra __169__:3 (1994) 874-890 [doi](https://doi.org/10.1006/jabr.1994.1314); _Hopf modules and the double of a quasi-Hopf algebra_, Trans. Amer. Math. Soc. __354__ (2002), 3349-3378 [doi](https://doi.org/10.1090/S0002-9947-02-02980-X) [pdf](https://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf); _Actions of monoidal categories, and generalized Hopf smash products_, Journal of Algebra __270__ (2003) 521-563, doi [ps](http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/amcghsp.ps)
* A. Borowiec, G. A. Vazquez Coutino, _Hopf modules and their duals_, [math.QA/0007151](http://arxiv.org/abs/math/0007151)
* H-J. Schneider, _Principal homogeneous spaces for arbitrary Hopf algebras_,
Israel J. Math. __72__ (1990), no. 1-2, 167--195 [MR92a:16047](http://www.ams.org/mathscinet-getitem?mr=1098988) [doi](https://doi.org/10.1007/BF02764619)
* Francesco d'Andrea, Alessandro de Paris, _On noncommutative equivariant bundles_, [arXiv:1606.09130](http://arxiv.org/abs/1606.09130)

* {#Vercruysse12} [[Joost Vercruysse]]. *Hopf algebras---Variant notions and reconstruction theorems*. (2012). ([arXiv:1202.3613](https://arxiv.org/abs/1202.3613))

category: algebra
[[!redirects Hopf modules]]
[[!redirects Hopf bimodule]]
[[!redirects Hopf bimodules]]


Hopf monad

+-- {: .rightHandSide}
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###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One can consider various compatibilities between a ([[comonad|co]])[[monad]] on a [[monoidal category]] and the underlying monoidal product; they are variants of the idea of a [[distributive law]]. Similarly, one can look at a compatibility between an action of a monoidal category and a (co)monad on the same category. A _Hopf monad_ satisfies conditions analogous to that of a _[[Hopf monoid]]_ ([Bruguières 06](#Bruguieres)).

## Definitions

Warning: more than one notion comes up with the names of [[bimonad]] and Hopf monad; there are also related notions of monoidal monad and opmonoidal monad as well of a [[strong monad]].

### Monoidal monads...

A __monoidal monad__ on a monoidal category is a [[monad]] whose underlying [[endofunctor]] is a [[lax monoidal functor]] and such that the unit and multiplication are monoidal natural transformations. Consequently, the [[Kleisli category]] of a monoidal monad has a canonical monoidal structure such that the [[forgetful functor]] is strict monoidal.

(...)

### Opmonoidal monads with invertible fusion

An **opmonoidal monad** on a monoidal category is a [[monad]] whose underlying [[endofunctor]] is a [[colax monoidal functor]], and such that the unit and multiplication are monoidal natural transformations.  An opmonoidal monad is also called a **bimonad**.

For an opmonoidal monad $T$, one can define the **left fusion operator** to be the natural transformation
$$ T(X\otimes T Y) \to T X \otimes T^2 Y \to T X \otimes T Y $$
of the opmonoidal constraint of $T$ and its monad multiplication.  Similarly, we have the **right fusion operator**
$$ T(T X\otimes Y) \to T^2 X \otimes T Y \to T X \otimes T Y $$
The fusion operators satisfy certain axioms.  In fact, an opmonoidal monad structure on $T$ is uniquely determined by a fusion operator (of either sort) along with the monad unit $Id\to T$ and the opmonoidal unit constraint $T I \to I$, satisfying appropriate axioms.

An opmonoidal monad is called a **Hopf monad** if both of its fusion operators are invertible.  More generally we have *left* and *right* Hopf monads where only one fusion operator is invertible.  This definition is in Bruguières-Lack-Virelizier.

### Distributing monad and comonad structures

Alternatively, one might be tempted to define a Hopf monad to be a [[Hopf monoid]] in the monoidal category of endofunctors, but that monoidal category is not [[symmetric monoidal category|symmetric]] or even [[braided monoidal category|braided]] or [[duoidal category|duoidal]], so this doesn't make sense.  However, we can instead ask for a "local" braiding in the form of a mixed [[distributive law]].

Thus, we might define a **bimonad** to be an endofunctor $H$ equipped with the structure of both a monad and a comonad, along with a [[distributive law]] $\lambda: H H \to H H$ satisfying suitable axioms analogous to those of a bimonoid.  A bimonad in this sense is a **Hopf monad** if it has an antipode $s:H\to H$ making the same diagrams commute as for a [[Hopf monoid]].  This definition is in Mesablishvili-Wisbauer.

