[[!redirects equivariant_related_illustration20170619]]
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equivariant

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

#Contents#
* table of contents
{:toc}

## Idea

An _equivariant function_ is a [[homomorphism]] between two [[G-sets]]:

$$
  f \;\colon\; X \longrightarrow Y
  \phantom{AAA}
  f(g \cdot x) = g \cdot f(x)
  \phantom{A}
  \forall g \in G
  \phantom{A}
  \forall x \in X
$$

More generally, an _equivariant morphism_ is a [[homomorphism]] of $G$-[[actions]].

## Related concepts

* [[equivariance]]

* [[equivariant structure]]

* [[equivariant cohomology]]

* [[equivariant homotopy theory]]

* [[equivariant stable homotopy theory]]

## References

* {#Bredon72} [[Glen Bredon]], Section I.2 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))

[[!redirects equivariant function]]
[[!redirects equivariant functions]]

[[!redirects equivariant map]]
[[!redirects equivariant maps]]

[[!redirects equivariant morphism]]
[[!redirects equivariant morphisms]]


equivariant Adams spectral sequence

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[Adams spectral sequence]] ([[Adams-Novikov spectral sequence]]) from [[stable homotopy theory]] to [[equivariant stable homotopy theory]].

> one may set up an equivariant analogue of the Adams spectral sequence $[...]$ We prefer not write $[$this$]$ out because this method needs time to mature. ([Adams-Gunawardena-Miller 85](#AdamsGunawardenaMiller85))

## References

* {#AdamsGunawardenaMiller85} [[John Frank Adams]], J. Gunawardena, [[Haynes Miller]], _The Segal conjecture for elementary abelian $p$-groups_, Topology Vol 24. No. 4, pp. 435-460 (1985)

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _On the non-existence of elements of Kervaire invariant one_, Annals of Mathematics ([journal version](http://annals.math.princeton.edu/articles/9563), arXiv:[0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf))


equivariant algebraic K-theory

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

#Contents#
* table of contents
{:toc}

## Idea

The [[equivariant cohomology]]/[[equivariant spectrum]]-version of [[algebraic K-theory]].

## Properties

### Rector completion theorem

[[!include Segal completion -- table]]

## Related concepts

* [[equivariant Adams conjecture]]

* [[equivariant K-theory]]

* [[equivariant elliptic cohomology]]

## References

* {#Lueck89} [[Wolfgang Lück]], _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

* Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of _Equivariant algebraic K-theory_, _Equivariant algebraic K-theory_, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 ([pdf](http://math.uchicago.edu/~may/PAPERS/40.pdf))

* Henning Hauschild, [[Stefan Waner]], theorem 0.1 of _The equivariant Dold theorem mod $k$ and the Adams conjecture_, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. ([euclid:1256065410](https://projecteuclid.org/euclid.ijm/1256065410))

* [[Kazuhisa Shimakawa]], _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi:10.2977/prims/1195167052](https://doi.org/10.2977/prims/1195167052))

* Christopher French, theorem 2.4 in _The equivariant $J$–homomorphism for finite groups at certain primes_, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 ([euclid:1513797069](https://projecteuclid.org/euclid.agt/1513797069))


equivariant bordism homology theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The refinement of [[bordism homology theory]] to [[equivariant stable homotopy theory]].

## Related concepts

* [[equivariant bordism theory]]

* [[equivariant cobordism cohomology theory]]

[[!include flavours of cobordism cohomology theories -- table]]

## References

Original discussion of equivariant [[cobordism]] classes of [[manifolds]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Differentiable periodic maps_, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86 ([euclid:bams/1183524501](https://projecteuclid.org/euclid.bams/1183524501))

* [[Pierre Conner]], [[Edwin Floyd]], _Maps of Odd Period_, Annals of Mathematics, Second Series, Vol. 84, No. 2 (1966), pp. 132-156 ([jstor:1970515](https://www.jstor.org/stable/1970515))

* {#Stong70} [[Robert Stong]], _Unoriented Bordism and Actions of Finite Groups_, Memoirs AMS 103, American Mathematical Society, 1970 ([ams:memo-1-103/4](https://bookstore.ams.org/memo-1-103/4))

Original discussion of [[equivariant spectrum|equivariant]] [[Thom spectra]]:

