topological G-space

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### Context
#### Topology
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[[!include topology - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[topology]], a _topological $G$-space_ (traditionally just _$G$-space_, for short, if the context is clear) is a [[topological space]] equipped with an [[action]] of a [[topological group]] $G$ -- the _[[equivariance group]]_ (usually taken to be the [[topological group]] underlying a [[compact Lie group]], such as a [[finite group]]).

The canonical [[homomorphisms]] of topological $G$-spaces are $G$-[[equivariant function|equivariant]] [[continuous functions]], and the canonical choice of [[homotopies]] between these are $G$-equivariant continuous homotopies (for [[trivial action|trivial]] $G$-[[action]] on the [[interval]]).

A $G$-[[equivariant Whitehead theorem|equivariant version]] of the [[Whitehead theorem]] says that on [[G-CW complexes]] these $G$-equivariant [[homotopy equivalences]] are equivalently those maps that induce [[weak homotopy equivalences]] on all [[fixed point]] spaces for all [[subgroups]] of $G$ (compact subgroups, if $G$ is allowed to be a [[Lie group]]).

By [[Elmendorf's theorem]], this, in turn, is equivalent to the [[(∞,1)-presheaves]] over the [[orbit category]] of $G$. See below at _[In topological spaces -- Homotopy theory](#HomotopyTheory)_.

See ([Henriques-Gepner 07](#HenriquesGepner07)) for expression in terms of [[topological groupoids]]/[[orbispaces]].

In the context of [[stable homotopy theory]] the [[stabilization]] of $G$-spaces is given by [[spectra with G-action]]; these lead to [[equivariant stable homotopy theory]]. See there for more details. (But beware that in this context one considers the richer concept of [[G-spectra]], which have a [[forgetful functor]] to [[spectra with G-action]] but better homotopy theoretic properties. ) The union of this as $G$ is allowed to vary is the [[global equivariant stable homotopy theory]].

## Properties

### Change of equivariance groups
 {#ChangeOfGroupsAndFixedLoci}

In the following $TopSp$ is a [[cartesian closed monoidal category|cartesian closed]] [[convenient category of topological spaces]], such as that of [[compactly generated topological spaces]].

We discuss how any homomorphism of topological groups induces an [[adjoint triple]] of [[functors]] between the corresponding $Topological G Spaces$ (e.g. [May 96, Sec. I.1](#May96), [DHLPS 19, p.8](#DHLPS19), see also at [induced representation](induced+representation#GeneralAbtractDefinition) for a formulation in [[homotopy type theory]])

\begin{tikzcd}
  G_1\mathrm{Spaces}
  \ar[
    rr,
    shift right=14pt,
    "{
      \mathrm{Maps}
      (
        G_2,
        -
      )^{G_1}
    }"{
      below
    },
    "\scalebox{.7}{$\bot$}"{above}
  ]
  \ar[
    rr,
    shift left=14pt,
    "{
      G_2 \times_{{}_{G_1}} (-)
    }"{
      above
    },
    "\scalebox{.7}{$\bot$}"{below}
  ]
  &&
  G_2\mathrm{Spaces}
  \mathrlap{\,,}
  \ar[
    ll,
    "\;\phi^\ast"{description}
  ]
\end{tikzcd}

Throughout, let $G_1, G_2 \in $ [[TopologicalGroups]]
and consider a [[continuous function|continuous]] [[homomorphism]] of [[topological groups]]

\[
  \label{TopologicalGroupHomomorphism}
  \phi
  \;\colon\;
  G_1
    \longrightarrow
  G_2
  \,.
\]

#### Pullback action

\begin{defn}\label{PullbackAction}
  **(pullback action)** \linebreak
Given a topological group homomorphism $\phi$ (eq:TopologicalGroupHomomorphism),
write
  $$
    Topological G_1 Spaces
      \overset{
        \;\;\;
        \phi^\ast
        \;\;\;
       }{\longleftarrow}
    Topological G_2 Spaces
  $$

for the [[functor]] which takes a [[topological G-space|topological G2-space]] $(X,\rho)$ to the same underlying [[topological space]] $\rho$, equipped with the $G_1$-action $ \rho(\phi(-))$.

\end{defn}

\begin{example}\label{RestrictedAction}
**(restricted action)** \linebreak

For $\phi \coloneqq i \colon H \hookrightarrow G$ a [[subgroup]]-inclusion, the pullback action in Def. \ref{PullbackAction} is obtained by restricting the $G$-action to that of its subgroup $H$.

\end{example}

#### Induced action

\begin{definition}\label{InducedAction}
**(induced action)**\linebreak
Given a topological group homomorphism $\phi$ (eq:TopologicalGroupHomomorphism), the _induced action_ functor

$$
  Topological G_1 Actions
  \overset{
     G_2 \times_{G_1} (-)
  }{\longrightarrow}
  Topological G_2 Actions
$$

sends $X \in Topological G_1 Spaces$ to the [[quotient space]] of its [[Cartesian product]] with $G_2$ by the [[diagonal action]] of $G_1$ (which  on $G_2$ is by inverse right multiplication through $\phi$):

$$
  G_2 \times_{G_1} X
  \;\coloneqq\;
  \big(
    G_2 \times X
  \big)
  \big/
  \big(
    (g_2, x) \sim (g_2 \cdot \phi(g_1)^{-1}, g_1 \cdot x)
    \,\vert\,
    g_1 \in G_1
  \big)
$$

and equipped with the $G_2$-action given by left multiplication on the $G_2$-factor.

\end{definition}

\begin{example}\label{QuotientsAsInducedActions}
**(Quotients as induced actions)** \linebreak

If $\phi \coloneqq G \longrightarrow 1$ is the [[terminal object|terminal]] morphims to the [[trivial group]], then its induced action (Def. \ref{InducedAction}) is the operation of forming $G$-[[quotient spaces]] (the $G$-[[orbit]]-space):

$$
  \array{
    Topological G Actions
    &\longrightarrow&
    Topological 1 Actions
    \mathrlap{  \;  = TopologicalSpaces }
    \\
    X &\mapsto &  1 \times_G X \mathrlap{ \; = X/G }
    \,.
  }
$$

\end{example}

\begin{prop}\label{InducedActionIsLeftAdjointToPullbackAction}
**(induced action is left adjoint to pullback action)** \linebreak
Given a topological group homomorphism $\phi$ (eq:TopologicalGroupHomomorphism), the _induced action_ functor (Def. \ref{InducedAction}) is [[left adjoint]] to the pullback action functor (Def. \ref{PullbackAction}):

$$
  G_1 Spaces
  \underoverset
    {\underset{\phi^\ast}{\longleftarrow}}
    {\overset{G_2 \times_{G_1} (-)}{\longrightarrow}}
    {\;\;\; \bot \;\;\; }
  G_2 Spaces
  \,.
$$

\end{prop}
\begin{proof}

For $X$ a $G_1$-space and $Y$ a $G_2$-space, consider a $G_2$-[[equivariant function]] out of the induced action

$$
  G_2 \times_{G_1} X
  \overset{\;\;\; f \;\;\;}{\longrightarrow}
  Y
  \,.
$$

Since the $G_2$-orbit of

$$
  X = G_1 \times_{G_1} X \hookrightarrow G_2 \times_{G_1} X
$$

is the entire domain space on the left, by $G_2$ equivariance, this function is completely determined by its restriction along this inclusion to a $G_1$-equivariant function $\tilde f \;\colon\; X \to Y$, hence to a homomorphism $\tilde f \;\colon\; X \to \phi^\ast Y$.  This correspondence

$$
  f \leftrightarrow \tilde f
$$

is clearly a [[natural bijection]] and hence establishes the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) characterizing the adjunction.

\end{proof}

#### Coinduced action
 {#CoinducedActions}

\begin{defn}\label{CoinducedAction}
**(coinduced action)** \linebreak

Given a topological group homomorphism $\phi$ (eq:TopologicalGroupHomomorphism), the _coinduced action_ functor

$$
  Topological G_1 Actions
  \overset{
     Maps(G_2,-)^{G_1}
  }{\longrightarrow}
  Topological G_2 Actions
$$

sends $X \in Topological G_1 Spaces$
to the $G_1$-[[fixed locus]] in the [[mapping space]] between [[topological spaces]] equipped with $G_1$-[[actions]] (on $G_2$ the $\phi$-induced left multiplication action)
and itself equipped with the $G_2$-[[group action|action]] given by

\[
  \label{ActionOnG1EquivariantMapsOutOfG2}
  \array{
    G_2 \times Maps(G_2,X)^{G_1}
    &
      \overset{}{\longrightarrow}
    &
    Maps(G_2,X)^{G_1}
    \\
    (g_2, h)
    &\mapsto&
    h\big(
      (-) \cdot g_2
    \big)
     \,.
  }
\]
\end{defn}

\begin{example}\label{FixedLociAsCoinducedActions}
**(Fixed loci as coinduced actions)**

Let $H \subset G$ be a topological subgroup and consider the group homomorphism (eq:TopologicalGroupHomomorphism) to be the $H$-[[quotient group]] [[coprojection]] from the [[normal subgroup]] $N(H) \subset G$:

$$
  \phi \coloneqq q \;\colon\; N(H) \longrightarrow N(H)/H
  \,.
$$

Then for any $N(H)$-space $X$ (which in practice will usually be a $G$-space after restriction of its action along $N(H) \hookrightarrow G$, Example \ref{RestrictedAction}) we have that its coinduced action (Def. \ref{CoinducedAction}) is its $H$-[[fixed locus]] $X^H$ equipped with its residual $N(H)$-action:

$$
  Maps
  \big(
    N(H)/H,
    X
  \big)^{N(H)}
  \;\;
  \simeq
  \;\;
   X^H
  \,.
$$