## Examples

* If $H$ is a [[bimonoid]] in a [[braided monoidal category]], then the monad $T_H X =H\otimes X$ (whose algebras are $H$-[[modules]]) is opmonoidal, with constraints induced by the comultiplication and counit of $H$.  If $H$ is moreover a [[Hopf monoid]], then $T_H$ is a (Bruguières-Lack-Virelizier) Hopf monad.

* In fact, if $H$ is a bimonoid as before, then $T_H$ is *also* a bimonad in the sense of Mesablishvili-Wisbauer, with monad and comonad structures induced by the monoid and comonoid structures of $H$.  Moreover, if $H$ is a Hopf monoid, then $T_H$ is a Hopf monad in their sense as well.

## Properties

If $T$ is a (Bruguières-Lack-Virelizier) Hopf monad on a [[closed monoidal category]], then its category of algebras is also closed and the monadic forgetful functor preserves internal-homs.

## Related concepts

* [[Hopf adjunction]]

* [[Hopf algebra]]

* [[Hopf monoid]]

* [[Hopf monoidal category]]

* [[Frobenius monad]]

## References

* [[Alain Bruguières]], [[Alexis Virelizier]], *Hopf monads*, Advances in Mathematics **215** 2 (2007) 679-733 [[doi:10.1016/j.aim.2007.04.011](https://doi.org/10.1016/j.aim.2007.04.011), [arXiv:math/0604180](https://arxiv.org/abs/math/0604180)]

* Masahito Hasegawa and Jean-Simon Pacaud Lemay, *Hopf Monads on Biproducts*, Theory and Applications of Categories **39** 28 (2023) 804-823 [[tac:3928](http://www.tac.mta.ca/tac/volumes/39/28/39-28abs.html)]

Monograph:

* [[Gabriella Böhm]], _Hopf algebras and their generalizations from a category theoretical point of view_, Lecture Notes in Math. __2226__, Springer 2018, [doi](https://link.springer.com/book/10.1007/978-3-319-98137-6)

See also:

* Wikipedia. *[Opmonoidal monad](http://en.wikipedia.org/wiki/Monoidal_monad)*

* [[Kornél Szlachányi]], _The monoidal Eilenberg-Moore construction and bialgebroids_, J. Pure Appl. Algebra __182__, no. 2–3 (2003) 287–315

* R. Wisbauer, _Bimonads and Hopf monads on categories_, [pdf](http://www.math.uni-duesseldorf.de/~wisbauer/Hopfmonad.pdf)

* Bachuki Mesablishvili, [[Robert Wisbauer]], _Notes on bimonads and Hopf monads_, Theory Appl. Cat. __26__:10, 2012, 281-303,  [abs](http://www.tac.mta.ca/tac/volumes/26/10/26-10abs.html) [pdf](http://www.tac.mta.ca/tac/volumes/26/10/26-10.pdf) [arxiv/1010.3628](http://arxiv.org/abs/1010.3628)

* D. Chikhladze, S. Lack, [[R. Street]], _Hopf monoidal comonads_, Theory and Appl. of Cat. __24__ (2010) No. 19, 554-563.[tac](http://www.tac.mta.ca/tac/volumes/24/19/24-19abs.html)

* {#Bruguieres} A. Bruguières, _Hopf monads_, [math.QA/0604180](http://arxiv.org/abs/math/0604180);

  * _Hopf monads: an introduction_, an exposè, [pdf](http://www.cirm.univ-mrs.fr/videos/2006/exposes/23/Bruguieres.pdf);

  * _Hopf monads, tensor categories and quantum invariants_, an exposè, [pdf](http://www.cirm.univ-mrs.fr/videos/2008/exposes/307/Bruguiere.pdf);

  * _Galois-Grothendieck duality, Tannaka duality and Hopf (co)monads_. talk at workshop "Hopf algebras and tensor categories", University of Almería, July 4-8, 2011, [pdf](http://www.ual.es/Congresos/hopf2010/charlas/bruguiertalk.pdf)

* Group = Hopf algebra, blog discussion, [sbseminar](http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra)
* Marek Zawadowski, _The formal theory of monoidal monads_, [arxiv/1012.0547](http://arxiv.org/abs/1012.0547)
* M. Zawadowski, _coMalcev monads_, slides from a talk, [pdf](http://duch.mimuw.edu.pl/~zawado/Talks/coMalcev_talk.pdf)
* [[Ieke Moerdijk]], _Monads on tensor categories_, J. Pure. Appl. Alg. __168__, 2-3, (2002), 189-208, doi
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Alg. __328__, n. 1, Feb 2011, 1-30, [doi](http://dx.doi.org/10.1016/j.jalgebra.2010.07.032), [arxiv/1002.4493](http://arxiv.org/abs/1002.4493)
[[G. Böhm]], _Weak bimonads and weak Hopf monads_, conference slides, 2010, [pdf](http://www.ual.es/congresos/hopf2010/charlas/bohm.pdf)