* [[Tammo tom Dieck]], _Kobordismentheorie und Transformationsgruppen_, Aaarhus University  Preprint Series 1968/69, No. 30 ([gbooks](https://books.google.de/books/about/Kobordismen_Theorie_und_Transformationsg.html?id=1ohGAAAAYAAJ&redir_esc=y))

* [[Tammo tom Dieck]], _Bordism of $G$-manifolds and integrality theorems_, Topology Volume 9, Issue 4, November 1970, Pages 345-358 (doi:10.1016/0040-9383(70)90058-3)

Original discussion of the relation between equivariant bordism classes of manifolds and equivariant Thom spectra:

* {#Wasserman69} [[Arthur Wasserman]], section 3 of _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

* [[Theodor Bröcker]], [[Edward Hook]], _Stable equivariant bordism_, Math Z 129, 269–277 (1972) ([doi:10.1007/BF01187353](https://doi.org/10.1007/BF01187353))

* [[Edward Hook]], _Equivariant cobordism and duality_, Trans. Amer. Math. Soc. 178 (1973), 241-258 ([doi:10.1090/S0002-9947-1973-0321120-4](https://doi.org/10.1090/S0002-9947-1973-0321120-4))

* {#tomDieck75} [[Tammo tom Dieck]], _Orbittypen und äquivariante Homologie II_, Arch. Math. 26 (1975), no. 6, 650–662 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tomdieck2.pdf))

* [[Bernhard Hanke]], _Geometric versus homotopy theoretic equivariant bordism_, Math. Ann. 332, 677–696 (2005) ([arXiv:math/0412550](https://arxiv.org/abs/math/0412550), [doi:10.1007/s00208-005-0648-0](https://doi.org/10.1007/s00208-005-0648-0))

On the [[Conner-Floyd isomorphism]] in [[equivariant cohomology theory]], relating relating [[equivariant cobordism cohomology]] to [[equivariant K-theory]]:

* {#Costenoble87} [[Steven Costenoble]], *Equivariant Conner-Floyd isomorphism*, Trans. Amer. Math. Soc. **304** 2 (1987) 801--818, [[jstor:2000742](http://www.jstor.org/stable/2000742)]

Survey:

* {#Schwede15} [[Stefan Schwede]], _Equivariant bordism from the global perspective_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/glasgow-handout.pdf), [[SchwedeGlobalEquivariantBordism.pdf:file]])

* [[Michael Wiemeler]], *Equivariant bordism and applications in Differential Geometry*, talk at Math Coll. Augsburg (Jan 2016) [[pdf](https://www.uni-muenster.de/IVV5WS/WebHop/user/wiemelem/slides/kol_augsburg.pdf), [[Wiemeler-EquivariantBordism.pdf:file]]]

Discussion in [[global equivariant homotopy theory]]:

* [[Stefan Schwede]], section 6.2 _[[Global homotopy theory]]_ New Mathematical Monographs, 34 Cambridge University Press, 2018 ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

* Ningchuan Zhang, *Equivariant cobordism* (2019) [[pdf](https://iwoat.github.io/2019/notes/Lecture-14.pdf)]

[[!redirects equivariant bordism]]

[[!redirects equivariant bordism cohomology theory]]

[[!redirects equivariant MO]]
[[!redirects equivariant MO-theory]]


equivariant bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Generally, for $G$ some [[group]], a _$G$-equivariant bundle_ is a [[bundle]], specifically a [[fiber bundle]] ([[principal bundle]], [[vector bundle]], etc.) all whose component spaces (total space $E$, base space $X$, [[fiber]] $F$, but also possibly the [[structure group]] $\Gamma$) are equipped with $G$-[[actions]], such that all structure morphisms (in particular the projection $E \overset{p}{\to} X$, but also the $\Gamma$-[[action]] for [[principal bundles]] etc.)  are $G$-[[equivariant functions]].

In short, this should mean ([SS 2021, Sec. 2.1](#SatiSchreiber21), Def. \ref{EquivariantPrincipalBundles} below) that $G$-equivariant (fiber-, principal-,...) bundles are (fiber-, principal, ...) bundles _[[internalization|internal]]_ to a [[category]] of [[G-spaces]] (e.g. [[topological G-spaces]], [[G-manifolds]] but also [[G-sets]] etc.).