\end{example}

\begin{prop}\label{CoinducedActionIsRightAdjointToPullbackAction}
**(coinduced action is right adjoint to pullback action)** \linebreak

Given a topological group homomorphism $\phi$ (eq:TopologicalGroupHomomorphism),
the pullback action functor (Def. \ref{PullbackAction}) is the [[left adjoint]] and the coinduced action functor (Def. \ref{CoinducedAction}) is the [[right adjoint]] in a pair of [[adjoint functors]]

$$
  G_1 Spaces
  \underoverset
    {
      \underset{
        Maps
        \big(
          G_2, -
        \big)^{G_1}
      }{\longrightarrow}
    }
    {
      \overset{
         \phi^\ast
      }{
         \longleftarrow
      }
    }
    {\bot}
  G_2 Spaces
$$

\end{prop}

\begin{proof}

To see the defining [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism), consider a $G_1$-[[equivariant function|equivariant]] [[continuous function]]

$$
  \phi^\ast X
  \overset{
    \;\;\;
    f
    \;\;\;
  }{
    \longrightarrow
  }
  Y
  \,.
$$

From this we obtain the following function

$$
  \array{
    X
    &
      \overset{
        \tilde f
      }{
        \longrightarrow
      }
    &
    Maps
    \big(
      G_2,
      Y
    \big)^{G_1}
    \\
    x
    &\mapsto&
    \big(
      g_2
      \mapsto
      f( g_2 \cdot x )
    \big)
    \,,
  }
$$

where $e \in G_2$ denotes the [[neutral element]].

This is manifestly:

* well-defined, due to the $G_1$-equivariance of $f$;

* continuous, being built from composition of continuous map;

* $G_2$-equivariant with respect to the action (eq:ActionOnG1EquivariantMapsOutOfG2).

Conversely, given a $G_2$-equivariant continuous function $X \overset{\tilde f}{\longrightarrow} Maps\big(G_2, Y\big)^{G_1}$, we obtain the following function

$$
  \array{
    \phi^\ast X
    &\overset{}{\longrightarrow}&
    Y
    \\
    x
    &\mapsto&
     \tilde f(x)(e)
    \,.
  }
$$

This is:

* continuous, being the composition of continuous functions;

* $G_1$-equivariant due to the equivariance properties of $\tilde f$:

  $$
    \begin{aligned}
      \phi(g_1) \cdot x
      & \mapsto
      \tilde f
      \big(
        \phi(g_1)\cdot x
      \big)
      (e)
      \\
      & =
      \tilde f
      (
        x
      )
      \big(
         e \cdot \phi(g_1)
      \big)
      \\
      & =
      \tilde f
      (
        x
      )
      \big(
         \phi(g_1)
         \cdot
         e
      \big)
      \\
      & =
      g_1
        \cdot
      \big(
      \tilde f
      (
        x
      )
      (
         e
      )
      \big)
    \end{aligned}
  $$

Finally, it is clear that these transformations $f \leftrightarrow \tilde f$ are [[natural transformation|natural]], hence it only remains to see that they are [[bijective]]:

Plugging in the above constructions we find indeed:

$$
 \widetilde
  {\tilde f}
  \;\colon\;
  x \mapsto f(e \cdot x) = f(x)
$$

and

$$
  \widetilde {\widetilde {\tilde f}}
  \;\colon\;
  x
  \mapsto
  \big(
    g_2
      \mapsto
    \underset{
        {= \tilde f(x)(e \cdot g_2)}
        \atop
        {= \tilde f(x)(g_2)}
    }{
      \underbrace{
        \tilde f(g_2\cdot x)(e)
      }
    }
  \big)
  \,.
$$

\end{proof}

\begin{remark}\label{AnalogousCofreeActionForSimplicialGroupActions}
  For the analogous statement of cofree-actions of [[simplicial groups]] on [[simplicial sets]] see at *[[Borel model structure]]* [this Prop.](Borel+model+structure#CofreeAction).
\end{remark}

### Fixed loci with residual Weyl group action
 {#FixedLociWithResidualWeylGroupAction}

Combining these change-of-equivariance grouo [[adjunctions]] (from [above](#ChangeOfGroupsAndFixedLoci))
to "pull-push" through the [[correspondence]]

\begin{tikzcd}[row sep=0pt]
  &
  N\!(H)
  \ar[
    dl, hook
  ]
  \ar[
    dr,
    ->>
  ]
  \\
  G
  &&
  W(H)
  \mathrlap{
    \; := N\!(H)\!/H
  }
\end{tikzcd}

we obtain the [[fixed locus]]-[[functor]] in the form in which it appears in [[Elmendorf's theorem]], namely with the residual [[Weyl group]]-[[action]] on the fixed loci:

\begin{example}\label{FixedLociWithResidualWeylGroupActionAsCoinducedAction}
**(Fixed loci with residual Weyl-group action as coinduced action)**

Let $H \subset G$ be a [[subgroup]]-inclusion. Write

* $N(H) \hookrightarrow G$ for the corresponding [[normalizer subgroup]] inclusion

* $N(H) \twoheadrightarrow N(H)/H = W(H)$ for the [[coprojection]] to its [[quotient group]] by $H$ (the "[[Weyl group]]" of $H$).

Then the composite of

1. forming the pullback action (Def. \ref{PullbackAction}) along $N(H) \hookrightarrow G$ (the restricted action, Example \ref{RestrictedAction})

1. forming the coinduced action (Def. \ref{CoinducedAction}) along $N(H) \twoheadrightarrow N(H)/H$ (the passage to fixed loci, Example \ref{FixedLociAsCoinducedActions})

is the passage to the $H$-[[fixed locus]] $(-)^H$ equipped with its residual [[Weyl group]]-action, and Prop. \ref{InducedActionIsLeftAdjointToPullbackAction} with Prop. \ref{CoinducedActionIsRightAdjointToPullbackAction} shows that this construction is a [[right adjoint]]:

\begin{tikzcd}[column sep=large]
  G\mathrm{Spaces}
  \ar[
    rr,
    shift right=7pt,
    "(N(H) \hookrightarrow G)^\ast"{below}
  ]
  \ar[
    rr,
    phantom,
    "\scalebox{.7}{$\bot$}"
  ]
  \ar[
    rrrr,
    rounded corners,
    to path={
         -- ([yshift=-20pt]\tikztostart.south)
         --node[below]{\scalebox{.7}{$(-)^H$}} ([yshift=-20pt]\tikztotarget.south)
         -- (\tikztotarget.south)}
  ]
  &&
  N\!(H)\mathrm{Spaces}
  \ar[
    ll,
    shift right=7pt,
    "G \times_{N\!(H)} (-)"{above}
  ]
  \ar[
    rr,
    shift right=7pt,
    "{
      \mathrm{Maps}
      (
        N\!(H)\!/H,
        -
      )^{N\!(H)}
    }"{below}
  ]
  \ar[
    rr,
    phantom,
    "\scalebox{.7}{$\bot$}"
  ]
  &&
  \big(
    N\!(H)\!/H
  \big)\mathrm{Spaces}
  \ar[
    ll,
    shift right=7pt,
    "{
      ( N\!(H) \twoheadrightarrow N\!(H)\!/H )^\ast
    }"{above}
  ]
  \ar[
    llll,
    rounded corners,
    to path={
         -- ([yshift=+20pt]\tikztostart.north)
         --node[above]{\scalebox{.7}{$G/H \times_{N\!(H)\!/H}(-)$}} ([yshift=+20pt]\tikztotarget.north)
         -- (\tikztotarget.north)}
  ]
\end{tikzcd}

\end{example}

### Limits and colimits
 {#LimitsAndColimits}

Recalling that the ambient category is assumed to be a [[cartesian closed monoidal category|cartesian closed]] [[convenient category of topological spaces]], such as that of [[compactly generated topological spaces]]:

\begin{proposition}
  \label{ForgetfulFunctorCreatesLimitsAndColimits}
**(forgetful functor to spaces creates all limits and colimits)**
\linebreak
 The [[forgetful functor]] from $G$-spaces to underlying spaces [[created limit|creates]] all [[limits]] and [[colimits]].
\end{proposition}
(e.g. [Schwede 2018, p. 736-737](#Schwede18))
\begin{proof}
  Topological $G$-spaces are equivalently the [[algebra over a monad|algebras]] over the [[monad]] $G \times (-)$, which is the composite of the two [[left adjoints]] in the change-of-group [[adjoint triple]] [above](#ChangeOfGroupsAndFixedLoci) along $1 \to G$.  Since [[left adjoints preserve colimits]], so does this monad.