* [[Simon Willerton]], _A diagrammatic approach to Hopf monads_, [arxiv/0807.0658](http://arxiv.org/abs/0807.0658)

* K. Dosen, Z. Petric, _Coherence for monoidal monads and comonads_, [arxiv/0907.2199](http://arxiv.org/abs/0907.2199)
* [[Anders Kock]], _Strong functors and monoidal monads_, [pdf](http://home.imf.au.dk/kock/SFMM.pdf), Archiv der Math. __23__: 113–120, (1972) [doi](http://dx.doi.org/10.1007%2FBF01304852); _Monads on symmetric monoidal categories_, [pdf](http://home.imf.au.dk/kock/MSMCC.pdf)
* B. J. Day, _Note on monoidal monads_, J. Austral. Math. Soc. A __23__, 292-311 (1977) [pdf](http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=4897004)
* Alain Bruguières, Sonia Natale, _Exact sequences of tensor categories_, [arXiv:math.QA/1006.0569](http://arxiv.org/abs/1006.0569) (generalizes Schneider's work on [[exact sequences of Hopf algebras]]; uses Hopf monads)

[[weak bimonad|Weak bimonad]] version

* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Algebra 328 (2011), 1-30, [arXiv:1002.4493](http://arxiv.org/abs/1002.4493)

[[!redirects Hopf monads]]
[[!redirects bimonad]]
[[!redirects bimonads]]
[[!redirects opmonoidal monad]]
[[!redirects opmonoidal monads]]


Hopf monoid

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# Hopf monoids

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## Idea

A Hopf monoid is the generalization of a [[Hopf algebra]] to an arbitrary [[symmetric monoidal category]].

## Definition

We work in an arbitrary symmetric monoidal category $\mathcal{C}$ (although one can generalize to a [[braided monoidal category]] or even a [[duoidal category]]).  Recall that a [[bimonoid]] in $\mathcal{C}$ is an object $H$ which is both a [[monoid]] object and a [[comonoid]] object in a compatible way, i.e. the comultiplication and the counit are morphisms of monoids, or equivalently the multiplication and the unit are morphisms of comonoids. (To make sense of this description, one requires the symmetry in order to define the appropriate monoid/comonoid structure on $H \otimes H$; e.g., if $\sigma$ is the symmetry and $\mu$ the monoid multiplication on $H$, the multiplication on $H \otimes H$ is

$$H \otimes H \otimes H \otimes H \stackrel{1 \otimes \sigma \otimes 1}{\to} H \otimes H \otimes H \otimes H \stackrel{\mu \otimes \mu}{\to} H \otimes H$$

where we are writing as if the monoidal product is [[coherence theorem for monoidal categories|strict]]. One can get away with an ambient context of a monoidal category provided that the monoid/comonoid structures in question can be viewed as living in the [[Drinfeld center|center]] of the monoidal category.)

+-- {: .un_defn}
###### Definition
A **Hopf monoid** is a bimonoid $H$ such that there exists a map $s:H\to H$, called the **antipode**, such that the following diagram commutes
$$
\array{
  H & \xrightarrow{\epsilon} & I & \xrightarrow{\eta} & H\\
  ^{\Delta}\downarrow &&&& \uparrow^{m}\\
  H\otimes H & & \xrightarrow{1\otimes s} & & H\otimes H.
  }
$$
as does the analogous diagram with $1\otimes s$ replaced by $s\otimes 1$.
=--

## Examples

* A Hopf monoid in $Vect$ is precisely a [[Hopf algebra]].

* In a [[cartesian monoidal category]], every [[monoid]] object is a bimonoid in a unique way.  Such a bimonoid is a Hopf monoid exactly when the monoid object is a [[group object]]. More generally, see the corresponding remark [there](https://ncatlab.org/nlab/show/group+object#GroupObjectsInGeneralMonoidalCategories).

## Properties

### Closed monoidal structure on modules
{#ClosedStructureOnModules}

For any bimonoid $H$ in $\mathcal{C}$, the category $Mod_H$ of $H$-[[modules]] inherits a monoidal structure such that the [[forgetful functor]] ("[[fiber functor]]") $Mod_H \to \mathcal{C}$ is [[strong monoidal functor|strong monoidal]]; see _[[bimonoid]]_.  If $H$ is moreover a Hopf monoid and $\mathcal{C}$ is a [[closed monoidal category]], then $Mod_H$ is also closed, and the forgetful functor is strong closed (preserves [[internal homs]]).