While the existing literature does not state the definition of equivariant bundles via [[internalization]], one sees (Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles} below) that the explicit definition in [tom Dieck 69](#tomDieck69), [tom Dieck 87, Sec. I.8](#tomDieck87) (for the case of [[principal bundles]]) is the equivalent external description, _including_, thereby, an [[action]] of the [[equivariance group]] $G$ on the [[structure group]] $\Gamma$ -- together with the respective compatibility conditions, which equivalently say (as highlighted in
[tom Dieck 69, Sec. 1.2](#tomDieck69), also [Murayama-Shimakawa 95, below 1.1](#MurayamaShimakawa95), see the discussion [here](category+of+G-sets#InternalGroupActions)) that the joint action is that of the [[semidirect product group]] $\Gamma \rtimes G$.

Beware that this [[action]] of the [[equivariance group]] $G$ on the [[structure group]] $\Gamma$ of an [[equivariant principal bundle]] is often and traditionally disregarded, i.e. implicitly taken to be the [[trivial action]] (e.g. [Lashof 82](#Lashof82), [Lashof-May-Segal 83](#LashofMaySegal83)), which equivalently means that the [[semidirect product group]] that acts is reduced to the [[direct product group]] $\Gamma \times G$, meaning that the [[action]] of the [[equivariance group]] _commutes_ with that of the [[structure group]]. This special case is the default meaning of _equivariant bundle_ in much of the literature!

(The definition of "generalized equivariant bundles" proposed in [Lashof-May 86](#LashofMay86) and advertized in [Lewis-May-Steinberger 86, Sec. IV.1](#LewisMaySteinberger86) allows any [[group extension]] of $G$ by $\Gamma$ to act. This reduces to [[semidirect products]] $\Gamma \rtimes G$ for [[split sequence|split extensions]] (see [there](group+extension#SplitExtensionsAndSemidirectProductGroups)) where $\Gamma$ is a [[normal subgroup]], and that is the case that [May 90](#May90), [Guillou-May-Merling 17](#GuillouMayMerling17) falls back to).

Much of the literature on equivariant bundles is interested

* in [[equivariant vector bundles]] regarded as representing [[cocycles]] for [[equivariant K-theory]] ([Segal 68, Sec 1](#Segal68), [Wasserman 69, Sec. 2](#Wasserman69),[HJJS 08, Sec. 13](#HJJS08));

and/or

* as [[twisted cohomology|twists]] for [[twisted equivariant cohomology theories]], and here most of the existing literature focuses on [[twisted equivariant K-theory]] and specifically on its equivariant degree-3 twist by equivariant [[projective bundles]] (e.g. [Barcenas-Espinoza-Joachim-Uribe 12](#BEJU14),  [Uribe & Lück 2014, Sec. 15](#UribeLück14)).

## Definition

We discuss  equivariant groups in/as [[topological spaces]], for definiteness and due to their relevance as models in [[equivariant homotopy theory]].  All of the discussion generalizes, say to [[smooth manifolds]] or general [[toposes]].

### Preliminaries

\begin{defn}
(**convenient category of topological spaces**)
\linebreak
We write
$$
  TopologicalSpaces
  \;\in\;
  Categories
$$
for any [[convenient category of topological spaces]] whose [[mapping space]] serves as an [[internal hom]], such as

* [[D-topological spaces]]

* [[compactly generated topological spaces]].

This means in particular that for $X,Y,A \,\in\, TopologicalSpaces$, we have a [[natural bijection]]

\[
  \label{HomAdjunctInTopologicalSpaces}
  X \times Y
    \longrightarrow
  A
  \;\;\;\;\;\;\;
  \overset{adjuncts}{\leftrightarrow}
  \;\;\;\;\;\;\;
  X
    \longrightarrow
  Maps(Y,A)
\]

between [[maps]] (meaning: [[continuous functions]]) out of the [[product topological space]] with $Y$ and maps into the [[mapping space]].