Now the [[forgetful functor]] in question is the respective [[monadic functor]]
and a general theorem ([this Prop.](created+limit#MonadicFunctorsCreateLimits))  says that monadic functors create all limits that exist in their codomain and those colimits which exist there and are preserved by the monad.
\end{proof}

\begin{proposition}
\label{CompactlyGeneratedGSpacesFormARegularCategory}
**(compactly generated topological $G$-spaces form a regular category)**
\linebreak
  For $TopSp$ denoting the category of [[compactly generated weakly Hausdorff spaces]] or [[compactly generated Hausdorff spaces]], and for $G \,\in\, Grp(TopSp)$ a [[topological group]], the category $G Act(TopSp)$ of topological $G$-spaces is a [[regular category]].
\end{proposition}
\begin{proof}
  By [this Prop.](compactly+generated+topological+space#kHausIsRegular) the given convenient category of topological spaces is regular. Since regularity is purely a condition on limits and colimits ([this Def.](regular+category#RegularCategory)) it transfers along any [[forgetful functor]] which [[created limits|creates]] limits and colimits. Therefore the statement follows by Prop. \ref{ForgetfulFunctorCreatesLimitsAndColimits}.
\end{proof}

\begin{lemma}\label{RecognitionOfCartesianQuotientProjection}
**(recognition of cartesian quotient projections)**
  Let $G$ be a [[compact topological group]] and let $f \colon X \longrightarrow Y$ be morphism of [[Hausdorff space|Hausdorff]] $G$-spaces.

Then its [[quotient topological space|quotient]] [[natural transformation|naturality]] square

\[
  \label{QuotientSpaceNaturalitySquare}
  \array{
    X
    &\overset{f}{\longrightarrow}&
    Y
    \\
    \big\downarrow
     &&
    \big\downarrow
    \\
    X/G
     &\overset{f/G}{\longrightarrow}&
    Y/G
  }
\]

is a [[pullback square]] if and only if $f$ preserves [[isotropy groups]], i.e. if and only if for each $x \in X$ we have

$$
  G_x \,\simeq\, G_{f(x)}
$$

as an [[isomorphism]] of [[stabilizer]] [[subgroups]] of $G$.
\end{lemma}

([Bykov-Flores 15, Prop. 4.1](#BykovFlores15))

\begin{remark}
The assumption in Lemma \ref{RecognitionOfCartesianQuotientProjection} is met in particular when the action on both sides is [[free action|free]], whence all [[isotropy groups]] are [[trivial group|trivial]].

This is the case in which $X$ and $Y$ are $G$-[[principal bundles]] without, however, necessarily needing to be [[locally trivial]] for Lemma \ref{RecognitionOfCartesianQuotientProjection} to apply ("[Cartan principal bundles](principal+bundle#CartanPrincipalBundles)").

On the other hand, even if $G$ is not [[compact topological group|compact]] but $X \to X/G$ and $Y \to Y/G$ are $G$-[[principal bundles]] which are [[locally trivial]], then it follows again that (eq:QuotientSpaceNaturalitySquare) is a [[pullback]] (since then the universal comparison morphism $X \to Y \times_{X/G} Y/G$ is a morphism of [[locally trivial]] [[principal bundles]] over the common base space $X/G$, which is an isomorphism since it is so on any [[open cover]] over which both $X$ and $Y \times_{X/G} Y/G$ trivialize).
\end{remark}

### Equivariant Tietze extension theorem

See at _[[equivariant Tietze extension theorem]]_

### Model structure and homotopy theory
 {#ModelStructureAndHomotopyTheory}

The standard [[homotopy theory]] on $G$-spaces used in [[equivariant homotopy theory]] considers [[weak equivalences]] which are [[weak homotopy equivalence]] on all (ordinary) [[fixed loci]] for all suitable [[subgroups]]. This is presented by the *[[fine model structure on topological G-spaces]]*, which, by [[Elmendorf's theorem]], is equivalent to [[(∞,1)-presheaves]] over the [[orbit category]] of $G$.

On the other hand there is also the standard homotopy theory of [[infinity-actions]], presented by the *[[Borel model structure]]*, in this context also called the "coarse" or "naive" equivariant model structure ([Guillou](#Guillou)).

## Examples
  {#Examples}

We discuss some classes of examples of [[G-spaces]].

### Euclidean $G$-spaces
 {#EuclideanGSpace}

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$.
Then the underlying [[Euclidean space]] $\mathbb{R}^V$ inherits the [[mathematical structure|structure]] of a [[G-space]]

\begin{xymatrix}
   \mathbb{R}^V\ar@(ul,ur)^G
\end{xymatrix}

We may call this the _[[Euclidean G-space]]_ associated with the linear representation $V$.

### Representation spheres

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$.
Then the [[one-point compactification]] of the underlying [[Euclidean space]] $\mathbb{R}^V$ inherits the [[mathematical structure|structure]] of a [[G-space]] with the [[one-point compactification|point at infinity]] a [[fixed point]]. This is called the _$V$-[[representation sphere]]_

\begin{xymatrix}
   S^V\ar@(ul,ur)^G
   \ar@{}[r]|-{:=}
   &
   \big( \mathbb{R}^V\ar@(ul,ur)^G\big)^{\mathrm{cpt}}
\end{xymatrix}

### Representation tori

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$.

If $G$ is the [[point group]] of a [[crystallographic group]] inside the [[Euclidean group]]

$$
  N \rtimes G
  \hookrightarrow
  Iso(\mathbb{R}^V)
$$

then the $G$-[[action]] on the Euclidean space $\mathbb{R}^V$ descends to the [[quotient]] by the [[action]] of the translational [[normal subgroup]] [[lattice in a vector space|lattice]] $N$ ([this Prop.](crystallographic+group#InducedPointGroupActionOnTorus)). The resulting $G$-space is an [[n-torus]] with $G$-action, which might be called the _[[representation torus]]_ of $V$

\begin{xymatrix}
   \mathbb{R}^V\ar@(ul,ur)^G
   \ar@{}[r]|-{:=}
   &
   \big( \mathbb{R}^V\ar@(ul,ur)^G\big)/N
\end{xymatrix}

<center>
<a href="https://arxiv.org/pdf/1909.12277.pdf#page=13">
<img src="https://ncatlab.org/schreiber/files/ExamplesOfLinearRepsAndInducedGSpaces.jpg" width="690">
</a>
</center>

> graphics grabbed from [SS 19](equivariant+Hopf+degree+theorem#SatiSchreiber19)

### Projective $G$-space

Let $G$ be a [[finite group]] (or maybe a [[compact Lie group]]) and let $V$ be a $G$-[[linear representation]] over some [[topological field|topological]] [[ground field]] $k$.

Then the corresponding _[[projective G-space]]_ is the [[quotient space]] of the [[complement]] of the origin in (the [[Euclidean space]] underlying) $V$ by the given [[action]] of the [[group of units]] of $k$ (from the $k$-[[vector space]]-[[mathematical structure|structure]] on $V$):

$$
  k P(V)
  \;:=\;
  \big(
    V \setminus \{0\}
  \big) / k^\times
$$

and equipped with the residual $G$-[[action]] on $V$ (which passes to the [[quotient space]] since it commutes with the $k$-action, by linearity).

### G-CW complexes

See at _[[G-CW complex]]_.

## Related concepts

* [[G-set]], [[G-manifold]], [[simplicial group action]]

* [[fixed point space]]

* [[slice theorem]]

* [[equivariant fundamental groupoid]]

* [[equivariant bundle]]

* [[cyclotomic space]]

[[!include equivariant homotopy theory -- table]]

See also

* [[rational topological G-space]]

* [[equivariant rational homotopy theory]]

  [[vector G-space]]

* [[equivariant stable homotopy theory]]

* [[equivariant rational stable homotopy theory]]

## References

(For basics see also the references at _[[group actions]]_.)

Early appearance of the notion (as "transformation groups"):

* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893  ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))

  English translation by M. W. Haskell:

  _A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full) [[KleinRetyped.pdf:file]])

See also:

* {#Koszul65} [[Jean-Louis Koszul]], _Lectures on Groups of Transformations_, Tata Institute 1965 ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr32.pdf), [[KoszulGroupsOfTransformations.pdf:file]])

Textbook accounts:

* {#Palais60} [[Richard Palais]], *The classification of $G$-spaces*, Memoirs of the AMS **36**, 1960  ([ISBN:978-0-8218-9979-3](https://bookstore.ams.org/memo-1-36) [pdf](http://vmm.math.uci.edu/PalaisPapers/ClassificationOfG-Spaces.pdf), [[Palais_ClassificationOfGSpaces.pdf:file]])

  > (the term "G-space" appears here)

* {#Bredon72} [[Glen Bredon]], _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* [[Tammo tom Dieck]], Chapter 8 in: _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979 ([doi:10.1007/BFb0085965](https://link.springer.com/book/10.1007/BFb0085965))

* {#tomDieck87} [[Tammo tom Dieck]], *[[Transformation Groups]]*, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#May96} [[Peter May]] et al., Section I.1 of: _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 ([ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91/?startBookmarkIdx=200)  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [[MayEtAlEquivariant96.pdf:file]])

* [[Alejandro Adem]], [[James Davis]], *Topics in Transformation Groups*, Chapter 1 in: *Handbook of Geometric Topology*, 2001, Pages 1-54 ([doi:10.1016/B978-044482432-5/50002-0](https://doi.org/10.1016/B978-044482432-5/50002-0), [pdf](https://personal.math.ubc.ca/~adem/topics.pdf))

* {#Schwede18} [[Stefan Schwede]], Appendix B of: *Global homotopy theory*, New Mathematical Monographs, 34, Cambridge University Press, 2018 ([doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161), [arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

More discussion in the context of [[equivariant homotopy theory]]:

* {#Guillou} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_ ([pdf](https://faculty.math.illinois.edu/~bertg/EquivModels.pdf), [[GuillouModelsForEquivariantHomotopyTheory.pdf:file]])

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

* {#BykovFlores15} Alexander Bykov, Raúl Juárez Flores _G-fibrations and twisted products_, Topology and its Applications Volume 196, Part B, December 2015, Pages 379-397 ([doi:10.1016/j.topol.2015.05.011](https://doi.org/10.1016/j.topol.2015.05.011))

Specifically in the context of [[proper equivariant homotopy theory]]:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_, Memoirs of the AMS ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

See also the references at _[[equivariant homotopy theory]]_.