This is a special case of [[Tannaka duality]] [for monoids/algebras](Tannaka+duality#ForAlgebraModules), see at _[[structure on algebras and their module categories - table]]_.

Specifically, given $H$-modules $M$ and $N$, we define an $H$-module structure on the internal-hom $Hom(M,N)$ in $\mathcal{C}$ to be the [[adjunct]] of the following composite:
$$
\begin{aligned}
  H \otimes Hom(M,N) \otimes M
  &\xrightarrow{\Delta} H\otimes H \otimes Hom(M,N) \otimes M\\
  &\xrightarrow{\cong} H \otimes Hom(M,N)\otimes H \otimes M\\
  &\xrightarrow{1\otimes 1\otimes s\otimes 1} H \otimes Hom(M,N)\otimes H \otimes M\\
  &\xrightarrow{act} H \otimes Hom(M,N)\otimes M\\
  &\xrightarrow{eval} H \otimes N\\
  &\xrightarrow{act} N.
\end{aligned}
$$
If $\mathcal{C}$ is cartesian and $H$ is a group object, then this is the "conjugation" action, with $g\in H$ sending $f:M\to N$ to $m\mapsto g\cdot f(g^{-1}\cdot m)$.  Diagram chases show that this makes $Hom(M,N)$ an $H$-module and $Mod_H$ a closed monoidal category.

This result can be generalized to algebras over any [[Hopf monad]] (in the sense of Bruguières-Lack-Virelizier).

## References

A few references emphasizing applications to combinatorics:

* [[Marcelo Aguiar]] and [[Swapneel Mahajan]]. [[Monoidal Functors, Species and Hopf Algebras]], 2010. ([author pdf](http://www.math.cornell.edu/~maguiar/a.pdf))

* [[Marcelo Aguiar]] and [[Swapneel Mahajan]]. Hopf monoids in the category of species. _Contemporary Mathematics_ 585 (2013), 17–124. ([author pdf](http://www.math.cornell.edu/~maguiar/hmonoid.pdf))

* [[Marcelo Aguiar]] and [[Federico Ardila]]. Hopf monoids and generalized permutahedra. [arXiv:1709.07504](https://arxiv.org/abs/1709.07504)

## Related concepts

* [[bimonoid]], [[Frobenius monoid]]

* [[Hopf algebra]]

* [[Hopf monoidal category]]

* [[Hopf monad]]

[[!redirects Hopf monoids]]


Hopf monoidal category

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## Idea

The lift of the notion of [[Hopf algebra|Hopf]] [[bialgebra]] from [[associative algebras]] to [[tensor categories]].

By the corresponding higher [[Tannaka duality]], the [[2-category]] of [[module categories]] over a [[Hopf monoidal category]] is a [[monoidal 2-category]] with duals.

Related to [[4d TQFT]] as [[Hopf algebras]] are related to [[3d TQFT]].

## Properties

### Relation to trialgebras

Just as [[monoidal categories]] with [[fiber functor]] are the [[categories of modules]] of a [[Hopf algebra]], so Hopf monoidal categories are supposed to be the categories of modules of a [[trialgebra]].

## Related concepts

* [[Hopf monoid]], [[Hopf algebra]]

* [[Hopf monad]]

* [[Hopf V-category]]

[[!include structure on algebras and their module categories - table]]

## References

The notion is due to

* [[Louis Crane]], [[Igor Frenkel]], _Four dimensional topological quantum field theory, Hopf categories, and the canonical bases_, J.Math.Phys. 35 (1994) 5136-5154, ([arXiv:hep-th/9405183](http://arxiv.org/abs/hep-th/9405183))
 {#CraneFrenkel}

A proposal for the corresponding definition of [[trialgebras]] is in

* [[Hendryk Pfeiffer]], _2-Groups, trialgebras and their Hopf categories of representations_ ([arXiv:math/0411468](http://arxiv.org/abs/math/0411468))
 {#Pfeiffer}

A survey of related references is in p. 98 of

* [[John Baez]], [[Aaron Lauda]], _A prehistory of $n$-categorical physics_, in _Deep beauty_, 13-128, Cambridge Univ. Press, Cambridge, 2011 ([arXiv:0908.2469](http://arxiv.org/abs/0908.2469))
 {#BaezLauda}

[[!redirects Hopf monoidal categories]]
[[!redirects Hopf category]]
[[!redirects Hopf categories]]

[[!redirects Hopf tensor category]]
[[!redirects Hopf tensor categories]]


Hopf-Galois extension

# Hopf--Galois extensions
* tic
{: toc}

##Definition##

Let $H$ be a $k$-[[bialgebra|bialgebra]] and $E$, say, a right $H$-comodule algebra (i.e. a [[monoid]] in the category of right $H$-[[comodule]]s) with coaction $\rho:E\to E\otimes H$.