\end{defn}

\begin{defn}\label{TopologicalGSpaces}
(**topological $G$-spaces**)
\linebreak
For $G$ be a [[topological group]] -- to be called the _[[equivariance group]]_ -- we write
$$
  G Actions(TopologicalSpaces)
  \;\in\;
  Categories
$$
for the [[category]] whose

* [[objects]] $(X,\rho)$ are [[topological spaces]] $X$ equipped with [[continuous function|continuous]] $G$-[[actions]]

  $$
    G \times X \overset{}{\longrightarrow} X
    \;\;\;\;\;\;
    \leftrightarrow
    \;\;\;\;\;\;
    G \overset{\rho}{\longrightarrow} Aut(X) \subset Maps(X,X)
    \,;
  $$

* [[morphisms]] are $G$-[[equivariant function|equivariant]] [[continuous functions]] between them ("maps"); i.e. for the category of "[[G-spaces]]", often denote "$G Spaces$" or even $G Sp$ or similar.

\end{defn}

\begin{remark}
(**further conditions on the equivariance group**)
\linebreak
For purposes of [[equivariant homotopy theory]] one typically assumes the [[topological group|topological]] [[equivariance group]] $G$ in Def. \ref{TopologicalGSpaces} to be that underlying a [[compact Lie group]], such as a [[finite group]] (as that guarantees that [[G-CW-complexes]] are well-behaved and that the [[equivariant Whitehead theorem]] holds). But for the plain [[point-set topology]] of equivariant bundles, this condition is not necessary.
\end{remark}

\begin{remark}\label{GSpacesAreCartesianMonoidal}
(**[[topological G-spaces]] are [[Cartesian monoidal category|cartesian monoidal]]**)
  The category of [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}) is a [[Cartesian monoidal category]]: The [[Cartesian product]] of two [[topological G-spaces]] $(X_i, \rho_i)$ is the underlying [[product topological space]] equipped with the [[diagonal action]] by $G$:
$$
  (X_1, \rho_1)
  \times
  (X_2, \rho_2)
  \;=\;
  \Big(
    X_1 \times X_2,
   \,
   \rho_{1,2}(g)(x_1,x_2)
   \,=\,
   \big(
     \rho_1(x_1), \rho_2(x_2)
   \big)
  \Big)
  \,.
$$
\end{remark}
In a [[Cartesian monoidal category]] there is a notion of [[internalization|internal]] [[group objects]]:

### Equivariant groups

We follow the terminology of [SS 2021, Def. 2.1.2](#SatiSchreiber21):

\begin{defn}\label{EquivariantTopologicalGroup}
(**equivariant topological groups**)
\linebreak
Given an [[equivariance group]] $G$ (Def. \ref{TopologicalGSpaces}),
a _$G$-[[equivariant topological group]]_ $(\Gamma, \alpha)$ is a [[group object]] [[internalization|internal]] to [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}):
  $$
    \big(
      \Gamma,
      \,
      \alpha
    \big)
    \;\in\;
    Groups
    \big(
      G Actions
      (
        TopologicalSpaces
      )
    \big)
    \,.
  $$
\end{defn}
(See Prop. \ref{EquivariantGroupsAsSemidirectProductGroups} below for the choice of notation used here.)

\begin{remark}\label{UnderlyingTopologicalGroupOfAnEquivariantTopologicalGroup}
Since the [[forgetful functor]] from [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}) to underlying [[topological spaces]]
$$
  \array{
    G Actions
    (
      TopologicalSpaces
    )
   &
     \overset{
     }{\longrightarrow}
   &
   TopologicalSpaces
  }
$$
[[preserved limit|preserves]] [[Cartesian products]] (explicitly so by Remark \ref{GSpacesAreCartesianMonoidal}), it preserves [[group objects]] and hence sends $G$-equivariant topological groups (Def. \ref{EquivariantTopologicalGroup}) to underlying plain [[topological groups]]:
\[
  \label{ForgetfulFunctorFromEquivariantTopologicalGroupsToTopologicalGroups}
  \array{
    Groups
    \big(
    G Actions
    (
      TopologicalSpaces
    )
   \big)
   &
     \overset{
     }{\longrightarrow}
   &
   Groups
   (
     TopologicalSpaces
   )
   \\
   \big(
     \Gamma
     ,
     \alpha
   \big)
   &\mapsto&
   \Gamma
   \,.
  }
\]
\end{remark}

### Equivariant bundles

We say that a $G$-equivariant principal bundle is a [[principal bundle]] [[internalization|internal]] to [[topological G-spaces]] ([SS 2021, Def. 2.1.3](#SatiSchreiber21)).