The special case of [[smooth function|smooth]] [[Lie group]]-[[group actions|actions]] on [[smooth manifolds]]:

Lecture notes:

* {#Meinrenken03} [[Eckhard Meinrenken]], *Group actions on manifolds*, Lecture Notes 2003 ([pdf](http://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf), [[Meinrenken_GroupActionsOnManifolds.pdf:file]])

[[!redirects topological G-spaces]]
[[!redirects topological G-spaces]]

[[!redirects G-space]]
[[!redirects G-spaces]]

[[!redirects TopologicalGActions]]

[[!redirects TopologicalGSpaces]]


topological group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _topological group_ is a [[topological space]] with a continuous [[group]] structure: a [[group object]] [[internalization|internal to]] the [[category]] [[Top]] of topological spaces and continuous functions between them.

## Definition

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

A _topological group_ is

1. a [[group]], hence

   1. a [[set]] $G$,

   1. a [[neutral element]] $e \in G$,

   1. a [[associativity|associative]] [[unital|unital]] [[function]]

   1. $(-)\cdot (-) \;\colon\; G \times G \to G$,

   1. a function $(-)^{-1} \;\colon\; G \to G$ such that $g \cdot g^{-1} = e = g^{-1} \cdot g$ for all $g \in G$;

1. a topology $\tau_G \subset P(G)$ giving $G$ the structure of a [[topological space]]

such that the operations $(-)^{-1}$ and $(-)\cdot (-)$ are [[continuous functions]] (the latter with respect to the [[product topology]]).

=--

Sometimes topological groups are understood to be [[Hausdorff space|Hausdorff]] (e.g. [Bredon 72, p. 1](#Bredon72)).

## Properties

### General

+-- {: .num_lemma #OpenSubgroupOfTopologicalGroupIsClosed}
###### Lemma
**(open subgroups of topological groups are closed)**

Every [[open subset|open]] [[subgroup]] $H \subset G$ of a topological group is [[closed subset|closed]], hence a [[closed subgroup]].

=--

(e.g [Arhangel'skii-Tkachenko 08, theorem 1.3.5](#ArhangelskiiTkachenko08))

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

The [[set]] of $H$-[[cosets]] is a [[cover]] of $G$ by [[disjoint subsets|disjoint]] [[open subsets]]. One of these cosets is $H$ itself and hence it is the [[complement]] of the [[union]] of the other cosets, hence the complement of an open subspace, hence closed.

=--

+-- {: .num_prop #ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact}
###### Proposition
**([[connected space|connected]] [[locally compact topological space|locally compact]] [[topological groups]] are [[sigma-compact topological space|sigma-compact]])**

Every [[connected topological space|connected]] [[locally compact topological space|locally compact]] topological group is [[sigma-compact topological space|sigma-compact]].

Every [[locally compact topological space|locally compact]] topological group is [[paracompact topological space|paracompact]].

=--

(e.g. [Arhangel'skii-Tkachenko 08, cor. 3.1.4, cor. 3.1.5](#ArhangelskiiTkachenko08))

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

By assumption of local compactness, there exists a [[compact topological space|compact]] [[neighbourhood]] $C_e \subset G$ of the [[neutral element]]. We may assume without restriction of generality that with $g \in C_e$ any element, then also the [[inverse element]] $g^{-1} \in C_e$.

For if this is not the case, then we may enlarge $C_e$ by including its inverse elements, and the result is still a compact neighbourhood of the neutral element: Since taking [[inverse elements]] $(-)^{-1} \colon G \to G$ is a [[continuous function]], and since [[continuous images of compact spaces are compact]], it follows that also the set of inverse elements to elements in $C_e$ is compact, and the union of two compact subspaces is still compact (obviously, otherwise see [this prop](compact+space#UnionsAndIntersectionOfCompactSubspaces)).

Now for $n \in \mathbb{N}$, write $C_e^n  \subset G$ for the [[image]] of $\underset{k \in \{1, \cdots n\}}{\prod} C_e \subset \underset{k \in \{1, \cdots, n\}}{\prod} G$ under the iterated group product operation $\underset{k \in \{1, \cdots, n\}}{\prod} G \longrightarrow G$.

Then

$$
  H \coloneqq
   \underset
   {n \in \mathbb{N}}
   {\cup}
  C_e^n
    \;\subset\;
  G
$$

is clearly a topological subgroup of $G$.

Observe that each $C_e^n$ is compact. This is because $\underset{k \in \{1, \cdots, n\}}{\prod}C_e$ is compact by the [[Tychonoff theorem]], and since [[continuous images of compact spaces are compact]]. Thus

$$
  H = \underset{n \in \mathbb{N}}{\cup} C_e^n
$$

is a countable union of compact subspaces, making it [[sigma-compact]]. Since [[locally compact and sigma-compact spaces are paracompact]], this implies that $H$ is paracompact.

Observe also that the subgroup $H$ is open, because it contains with the [[interior]] of $C_e$ a non-empty open subset $Int(C_e) \subset H$ and we may hence write $H$ as a union of open subsets

$$
  H = \underset{h \in H}{\cup} Int(C_e) \cdot h
  \,.
$$

Finally, as indicated in the proof of Lemma \ref{OpenSubgroupOfTopologicalGroupIsClosed}, the cosets of the open subgroup $H$ are all open and partition $G$ as a [[disjoint union space]] ([[coproduct]] in [[Top]]) of these open cosets. From this we may draw the following conclusions.

* In the particular case where $G$ is connected, then there is just one such coset, namely $H$ itself. The argument above thus shows that a connected locally compact topological group is $\sigma$-compact and (by local compactness) also paracompact.

* In the general case, all the cosets are homeomorphic to $H$ which we have just shown to be a paracompact group. Thus $G$ is a [[disjoint union space]] of paracompact spaces. This is again paracompact (by [this prop.](paracompact+topological+space#ParacompactnessPreservedByDisjointUnion)).

=--

### Uniform structure

A topological group $G$ carries two canonical [[uniform space|uniformities]]: a right and left uniformity. The **right uniformity** consists of entourages $\sim_{l, U}$ where $x \sim_{l, U} y$ if $x y^{-1} \in U$; here $U$ ranges over neighborhoods of the identity that are symmetric: $g \in U \Leftrightarrow g^{-1} \in U$. The **left uniformity** similarly consists of entourages $\sim_{r, U}$ where $x \sim_{r, U} y$ if $x^{-1} y \in U$. The uniform topology for either coincides with the topology of $G$.

Obviously when $G$ is commutative, the left and right uniformities coincide. They also coincide if $G$ is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.

Let $G$, $H$ be topological groups, and equip each with their left uniformities. Let $f: G \to H$ be a group homomorphism.

+-- {: .num_prop}
######Proposition
The following are equivalent:

* The map $f$ is continuous at a point of $G$;

* The map $f$ is continuous;

* The map $f$ is uniformly continuous.

=--

+-- {: .proof}
######Proof
Suppose $f$ is continuous at $g \in G$. Since neighborhoods of a point $x$ are $x$-translates of neighborhoods of the identity $e$, continuity at $g$ means that for all neighborhoods $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that

$$f(g U) \subseteq f(g) V$$

Since $f$ is a homomorphism, it follows immediately from cancellation that $f(U) \subseteq V$. Therefore, for every neighborhood $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that

$$x y^{-1} \in U \Rightarrow f(x) f(y)^{-1} = f(x y^{-1}) \in V$$

in other words such that $x \sim_U y \Rightarrow f(x) \sim_V f(y)$. Hence $f$ is uniformly continuous with respect to the right uniformity. By similar reasoning, $f$ is uniformly continuous with respect to the right uniformity.
=--

### Unitary representation on Hilbert spaces

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

A unitary [[representation]] $R$ of a topological group $G$ in a [[Hilbert space]] $\mathcal{H}$ is a continuous [[group homomorphism]]

$$
   R \colon G \to \mathcal{U}(\mathcal{H})
$$

where [[U(H)|$\mathcal{U}(\mathcal{H})$]] is the group of [[unitary operators]] on $\mathcal{H}$ with respect to the [[strong topology]].

=--

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

Here [[U(H)|$\mathcal{U}(\mathcal{H})$]] is a complete, metrizable topological group in the [[strong topology]], see ([Schottenloher, prop. 3.11](#Schottenloher)).

=--

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

In [[physics]], when a [[classical mechanical system]] is symmetric, i.e. invariant in a proper sense, with respect to the action of a topological group $G$, then an unitary representation of $G$ is sometimes called a **quantization** of $G$. See at _[[geometric quantization]]_ and _[[orbit method]]_ for more on this.

=--

#### Why the strong topology is used

The reason that in the definition of a [[unitary representation]], the [[strong operator topology]] on $\mathcal{U}(\mathcal{H})$ is used and not the [[norm topology]], is that only few [[homomorphisms]] turn out to be [[continuous map|continuous]] in the norm topology.