The subalgebra $U = E^{\mathrm{co}H}$ of $H$-coinvariants in $E$ consists of all $u\in E$ such that $\rho(u)=u\otimes 1$.

The $k$-algebra **extension** $U\hookrightarrow E$ is __Hopf--Galois__ over $H$ if the natural map $E\otimes_U E\to E\otimes H$ given by the $k$-linear extension of the formula $e\otimes e'\mapsto (e\otimes_k 1)\rho(e')$ is a bijection (hence a $k$-module isomorphism).

A **Hopf--Galois object** over a $k$-bialgebra $H$ is any Hopf-Galois extension $k\hookrightarrow E$ over $H$ of the ground field (or ring) $k$. It is a dual (and noncommutative) analogue to a torsor over a point.

##Classical Galois extensions as a special case##

If $k\subset U=E^G$, $k,E$ are [[field]]s, $G$ a finite group and $H = (k[G])^*$ is the dual [[Hopf algebra]] to the [[group algebra]] of $G$, then $E^G\hookrightarrow E$ is (classically) a [[Galois extension]] iff it is a $H$-Hopf--Galois extension, where the coaction of $H$ is induced by the action of $k[G]$, hence of $G$. One uses the Dedekind lemma on independence of automorphisms to prove this equivalence. It is possible however that $E^G\subset E$ is not (classically) Galois, but it is $K$-Hopf--Galois for some Hopf algebra $K\neq (k[G])^*$.

##Role in geometry##

In [[algebraic geometry]], given an affine algebraic $k$-group [[scheme]] $G$, the algebra $E$ of regular functions over the total scheme $X$ of an affine $G$-[[torsor]] $X\to Y$, whose base $Y$ also happens to be affine, is a commutative $H$-Hopf--Galois extension of the algebra of regular functions $U$ on the base $Y\cong X/G$, where $H$ is the $k$-Hopf algebra of global regular functions on $G$. In algebraic topology, a generalization to [[spectrum|spectra]] (with the [[smash product]] of spectra in the role of tensor product) was studied by Rognes and others (see ([Rognes 08](#Rognes08))). In [[noncommutative geometry]], Hopf--Galois extensions are studied as affine [[noncommutative principal bundle]]s, with interesting [[descent]] theorems for [[Hopf module|Hopf modules]] like the [[Schneider's theorem]]. Given a right $H$-Hopf-Galois extension $U\hookrightarrow E$ and a left $H$-comodule $V$ the cotensor product $k$-module $E\Box^H V$ is interpreted as a space of sections of the associated [[fiber bundle]] with structure group $Spec H$ (in noncommutative sense) and fiber $V$.

##Literature##

* H. F. Kreimer, M. Takeuchi, _Hopf algebras and Galois extensions of an algebra_, Indiana Univ. Math. J. 30 (1981), 615-692 [web](https://www.iumj.indiana.edu/IUMJ/fulltext.php?year=1981&volume=30&artid=30052) [pdf](https://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30052/pdf) [djvu](https://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30052/djvu)

* Hans-Jürgen Schneider, _Principal homogeneous spaces for arbitrary Hopf algebras_, Israel J. Math. __72__ (1990) 1-2, 167–195 [MR92a:16047](http://www.ams.org/mathscinet-getitem?mr=1098988) [doi](http://dx.doi.org/10.1007/BF02764619)

* [[Susan Montgomery]], _Hopf Galois theory: a survey_, Geometry and topology monographs 16 (2009) 367--400; [link](http://www.msp.warwick.ac.uk/gtm/2009/16/p012.xhtml), [doi](https://doi.org/10.2140/gtm.2009.16.367)

Discussion for [[ring spectra]]:

* {#Rognes08} [[John Rognes]], _Galois extensions of structured ring spectra/Stably dualizable groups_, Memoirs of the American Mathematical Society, 192(898), 2008,  ([partial pdf](https://folk.uio.no/rognes/papers/galois.pdf), [doi:10.1090/memo/0898](http://dx.doi.org/10.1090/memo/0898))

category: algebraic geometry
[[!redirects Hopf-Galois extensions]]
[[!redirects Hopf–Galois extension]]
[[!redirects Hopf–Galois extensions]]
[[!redirects Hopf--Galois extension]]
[[!redirects Hopf--Galois extensions]]


Hopf-Wess-Zumino term

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## Idea

The [[Green-Schwarz sigma-model]] of the [[M5-brane]] contains a 6d [[higher WZW term]].  The first summand of this term was called the _Hopf-Wess-Zumino term_ in [Intriligator 00](#Intriligator00), due to its resemblance of the [[Whitehead integral formula]] for the [[Hopf invariant]] if the [[C-field]] is assumed to be classified by a [[smooth function]] to the [[4-sphere]]. But in fact the full term is a [[homotopy Whitehead integral formula]] ("[[functional cup product]]" or "[[homotopy period]]") still computing the [[Hopf invariant]] if the [[C-field]] is assumed to be a cocycle in [[twisted Cohomotopy|twisted]] [[cohomotopy]] in degree 4 ([FSS 19](#FSS19WZ)).

Similar Hopf terms can be considered in all dimensions $4k+2$ ([TN 89](#TN89)).

## Related concepts

* [[Hopf invariant]], [[Hopf invariant one theorem]]

* [[Whitehead integral formula]]

* [[functional cup product]]

* [[M-theory]]

## References

The general concept of Hopf-Wess-Zumino terms was considered in

* {#WilczekZee84} [[Frank Wilczek]] and [[Anthony Zee]], _Linking Numbers, Spin, and Statistics of Solitons_, Phys. Rev. Lett. 51, 2250, 1983 ([doi:10.1103/PhysRevLett.51.2250](https://doi.org/10.1103/PhysRevLett.51.2250))

* {#TN89} Hsiung Chia Tze, Soonkeon Nam, _Topological Phase Entanglements of Membrane Solitons in Division Algebra $\sigma$ Models With a Hopf Term_, Annals Phys. 193 (1989) 419-471 ([spire:25058](http://inspirehep.net/record/25058/), doi:10.1016/0003-4916(89)90005-5)

The [[higher dimensional WZW model|higher WZW term]] of the [[M5-brane]] was first proposed in

* {#Aharony94} [[Ofer Aharony]], p. 10 (11 of 18) in *String theory dualities from M theory*, Nucl. Phys. B **476** (1996) 470-483 [[arXiv:hep-th/9604103](https://arxiv.org/abs/hep-th/9604103), doi:10.1016/0550-3213(96)00321-5]

and had been settled by the time of

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], (1) in: _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. **78** (1997) 4332-4334 [[arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149), [doi:10.1103/PhysRevLett.78.4332](https://doi.org/10.1103/PhysRevLett.78.4332)]

* [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], (17) in: _Covariant Action for a $D=11$ Five-Brane with the Chiral Field_, Phys. Lett. B **398** (1997) 41 [[arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037), doi:10.1016/S0370-2693(97)00188-3]

The resemblence of the first summand of the term to the [[Whitehead integral formula]] for the [[Hopf invariant]] was noticed in

* {#Intriligator00} [[Kenneth Intriligator]], _Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, $\mathcal{N}= (2,0)$ Field Theories_, Nucl. Phys. B **581** (2000) 257-273 [[arXiv:hep-th/0001205](https://arxiv.org/abs/hep-th/0001205), doi:10.1016/S0550-3213(00)00148-6]

which hence introduced the terminology "Hopf-Wess-Zumino term". Followup to this terminology includes

* {#KalkkinenStelle02} [[Jussi Kalkkinen]], [[Kellogg Stelle]], Section 3.2 of: _Large Gauge Transformations in M-theory_, J. Geom. Phys. **48** (2003) 100-132 [[arXiv:hep-th/0212081](https://arxiv.org/abs/hep-th/0212081), doi:10.1016/S0393-0440(03)00027-5]

* Shan Hu, [[Dimitri Nanopoulos]], _Hopf-Wess-Zumino term in the effective action of the 6d, $(2, 0)$ field theory revisted_, JHEP 1110:054 (2011) [[arXiv:1110.0861](https://arxiv.org/abs/1110.0861), doi:10.1007/JHEP10(2011)054]

* [[Alex Arvanitakis]], Section 4.1 of _Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 ([arXiv:1804.07303](https://arxiv.org/abs/1804.07303), [doi:10.4310/ATMP.2019.v23.n5.a1](https://dx.doi.org/10.4310/ATMP.2019.v23.n5.a1))

More on the relation to the Hopf invariant in

* {#Sati13} [[Hisham Sati]], _Framed M-branes, corners, and topological invariants_, J. Math. Phys. 59 (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

Discussion of the full 6d WZ term in view of [[schreiber:Hypothesis H|Hypothesis H]]:

* {#FSS19WZ} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[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 Hopf-Wess-Zumino terms]]

[[!redirects Hopf-WZ term]]
[[!redirects Hopf-WZ terms]]

[[!redirects Hopf WZ term]]
[[!redirects Hopf WZ terms]]


Hopfian group

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## Definition

### In sets

\begin{definition}
A [[discrete  group]] $G$ is called a __Hopfian group__ if every [[surjection|surjective]] [[endomorphism]] $\phi \colon G\to G$ is an [[isomorphism]].