Example: let $G$ be a [[compact Lie group]] and $L^2(G)$ be the [[Hilbert space]] of square integrable [[measurable function]]s with respect to its [[Haar measure]]. The right [[regular representation]] of $G$ on $L^2(G)$ is defined as

$$
      R: G \to \mathcal{U}(L^2(G))
$$

$$
         g \mapsto (R_g: f(x) \mapsto f(x g))
$$

and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.

### Relation to Lie groups

* *[[Hilbert's fifth problem]]*: "Which topological groups admit Lie group structure?"

* [[continuous homomorphisms of Lie groups are smooth]]

### Protomodularity
 {#Protomodularity}

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

The [[category]] [[TopGrp]] of topological groups and [[continuous function|continuous]] [[group homomorphisms]] between them is a [[protomodular category]].

=--

A proof is spelled out by [[Todd Trimble]] [here on MO](http://mathoverflow.net/questions/133110/why-is-topgrp-the-category-of-topological-groups-and-continous-homomorphisms-prot/133601#133601).

## Examples

* The [[classical Lie groups]] are in particular topological groups, such as the [[general linear group]] and its subgroups.

* ...

## Related concepts

* [[group]]

* [[discrete group]], [[finite group]]

* [[topological monoid]]

* **topological group**,

  * [[locally compact topological group]]

  * [[compact topological group]]

  * [[amenable topological group]]

  * [[maximal compact subgroup]]

  * [[closed subgroup]]

  * [[loop group]]

* [[Lie group]]

## References

* {#Bourbaki71} [[Nicolas Bourbaki]], *Topological Groups*, Chapter 3 in: *General topology*, Elements of Mathematics III, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

* {#Bredon72} [[Glen Bredon]], Chapter 0 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#ArhangelskiiTkachenko08} Alexander Arhangel'skii, Mikhail Tkachenko, _Topological Groups and Related Structures_, Atlantis Press 2008 ([doi:10.2991/978-94-91216-35-0](https://doi.org/10.2991/978-94-91216-35-0))

The following monograph is not particulary about group representations, but some content of this page is based on it:

* {#Schottenloher} [[Martin Schottenloher]], _A mathematical introduction to conformal field theory._ Springer, 2nd edition 2008 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1161.17014&format=complete))

[[!redirects topological groups]]

[[!redirects TopologicalGroups]]


topological groupoid

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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### General

A **topological groupoid** is an [[internal groupoid]] in the [[category]] [[Top]].

So this is a [[groupoid]] with a [[topological space]] of [[object]]s and one of [[morphism]]s, and all structure maps (source, target, identity, composition, inverse) are [[continuous maps]]. Composition here refers to the map defined on the space of all composable morphisms.

A topological groupoid $C$ is  called an **open topological groupoid** if the source map $s : Mor C \to Obj C$ is an [[open map]].

It is called an **[[étale groupoid]]** if in addition $s$ is a [[local homeomorphism]].

## Properties

### Relation to toposes

Every [[topos]] ([[Grothendieck topos]]) with [[point of a topos|enough points]] is the _[[classifying topos of a topological groupoid]]_. See there for more.

## Related entries

* [[Lie groupoid]]

* [[topological group]]

* [[topological stack]]

* [[amenable topological groupoid]]

* [[localic groupoid]]

## References

The notion of topological categories, hence of topological groupoids, goes back to

* [[Charles Ehresmann]], _Catégories topologiques et categories différentiables_, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 ([[EhresmannCategoriesTopologiques.pdf:file]], [zbMath:0205.28202](https://zbmath.org/?q=an:0205.28202))

Their understanding as [[internal groupoids]] [[internalization|internal]] to [[TopologicalSpaces]] is often attributed to

* [[Charles Ehresmann]], _Catégories structurées_, Annales scientifiques de l'École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 ([numdam:ASENS_1963_3_80_4_349_0](http://www.numdam.org/item/ASENS_1963_3_80_4_349_0))

but the simple notion of [[internalization]] and [[internal groupoids]] ([Grothendieck 1960](internalization#Grothendieck60), [61](internalization#Grothendieck61)) is hardly recognizable in this account.

Exposition:

* [[Alan Weinstein]], p. 6 of: _Groupoids: Unifying Internal and External Symmetry -- A Tour through some Examples_, Notices of the AMS volume 43, Number 7 (1996) ([pdf](http://www.ams.org/notices/199607/weinstein.pdf), [[Weinstein_Groupoids.pdf:file]])

Textbook account:

* [[Kirill Mackenzie]], Chapter II in: _Lie groupoids and Lie algebroids in differential geometry_, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp ([doi:10.1017/CBO9780511661839](https://doi.org/10.1017/CBO9780511661839), [MR:896907](http://www.ams.org/mathscinet-getitem?mr=896907))

Many references on topological groupoids deal with them as models for [[topological stacks]], see there for more.

On topological groupoids as a model for [[orbispaces]]:

* {#Haefliger84} [[André Haefliger]], _Groupoides d'holonomie et classifiants_, Astérisque no. 116  (1984), p. 70-97 ([numdam:AST_1984__116__70_0](http://www.numdam.org/item/?id=AST_1984__116__70_0))

* {#Haefliger91} [[André Haefliger]], _Complexes of Groups and Orbihedra_, in: E. Ghys, A. Haefliger, A Verjovsky (eds.), _Proceedings of the Group Theory from a Geometrical Viewpoint_, ICTP, Trieste, Italy , 26 March – 6 April 1990, World Scientific 1991 ([doi:10.1142/1235](https://doi.org/10.1142/1235))

On topological groupoids as presenting [[toposes]] with [[point of a topos|enough points]]:

* [[Carsten Butz]], [[Ieke Moerdijk]], *Representing topoi by topological groupoids*, Journal of Pure and Applied Algebra Volume 130, Issue 3, 17 September 1998, Pages 223-235 (doi:10.1016/S0022-4049(97)00107-2)

On [[exponential objects]] among [[topological groupoids]]:

* [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

[[!redirects topological groupoids]]

[[!redirects open topological groupoid]]
[[!redirects open topological groupoids]]


topological homotopy theory

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

#Contents#
* table of contents
{:toc}

## Idea

_Topological homotopy theory_ is [[homotopy theory]] for or modeled on [[topological spaces]] (in contrast to [[simplicial homotopy theory]] or [[localic homotopy theory]]).

The key tools are the [[Strøm model category]] and the [[classical model structure on topological spaces]] on the category [[Top]] of all topological spaces, or on some [[convenient category of topological spaces]].

## Related concepts

* [[left homotopy]], [[right homotopy]]

* [[homotopy equivalence]], [[deformation retract]]

* [[fundamental group]], [[covering space]]

* [[fundamental theorem of covering spaces]]

* [[homotopy group]]

* [[weak homotopy equivalence]]

* [[Whitehead's theorem]]

* [[Freudenthal suspension theorem]]

* [[nerve theorem]]

* [[homotopy extension property]], [[Hurewicz cofibration]]

* [[topological cofiber sequence|cofiber sequence]]

* [[Strøm model category]]

* [[classical model structure on topological spaces]]


topological homotopy type is cohesive shape of continuous diffeology -- proposition

\begin{proposition}\label{TopologicalHomotopyTypeIsCohesiveShapeOfCdfflg}
**([[topological space|topological]] [[homotopy type]] is [[cohesive shape]] of [[continuous diffeology]])**
\linebreak
For every $X \in$ [[TopologicalSpaces]], the [[cohesive shape]]/[[path ∞-groupoid]] presented by its _diffeological singular simplicial set_ (Def. \ref{DiffeologicalSingularSimplicialSet}, Remark \ref{DiffeologicalSingularSetIsPathGroupoid}) of its [[continuous diffeology]] is [[natural transformation|naturally]]$\,$[[weak homotopy equivalence|weak homotopy equivalent]] to the [[homotopy type]] of $X$ presented by the ordinary [[singular simplicial set]]:

$$
  Sing_{diff}
  \big(
    Cdfflg(X)
  \big)
  \underoverset
    { \in \mathrm{W}_{wh} }
    {}
    {\longrightarrow}
  Sing(X)
  \,.
$$

\end{proposition}

([Christensen & Wu 2013, Prop. 4.14](diffeological+space#ChristensenWu13))


topological index

+-- {: .rightHandSide}
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### Context
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For a given [[manifold]] $X$ of finite [[dimension]] there exists an [[embedding]] $i : X \to \mathbb{R}^n$ into some [[Cartesian space]]. Using the [[Pontrjagin-Thom collapse map]] this induces a morphism in [[topological K-theory]]

$$
  i_! : K(T X) \to K(T \mathbb{R}^n)
  \,.
$$

Similarly for any point inclusion $j : * \to \mathbb{R}^n$ there is such a morphism $j_! : \mathbb{Z} = K(*) \to K(T \mathbb{R}^n)$ which is an [[isomorphism]] -- the [[Thom isomorphism]].

The **topological index** of [[topological K-theory]] on $X$ is the composite

$$
  ind_{top} : \mathbb{Z} = K(T X) \stackrel{i_!}{\to}
   K(T \mathbb{R}^n) \stackrel{j_!^{-1}}{\to} K(*) = \mathbb{Z}
  \,.
$$

One can prove that this is independent of all the occurring choices. In particular it does not depend on the specific choice of embedding of the manifold $X$ into to the [[Euclidean space]]. The _topological index function_ is uniquely fixed by two properties (this is the content of the [[Atiyah-Singer index theorem]]):

1. For $X$ a point we have $ind_t=id$.

1. Index functions commute with the maps $i_!$.

From this one defines the _topological index of an [[elliptic operator]]_ . The [[principal symbol]] of the operator defines a homogenous length-one [[chain complex]] of [[bundles]] on $T X$ [[exact sequence|exact]] outside the null section. Elements of this kind are precisely cycles for the compactly supported [[K-theory]] of $T X$ hence an elliptic operator $D$ has a topological index only depending on its principal symbol.

On the other hand, analysis associates to $D$ its [[analytical index]] that is

$$
\operatorname{dim}\operatorname{Ker}(D)-\operatorname{dim}\operatorname{coKer}(D)
$$

on one (hence all) [[Sobolev space]] that $D$ is defined on.

The [[Atiyah-Singer index theorem]] states that the [[analytical index]] of $D$ is equal to its topological index.

### More on the Thom map

The story starts with an embedding $i:X\to Y$ of [[compact space|compact]] [[manifolds]]. In this situation one can construct a [[homomorphism]]

$$
  i_{!} : K(T X)\longrightarrow K(T Y)
$$

between the [[compact support|compactly supported]] [[K-theory|K-theories]] of their [[tangent bundles]].

Notice here the reverse [[functor|functoriality]]: for the base space $K$ is contravariant while for the total spaces of the tangent bundles it is covariant. This uses the [[Pontrjagin-Thom collaps map|Thom mapping]]: if $X$ is a compact manifold and $V$ a real [[vector bundle]] over $X$ there is a natural map

$$
  \varphi:K(X)\longrightarrow K(V)
  \,.
$$

One of the most important results of [[K-theory]], namely [[Bott periodicity]], can be seen as the statement of the fact that this map is an [[isomorphism]]. Now apply this construction to the [[normal bundle]] $N$ of $X$ in $Y$ to get

$$
  \varphi
  :
  K(T X)\longrightarrow K(T N)
$$

and (looking at $N$ as a [[tubular neighbourhood]] of $X$ in $Y$) compose it with the natural map

$$
  K_*
  :
  K(T N)\longrightarrow K(T Y)
$$

to get $i_!$.

Now given a manifold $X$, embed it in a [[Euclidean space]] $\mathbb{R}^n$ for some suitable $n$ and consider the inclusion $\{0\}\to \mathbb{R}^n$. This induces the ([[Thom isomorphism]]) mapping $j_!:\mathbb{Z}=K(\{0\}) \longrightarrow K(T\mathbb{R}^n)$.

The _topological index_ is defined to be

$$
  ind_t
  :=
  j_!^{-1}\circ i_!
  \,.
$$

## Related concepts

* [[index theory]]

  * **topological index**

  * [[analytical index]]

  * [[Atiyah-Singer index theorem]]

* [[eta invariant]], [[zeta function of an elliptic differential operator]], [[functional determinant]]

* [[fiber integration in differential K-theory]]

[[!redirects topological indices]]

[[!redirects index of an operator]]


topological infinity-groupoid

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

#Contents#
* table of contents
{:toc}

## Idea

A _topological ∞-groupoid_ is meant to be an [[∞-groupoid]] that is equipped with [[topology|topological]] structure. For instance a [[0-truncated object|0-truncated]] topological $\infty$-groupoid should just be a [[topological space]], and 1-truncated topological $\infty$-groupoids should reproduce [[topological groupoids]]/[[topological stacks]], etc.

A generic way to make sense of this in a _[[gros topos]]_ perspective is to pick some small [[subcategory]] $Top_{sm}$ of the category [[Top]] of [[topological spaces]], regard it as a [[site]] with respect to an evident [[coverage]] by [[open covers]], and then take the [[(∞,1)-topos]] of generalized topological $\infty$-groupoids to be the [[(∞,1)-category of (∞,1)-sheaves]] on this site:

$$
  Top \infty Grpd \coloneqq Sh_\infty(Top_{sm})
  \,.
$$

This gives a nice category with very general objects. In there one may find smaller, less nice categories of nicer objects.

There are different choices of [[sites]] $Top_{sm}$ to make. For instance

1. taking $Top_{sm}$ to be a small site of [[locally contractible topological spaces]] yields a concept of _[[locally contractible topological infinity-groupoids]]_;

1. taking $Top_{sm}$ be the the subcategory of topological [[Cartesian spaces]] yield a concept of [[Euclidean-topological infinity-groupoids]].

These two happen to constitute [[cohesive ∞-toposes]], due to the local contractibility of the objects in the site.

## Related concepts

* [[discrete ∞-groupoid]]

* [[D-topological ∞-groupoid]]

* [[smooth ∞-groupoid]]

* [[formal smooth ∞-groupoid]]

* [[super formal smooth ∞-groupoid]]

[[!redirects topological infinity-groupoid]]
[[!redirects topological infinity-groupoids]]
[[!redirects topological ∞-groupoid]]
[[!redirects topological ∞-groupoids]]

[[!redirects ?TopGrpd]]

[[!redirects continuous infinity-groupoid]]
[[!redirects continuous infinity-groupoids]]
[[!redirects continuous ∞-groupoid]]
[[!redirects continuous ∞-groupoids]]

[[!redirects topological infinity-group]]
[[!redirects topological infinity-groups]]
[[!redirects topological ∞-group]]
[[!redirects topological ∞-groups]]

[[!redirects topological infinity-stack]]
[[!redirects topological infinity-stacks]]

[[!redirects topological ∞-stack]]
[[!redirects topological ∞-stacks]]


topological insulator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For the moment see at *[[topological phase of matter]]*.

## Examples

* [[quantum Hall effect]]

* [[Chern insulator]]

* [[Haldane model]]

## Related concepts

* [[topological crystalline insulator]]

* [[photonic topological insulator]]

* [[gapped Hamiltonian]]

* [[topological phase of matter]]

* [[Chern insulator]]

* [[topological photonics]]

* [[quantum material]]

* [[Majorana zero mode]]

## References

### General

The term "topological insulator" originates with:

* [[Joel E. Moore]], [[Leon Balents]], *Topological invariants of time-reversal-invariant band structures*, Phys. Rev. B **75** (2007) 121306(R) $[$[doi:10.1103/PhysRevB.75.121306](https://doi.org/10.1103/PhysRevB.75.121306)$]$

Reviews:

* [[Liang Fu]], [[Charles Kane]], [[Eugene Mele]], *Topological Insulators in Three Dimensions*, Phys. Rev. Lett. **98** (2007) 106803 $[$[doi:10.1103/PhysRevLett.98.106803](https://doi.org/10.1103/PhysRevLett.98.106803)$]$

* Rahul Roy, *Topological phases and the quantum spin Hall effect in three dimensions*, Phys. Rev. B **79** (2009) 195322 $[$[doi:10.1103/PhysRevB.79.195322](https://doi.org/10.1103/PhysRevB.79.195322)$]$

* [[M. Zahid Hasan]], [[Charles Kane]], *Colloquium: Topological insulators*, Rev. Mod. Phys. **82** (2010) 3045 $[$[doi:10.1103/RevModPhys.82.3045](https://doi.org/10.1103/RevModPhys.82.3045)$]$

* [[Joel E. Moore]], *The birth of topological insulators*, Nature volume 464, pages 194–198 (2010) ([doi:10.1038/nature08916](https://doi.org/10.1038/nature08916))

* Xiao-Liang Qi, Shou-Cheng Zhang, *Topological insulators and superconductors*, Rev. Mod. Phys. **83** (2011) 1057-1110 $[$[arXiv:1008.2026](https://arxiv.org/abs/1008.2026), [doi:10.1103/RevModPhys.83.1057](https://doi.org/10.1103/RevModPhys.83.1057)$]$

* [[M. Zahid Hasan]], [[Joel E. Moore]], *Three-Dimensional Topological Insulators*, Ann. Review.Condensed Matter Physics **2** (2011) 55-78 $[$[arXiv:1011.5462](https://arxiv.org/abs/1011.5462), [doi:10.1146/annurev-conmatphys-062910-140432](https://doi.org/10.1146/annurev-conmatphys-062910-140432)$]$

*  [[Marcel Franz]], [[Laurens Molenkamp]] (eds.), *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) $[$[ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$

* Michel Fruchart, David Carpentier, *An introduction to topological insulators*, Comptes Rendus Physique **14** 9–10, (2013) 779-815 ([doi:10.1016/j.crhy.2013.09.013](https://doi.org/10.1016/j.crhy.2013.09.013))

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 5 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

With focus on the case [[symmetry protected topological phase|protected by]] [[crystallographic group]]-[[symmetry]]:

* [[Yoichi Ando]], [[Liang Fu]], *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Annual Review of Condensed Matter Physics **6** (2015) 361-381 $[$[arXiv:1501.00531](https://arxiv.org/abs/1501.00531), [doi:10.1146/annurev-conmatphys-031214-014501](https://doi.org/10.1146/annurev-conmatphys-031214-014501)$]$

* [[Tanmoy Das]], *A pedagogic review on designing model topological insulators*, Journal of the Indian Institute of Science **96** 77-106 (2016) $[$[arXiv:1604.07546](https://arxiv.org/abs/1604.07546), [ISSN:0970-4140](http://journal.library.iisc.ernet.in/index.php/iisc/article/view/4606)$]$

Textbook account:

* Shun-Qing Shen, *Topological Insulators*, Springer 2012 ([doi:10.1007/978-3-642-32858-9](https://link.springer.com/book/10.1007/978-3-642-32858-9))

* Panagiotis Kotetes, *Topological Insulators*, IOP Science 2019 ([ISBN:978-1-68174-517-6](https://iopscience.iop.org/book/978-1-68174-517-6))