[[formal duality|Dually]], it is called __coHopfian__ if any [[injection|injective]] endomorphism of $G$ is an isomorphism.
\end{definition}

### In concrete categories

As the [[epimorphisms]] and [[monomorphisms]] in [[Grp]] are precisely the surjections and injections (see [[epimorphisms of groups are surjective]]), the definition generalises immediately to that of a [[Hopfian object]] in any [[category]]. In other words, we *could* define an object $X$ to be Hopfian if every epimorphic endomorphism is an isomorphism (cf. *[[Dedekind finite object]]*).

Whatever that generalization is worth, much of the literature (such as [Varadarajan 1992](#Varadarajan92)) adopts the more concrete notion:

\begin{definition}
Given a [[concrete category]] $U \colon C \to Set$ with $U$ a [[faithful functor]], say that an [[object]] $X$ of $C$ is *Hopfian* if every [[morphism]] $\phi \colon X \to X$ in $C$ with $U(\phi)$ surjective is an isomorphism.
\end{definition}

\begin{remark}
Notice that if $U$ if [[faithful functor|faithful]], then $\phi$ is epimorpic if $U(\phi)$ is surjective.
\end{remark}

\begin{remark}
For [[monadic functors]] $U \colon C \to Set$, this surjectivity assumption is the same as the assumption that $\phi$ is a [[regular epimorphism]].
\end{remark}

Of course there is a dual notion of being co-Hopfian; here the hypothesis that $U(\phi)$ is injective frequently coincides simply with $\phi$ being monomorphic -- certainly that is true if $U$ [[preserved limit|preserves]] [[finite limits]] (which is frequently the case in application).

## Examples

\begin{example}
Clearly all [[finite groups]] are both Hopfian and coHopfian.
\end{example}

\begin{example}
Using Nielsen's method, one can show that every finitely generated [[free group]] and the union of any ascending chain of such free groups (for example, $\mathbb{Q}$) are Hopfian.
\end{example}

\begin{example}
Every [[torsion-free group|torsion-free]] [[hyperbolic group]] is Hopfian.
\end{example}

## References

* {#Varadarajan92} K. Varadarajan, _Hopfian and Co-Hopfian Objects_, Publicacions Matemátiques, Vol. 36 (1992), 293-317. ([web](http://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37700/37574))

[[!redirects Hopf group]]
[[!redirects coHopfian group]]
[[!redirects co-Hopfian group]]


hopfish algebra

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of _hopfish algebra_ is a generalization of that of [[Hopf algebra]] designed to behave better with respect to [[Morita equivalence]] of algebras. It is defined to be a [[sesquialgebra]] (hence a 2-algebra/[[3-module]]) which is grouplike in a suitable sense.

The notion subsumes [[Hopf algebras]] and [[weak Hopf algebras]].

## Definition

Let $R$ be some [[commutative ring]] (or [[E-infinity ring]]).

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

A [[sesquiunital sesquialgebra]] over $R$ is an [[associative algebra]] $A$ over $R$ equipped with the structure of an [[algebra object]] internal to the [[2-category]] [[2Mod]] of [[associative algebras]], [[bimodules]] and [[bimodule intertwiners]].

This means that it is an $R$-algebra $A$ equipped with

* a product $A \otimes_R A$-$A$-[[bimodule]] $\Delta$;

* a unit $R$-$A$-[[bimodule]] $\epsilon$

satisfying the evident [[associative law]] and [[unit law]].

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

A **preantipode** for a [[sesquiunital sesquialgebra]] $A$ is a left $A \otimes A$-[[module]] $S$ equipped with an [[isomorphism]] of right $A \otimes A$-[[modules]]

$$
  S^* \simeq Hom_A(\epsilon, \Delta)
  \,.
$$

An preantipode is an **antipode** if it is a [[free module]] over $A$ of [[rank]] 1 when regarded as an $A$-$A^{op}$-[[bimodule]].

A sesquiunital sesquialgebra equipped with such an antipode is a **hopfish algebra**.

=--

This is ([TWZ, def. 3.1, def. 3.2](#TWZ)).

## Properties

### Module categories and Tannaka duality
 {#ModuleCategoriesAndTannakaDuality}

The notion of [[sesquialgebra]] generalizes that of [[bialgebra]] such that under [[Tannaka duality]] sesquialgebras corespondond to [[monoidal categories]] generally, while the strictness of bialgebras means that there their monoidal category of [[modules]] is equipped with a [[fiber functor]].

Since moreover [[Hopf algebras]] correspond to [[rigid monoidal categories]] with fiber functor under [[Tannaka duality]], the correct sesqui-algebra generalization of Hopf algebras should have exactly the rigid monoidal categories as module categories, up to equivalence, without necessarily a fiber functor. This is expressed by the following table

[[!include structure on algebras and their module categories - table]]

## References

The notion was introduced in

* Xiang Tang, [[Alan Weinstein]], [[Chenchang Zhu]], _Hopfish algebras_, Pacific J. Math. 231 (2007), no. 1, 193--216. ([arXiv:math/0510421](http://arxiv.org/abs/math/0510421))
 {#TWZ}

[[!redirects Hopfish algebra]]

[[!redirects hopfish algebras]]
[[!redirects Hopfish algebras]]


Hopkins-Miller theorem

## References

* [[Charles Rezk]], _Notes on the Hopkins-Miller theorem_ ([pdf](http://www.math.uiuc.edu/~rezk/hopkins-miller-thm.pdf))


Horacio Porta

* [webpage](https://faculty.math.illinois.edu/~horacio/)

## related $n$Lab entries

* [[topological algebra]]

* [[Gelfand duality]]

category:people


Horatiu Nastase

* [webpage](http://www.het.brown.edu/~nastase/)

## Selected writings

On [[Inönü-Wigner contraction]] of [[anti de Sitter spacetimes]] to [[pp-wave spacetimes]], [[single trace operators]] in the [[AdS/CFT correspondence]], and introducing the [[BMN matrix model]]:

* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], Section 2 of: _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP 0204 (2002) 013 ([arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021))

On [[D4-D8 brane bound states]] as [[black branes]], and [[holographic QCD]]:

* {#Nastase03} [[Horatiu Nastase]], Sections 2, 3 of: _On Dp-Dp+4 systems, QCD dual and phenomenology_ ([arXiv:hep-th/0305069](https://arxiv.org/abs/hep-th/0305069))

On the [[AdS/CFT correspondence]]:

* [[Horatiu Nastase]], _Introduction to AdS-CFT_ ([arXiv:0712.0689](http://arxiv.org/abs/0712.0689))

* [[Horatiu Nastase]], _Introduction to AdS/CFT correspondence_, Cambridge University Press, 2015 ([cds:1984145](http://cds.cern.ch/record/1984145), [doi:10.1017/CBO9781316090954](https://doi.org/10.1017/CBO9781316090954))

* {#Nastase18} [[Horatiu Nastase]], _Towards deriving the AdS/CFT correspondence_ ([arXiv:1812.10347](https://arxiv.org/abs/1812.10347))

On the [[AdS/CMT correspondence]]:

* [[Horatiu Nastase]], *String Theory Methods for Condensed Matter Physics*, Cambridge University Press (2017) [[doi:10.1017/9781316847978](https://doi.org/10.1017/9781316847978) ]

On [[M2-M5 brane bound states]] in relation to [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]] via the [[ABJM model]]:

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], [[Sanjaye Ramgoolam]], _The fuzzy $S^2$ structure of M2-M5 systems in ABJM membrane theories_, JHEP 0905:123, 2009 ([arXiv:0903.3966](https://arxiv.org/abs/0903.3966))

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], _Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors_, SIGMA 6:058, 2010 ([arXiv:1003.5590](https://arxiv.org/abs/1003.5590))

On the [[ABJM model]] as a [[matrix model]]:

* {#MohammedMuruganNastase10} Asadig Mohammed, Jeff Murugan, [[Horatiu Nastase]], _Looking for a Matrix model of ABJM_, Phys. Rev. D82:086004, 2010 ([arXiv:1003.2599](https://arxiv.org/abs/1003.2599))

On a [[TT-deformation]] of the [[Skyrme model]] applied to [[nucleon]] [[scattering]]:

* [[Horatiu Nastase]], [[Jacob Sonnenschein]], _A $T \bar T$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering_ ([arXiv:2101.08232](https://arxiv.org/abs/2101.08232))

## Related $n$Lab entries

* [[Chern-Simons gravity]]

category: people


horizon > history

< [[horizon]]