See also:

* Wikipedia, _[Topological insulator](http://en.wikipedia.org/wiki/Topological_insulator)_

Review in the more general context of [[topological phases of matter]]

* Shou-cheng Zhang, _Viewpoint: Topological states of quantum matter_,  APS Physics 1, 6 (2008) [doi:10.1103/Physics.1.6](http://dx.doi.org/10.1103/Physics.1.6)

* Vishal Bhardwaj, Ratnamala Chatterjee, *Topological Materials -- New Quantum Phases of Matter*, Resonance **25** (2020) 431–441  ([doi:10.1007/s12045-020-0955-5](https://doi.org/10.1007/s12045-020-0955-5), [pdf](https://www.ias.ac.in/article/fulltext/reso/025/03/0431-0441))

* [[Tudor D. Stanescu]], Section II.5 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

See also:

* [[Liang Fu]], [[Charles L. Kane]], _Topological insulators with inversion symmetry_, Physical Review B 76 (4): 045302. [arXiv:cond-mat/0611341](http://arxiv.org/abs/cond-mat/0611341) [doi](http://dx.doi.org/10.1103%2FPhysRevB.76.045302);

* _Superconducting proximity effect and Majorana fermions at the surface of a topological insulator_, Phys. Rev. Lett. __100__: 096407, [arXiv:0707.1692](http://arxiv.org/abs/0707.1692) [doi](http://dx.doi.org/10.1103%2FPhysRevLett.100.096407)

* Jeffrey C. Y. Teo, [[Liang Fu]], [[Charles L. Kane]], _Surface states and topological invariants in three-dimensional topological insulators: Application to_ $Bi_{1-x}Sb_x$, Phys. Rev. __B 78__, 045426 (2008) [doi](http://link.aps.org/doi/10.1103/PhysRevB.78.045426)

* J. Kellendonk, _On the $C^\ast$-algebraic approach to topological phases for insulators_, [arxiv/1509.06271](http://arxiv.org/abs/1509.06271)

* A. Kitaev, Periodic table for topological insulators and superconductors. (Advances in Theoretical Physics: Landau Memorial Conference) AIP Conference Proceedings 1134, 22-30 (2009).

Interacting topological insulators:

* Stephan Rachel, *Interacting topological insulators: a review*, Rep. Prog. Phys. **81** 116501 (2018) $[$[arXiv:1804.10656](https://arxiv.org/abs/1804.10656), [doi:10.1088/1361-6633/aad6a6](https://doi.org/10.1088/1361-6633/aad6a6)$]$

The topological insulator in 2D exhibiting a [[quantum spin Hall effect]] has been first proposed in

* B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science __314__, n. 5806, pp. 1757-1761, Dec 2006 [doi](http://dx.doi.org/10.1126/science.1133734)

* Y. L. Chen et al. _Experimental Realization of a Three-Dimensional Topological Insulator, $Bi_2 Te_3$_, Science __325__, no. 5937 pp. 178-181, July 2009, [doi](http://dx.doi.org/10.1126/science.1173034)

> Comment by X.-G. Wen: In fact, none of the above materials have quantum spin Hall effect since the spin is not conserved due to the spin-orbital interaction that makes those materials non trivial.

* Ricardo Kennedy, Charles Guggenheim, _Homotopy theory of strong and weak topological insulators_, [arxiv/1409.2529](http://arxiv.org/abs/1409.2529)

* L. Wu et al. _Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator_, Science (2016). [doi](http://dx.doi.org/10.1126/science.aaf5541)

Discussion via [[AdS/CFT in solid state physics]]:

* Georgios Linardopoulos, *Modelling Topological Materials with D-branes*, INPP Annual Meeting 2017 ([pdf](https://indico.cern.ch/event/680759/contributions/2789343/attachments/1559809/2454998/Linardopoulos.pdf), [[LinardopoulosTopologicalMaterialsWithDBranes.pdf:file]])

Higher order topological insulators (with protected corner-modes beyond the edge-modes):

* {#SchindlerEtAl18} [[Frank Schindler]], Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, [[B. Andrei Bernevig]], [[Titus Neupert]], *Higher-Order Topological Insulators*, Science Advances 01  **4** 6 (2018) eaat0346 [[arXiv:1708.03636](https://arxiv.org/abs/1708.03636), [doi:10.1126/sciadv.aat0346](https://doi.org/10.1126/sciadv.aat0346)]

### Experimental realization

* Bo Song, Long Zhang, Chengdong He, Ting Fung Jeffrey Poon, Elnur Hajiyev, Shanchao Zhang, Xiong-Jun Liu, Gyu-Boong Jo, *Observation of symmetry-protected topological band with ultracold fermions*, Science Advances **4** eaao4748 (2018) $[$[doi:10.1126/sciadv.aao4748](https://doi.org/10.1126/sciadv.aao4748)$]$

{#ReferencesOnExternalManipulationViaStrain} External manipulation of topological phases via [[strain]] (see also the references [here](graphene#ReferencesMovementOfDiracPoints) at *[[graphene]]*):

* [[Marwa Mannaï]], [[Sonia Haddad]], *Strain tuned topology in the Haldane and the modified Haldane models*., J of Physics: Condens. Matter **32** 225501 (2020) $[$[arXiv:1907.11213](https://arxiv.org/abs/1907.11213), [doi:10.1088/1361-648X/ab73a1](https://doi.org/10.1088/1361-648X/ab73a1)$]$

* [[Marwa Mannaï]], [[Sonia Haddad]], *Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides*, Phys. Rev. B **103** 201112 (2021) $[$[arXiv:2011.08818](https://arxiv.org/abs/2011.08818), [doi:10.1103/PhysRevB.103.L121112](https://doi.org/10.1103/PhysRevB.103.L121112)$]$

* Jiesen Li, Wanxing Lin, D. X. Yao, *Strain-induced topological phase transition in two-dimensional platinum ditelluride* $[$[arXiv:2106.16212](https://arxiv.org/abs/2106.16212)$]$

* T. Kondo et al., *Visualization of the strain-induced topological phase transition in a quasi-one-dimensional superconductor $TaSe_3$*, Nature Materials **20** 1093–1099 (2021) $[$[doi:10.1038/s41563-021-01004-4](https://doi.org/10.1038/s41563-021-01004-4)$]$

* Phil D. C. King, *Controlling topology with strain*, Nat. Mater. **20** (2021) 1046–1047   $[$[doi:10.1038/s41563-021-01043-x](https://doi.org/10.1038/s41563-021-01043-x)$]$

### Interacting TIs

Discussion of topological insulators with non-negligible [[interactions]]:

* AtMa P. O. Chan, Thomas Kvorning, Shinsei Ryu, and Eduardo Fradkin, *Effective hydrodynamic field theory and condensation picture of topological insulators*, Phys. Rev. B **93** 155122 (2016) $[$[doi:10.1103/PhysRevB.93.155122](https://doi.org/10.1103/PhysRevB.93.155122)$]$

* Benjamin Moy, Hart Goldman, Ramanjit Sohal, Eduardo Fradkin, *Theory of oblique topological insulators* $[$[arXiv:2206.07725](https://arxiv.org/abs/2206.07725)$]$

category: physics

[[!redirects topological insulators]]


topological interval

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

#Contents#
* table of contents
{:toc}

## Idea

Generally, a _topological interval_ is a ([[bounded set|bounded]]) [[interval]] in the [[real line]] (an [[open interval]] $(a,b)$ or a [[closed interval]] $[a,b]$ or a [[half-open interval]] $(a,b]$ or $[a,b)$) equipped with the [[subspace topology]] of the [[Euclidean space|Euclidean]] [[metric topology]].

Specifically in the context of [[topological homotopy theory]], the _standard topological [[interval object]]_ is the [[closed interval]] $[0,1]$ equipped with the [[continuous functions]]

1. $const_0 \;\colon\; \ast \to [0,1]$

1. $const_1 \;\colon\; \ast \to [0,1]$

which include the [[point space]] as the two endpoints, respectively.

Together with the unique function $[0,1] \to \ast$ this yields the factorization of the [[codiagonal]] on the [[point space]]

$$
  \nabla_{\ast}
  \;\colon\;
  \ast \sqcup \ast \overset{(const_0,const_1)}{\longrightarrow} [0,1] \overset{\exists!}{\longrightarrow} \ast
$$

which exhibits an example of an _[[interval object]]_ in the general sense of [[model category]] theory with respect to the [[classical model structure on topological spaces]].

## Induced constructions

### Topological cylinder

For $X$ a [[topological space]], then the [[product topological space]] $X \times [0,1]$ with the topological interval is the _standard topological [[cylinder]]_ over $X$. Via the above inclusion functions, this inherits a factorization of the [[codiagonal]] of $X$ (the canonical continuous function out of the [[disjoint union space]] of $X$ with itself to $X$):

$$
  \nabla_X
  \;\colon\;
  X \sqcup X
   \overset{(id_X \times const_0, id_X \times const_1)}{\longrightarrow}
  X \times [0,1]
   \overet{pr_1}{\longrightarrow}
  X
  \,.
$$

Accordingly, with respect to the [[classical model structure on topological spaces]] this is an example a of _[[cylinder object]]_.

### Left homotopy

Given $X,Y$ two [[topological spaces]] and $f,g \;\colon\; X \longrightarrow Y$ two [[continuous functions]],then a **[[left homotopy]]**

$$
  \eta \colon f \,\Rightarrow_L\, g
$$

is a [[continuous function]]

$$
  \eta \;\colon\; X \times I \longrightarrow Y
$$

out of the [[product topological space]] of $X$ with the topological interval, such that this fits into a [[commuting diagram]] of the form

$$
  \array{
     X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     X \times I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     X
  }
  \,.
$$

(graphics grabbed from J. Tauber [here](http://jtauber.com/blog/2005/07/01/path_homotopy/))

### Path space

For $X$ a [[topological space]], then its _path space_ is the [[mapping space]] $X^{[0,1]}$, out of the topological interval into $X$, i.e. the set of [[continuous function]] $\gamma \;\colon\; [0,1] \to X$ equipped with the [[compact-open topology]].

The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions

$$
  X
    \overset{}{\longrightarrow}
  X^{[0,1]}
   \overset{X^{(const_0,const_1)}}{\longrightarrow}
  X \times X
$$

(inclusion of constant paths and endpoint evaluation of paths).

## Properties

### Freyd's characterization

The topological interval $[0, 1]$ may be characterized by a [[coalgebra for an endofunctor|coalgebraic]] definition first identified by [[Freyd]]:

Let $Top_{\ast, \ast}$ be the [[category]] of [[topological spaces]] $X$ equipped with a [[pair]] $x_0, x_1$ of distinct points, for example $I = ([0, 1]; 0, 1)$. Let $F \colon Top_{\ast, \ast} \to Top_{\ast, \ast}$ be the [[functor]] defined on [[objects]] by

$$
  F(X; x_0, x_1) = (X \vee X, y_0, y_1)
  \,,
$$

where $X \vee X$ denotes the [[quotient space]] of the [[disjoint union space]] $X \sqcup X$ formed by identifying the element $x_1$ of the first copy of $X$ with $x_0$ of the second copy of $X$, where $y_0$ is identified with $x_0$ of the first copy, and $y_1$ is identified with $x_1$ of the second copy.

For $I = ([0, 1]; 0, 1)$ there is an evident identification $F(I) = ([0, 2]; 0, 2)$, and moreover there is an $F$-[[coalgebra of an endofunctor|coalgebra]] structure $I \to F(I)$ given by "multiplication by $2$".

+-- {: .num_theorem}
###### Theorem
**(Freyd)**

The topological interval $I = ([0, 1], 0, 1)$ is the [[terminal coalgebra of an endofunctor|terminal F-coalgebra]].

=--

For more information, see [[coalgebra of the real interval]], which shows in particular how the interval structure of $[0, 1]$ may be defined by coinduction.

[[!redirects topological intervals]]


topological invariance of dimension

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

# Contents
* table of contents
{: toc}

## Idea

A fundamental theorem of [[topology]]: for $n_1, n_2 \in \mathbb{N}$ two different natural numbers, $n_1 \neq n_2$, then the [[Cartesian spaces]] $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ of [[dimension]] $n_1$ and $n_2$, respectively, are _not_ [[homeomorphism|homeomorphic]] (while for $n_1 = n_2$ then of course they are). Hence [[dimension]] of [[topological manifolds]] is a _topological [[invariant]]_.

This seems intuitively obvious, but the [[formal proof]] (due to [[Brouwer]] around 1910) is comparatively hard, and the problem was an important open problem in the 19th century. To appreciate that the problem is harder than it may superficially seem it serves to notice that for every $n \in \mathbb{N}$ there are [[surjective]] [[continuous maps]] $\mathbb{R}^1 \to \mathbb{R}^n$ (the [[Peano curves]]), in addition to the obvious [[injective]] continuous maps $\mathbb{R}^1 \to \mathbb{R}^n$ for $n \geq 1$ (or indeed $\mathbb{R}^m \to \mathbb{R}^n$ for $m \leq n$), because of which there are also [[bijective]] (but discontinuous) maps $\mathbb{R}^m \to \mathbb{R}^n$ for $m, n \geq 1$ (by the [[Schroeder–Bernstein theorem]]).

## Statement
 {#Statement}

+-- {: .num_theorem #InvarianceOfDimension}
###### Theorem
**(topological invariance of dimension)**

For $n_1, n_2 \in \mathbb{N}$ then the [[Euclidean space]] $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ (regarded as [[topological spaces]] with their [[metric topology]]) are [[homeomorphism|homeomorphic]] if and only if $n_1 = n_2$.

=--

More generally:

+-- {: .num_theorem #InvarianceOfDomain}
###### Theorem
**(Brouwer invariance of domain theorem)**

For $U \subset \mathbb{R}^n$ an [[open subset]] of a [[Cartesian space]], for any $n \in \mathbb{N}$, and for $f \colon U \to \mathbb{R}^n$ a [[continuous map|continuous]] [[injective map]], then its [[image]] $f(U)$ is also an [[open subset]].

In other words, continuous injections between Cartesian spaces of the same [[dimension]] are [[open maps]].

=--

This implies theorem \ref{InvarianceOfDimension}:

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

For $n_1, n_2 \in \mathbb{N}$, there is a [[continuous map|continuous]] [[injection]] from $\mathbb{R}^{n_1}$ to $\mathbb{R}^{n_2}$ if and only if $n_1 \leq n_2$.

=--

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

If $n_1 \leq n_2$, use $\iota_{n_1,n_2}\colon (x_1, \ldots, x_{n_1}) \mapsto (x_1, \ldots, x_{n_1}, \vec{0})$, where $\vec{0}$ consists of $n_2 - n_1$ copies of $0$.  If $n_1 \gt n_2$, then supposing a continuous injection $f\colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$, compose $f$ with $\iota_{n_2,n_1}$ to get a map from $\mathbb{R}^{n_1}$ to itself, also a continuous injection.  By invariance of domain (theorem \ref{InvarianceOfDomain}), the [[image]] of this map is open in $\mathbb{R}^{n_1}$, yet contained in the range of $\iota_{n_2,n_1}$, and the only open subset of that range is [[empty subset|empty]], a contradiction since $\mathbb{R}^{n_1}$ is not empty.
=--

+-- {: .proof}
###### Proof
of theorem \ref{InvarianceOfDimension} from theorem \ref{InvarianceOfDomain}

A homeomorphism is a continuous injection both ways, so $n_1 \leq n_2$ and $n_2 \leq n_1$ by lemma \ref{LemmaForTheCorollary}.

=--

This only-if half of this argument immediately generalizes to any [[inhabited subset|inhabited]] open subsets of $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$.

## Proofs
 {#Proofs}

We discuss various [[proofs]] of the topological invariance of dimension (theorem \ref{InvarianceOfDimension}).

### Via ordinary cohomology

+-- {: .proof}
###### Proof
of theorem \ref{InvarianceOfDimension}

Consider [[relative cohomology|relative]] [[ordinary cohomology]] $H_\bullet$
with [[coefficients]] in, says, the [[integers]] $\mathbb{Z}$. Recall that for $A \hookrightarrow X$ an inclusion of [[CW-complexes]], then there is a [[long exact sequence]] of the form

$$
  \cdots
   \to H^{n-1}(X) \longrightarrow H^{n-1}(A) \longrightarrow H^n(X,A) \longrightarrow H^n(X) \to \cdots
$$

Applied to the point inclusion $\{0\} \hookrightarrow \mathbb{R}^n$ and using that

1. $H^k(\mathbb{R}^n) = \left\{  \array{ \mathbb{Z} & \vert\, \text{if}\, k = 0 \\ 0 & \vert\, \text{otherwise}} \right.$

   because $\mathbb{R}^n$ is [[homotopy equivalence|homotopy equivalent]] to the point;

1. $H^k(\mathbb{R}^n \backslash \{0\}) = \left\{ \array{ \mathbb{Z} &\vert\, \text{if} \, k \in \{0,n-1\}  \\ 0 & |\, \text{otherwise} } \right.$

   because $\mathbb{R}^n \backslash \{0\}$ is [[homotopy equivalence|homotopy equvaent]] to th [[n-sphere|(n-1)-sphere]] $S^{n-1}$

this gives for $n \geq 2$ that $H^k(\mathbb{R}^n , \mathbb{R}^n - \{0\}) = \left\{ \array{ \mathbb{Z} &\vert\, \text{if} \, k \in \{0,n\} \\ 0 & \vert\, \text{otherwise}  }\right.$

Hence $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ do not have the same [[relative cohomology]] unless $n_1 = n_2$. Since  every [[homeomorphism]] induces an isomorphism on (relative) cohomology, they cannot be homeomorphic (using [[excluded middle]] here).

=--

### Via K-Theory

A proof also follows from the properties of the [[Adams operations]] on [[topological K-theory]], see for instance ([Wirthmuller 12, p. 46](#Wirthmuller12)).

## Related concepts

* [[Tietze extension theorem]]

* [[Brouwer's fixed point theorem]]

## References

The first proof is due to [[Brouwer]] around 1910.

* [[Terry Tao]], _[Brouwer's fixed point and invariance of domain theorems, and Hilbert's fifth problem](https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/)_

* Wikipedia, _[Invariance of domain](https://en.wikipedia.org/wiki/Invariance_of_domain)_

A proof also follows from the properties of the [[Adams operations]] on [[topological K-theory]], see for instance

* {#Wirthmuller12} [[Klaus Wirthmüller]], p. 46 of _Vector bundles and K-theory_, 2012 ([pdf](ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf))

[[!redirects invariance of dimension]]
[[!redirects invariance of domain]]