(Beware that, following tradition in equivariant bundle theory, we do not impose a [[local trivialization|local trivializability condition]] at this point, but add this as an extra clause later and then speak explicitly of _locally trivial equivariant bundles_ -- for more on this see at _[Notions of equivariant local triviality](#NotionsOfEquivariantLocalTriviality)_.)

\begin{defn}\label{EquivariantPrincipalBundles}
(**[[equivariant principal bundle]]**)
  \linebreak
Given

  * an [[equivariance group]] $G \in Groups(TopologicalSpaces)$;

  * an [[equivariant topological group]] (Def. \ref{EquivariantTopologicalGroup})

    $
      (\Gamma, \alpha)
      \;\in\;
      Groups\big(  G Actions(TopologicalSpaces) \big)
    $

    to be called the $G$-equivariant _[[structure group]]_;

a _$G$-equivariant $(\Gamma,\alpha)$-principal bundle over $(X,\rho)$ is:

* a [[internalization|internal]] [[bundle]], namely a [[morphism]] $P \overset{p}{\to} X$ in [[topological G-spaces]];

* the [[mathematical structure|structure]] of a $(\Gamma,\alpha)$-[[action object]] on the total space $P$:

such that the following conditions holds:

* **(fiberwise action)** the [[action]] is over $X$, in that the following [[commuting diagram|diagram commutes]]:

\begin{tikzcd}
  \Gamma \times P
  \ar[
    rr,
    "\rho"
  ]
  \ar[
    dr,
    "p\circ \mathrm{pr}_2\;"{left}
  ]
  &&
  P
  \ar[
    dl,
    "\;p"{right}
  ]
  \\
  & X
\end{tikzcd}

* {#EquivariantPrincipality} **(principality)** the _[[shear map]]_

  $$
    \array{
      \Gamma \times P
      &\longrightarrow&
      P \times_X P
      \\
      (\gamma, p)
      &\mapsto&
      (\rho(\gamma)(p), p)
    }
  $$

  is an [[isomorphism]], hence this [[commuting diagram]] is moreover a [[pullback square]] (in [[topological G-spaces]]):

\begin{tikzcd}
  \Gamma \times P
  \ar[
    drr,
    bend left=20,
    "\rho"
  ]
  \ar[
    ddr,
    bend right=20,
    "\mathrm{pr}_2\;\;\;"{below}
  ]
  \ar[
    dr,
    dashed,
    "\simeq"{description}
  ]
  \\
  &
  P \times_X P
  \ar[
    r
  ]
  \ar[d]
  \ar[
    dr,
    phantom,
    "\mbox{\tiny(pb)}"{description}
  ]
  &
  P
  \ar[
    d,
    "p"
  ]
  \\
  &
  P
  \ar[
    r,
    "p"{below}
  ]
  &
  X
\end{tikzcd}

\end{defn}

\begin{remark}\label{OnTheInternalDefinitionOfPrincipality}
**(on the internal definition of principality)**
Notice that in Def. \ref{EquivariantPrincipalBundles} we do *not* explicitly demand that the base space is the [[quotient space]] of the total space by the [[group action]] (it defines an [[internalization|internal]] [[pseudo-torsor]] instead of a [[torsor]]). This has several reasons:

1. The way the definition is stated, it involves only [[finite limits]] (no [[colimits]]), which means that this notion of equivariant principal bundle will be preserved by all [[right adjoint]] functors on the ambient category, such as notably the [[fixed locus]]-functors (see [there](topological+G-space#FixedLociWithResidualWeylGroupAction)).

1. Nevertheless, the expected quotient condition is both implied as well as circumvented, as need be:

   Once equivariant local trivializability is imposed [below](#NotionsOfEquivariantLocalTriviality) it follows that the underlying [[continuous function]] $P \to X$ is a [[locally trivial]] [[fiber bundle]]. There are then two cases:

   1. Either the [[typical fiber]] is [[inhabited set|inhabited]]. This implies that $P \to X$ is an [[effective epimorphism]], and with that property the shear map condition [above](#EquivariantPrincipality) implies that $X$ is the quotient of $P$ by $G$.

   1. Or the [[typical fiber]] is the [[empty space]]. This case, often disregarded in discussion of fiber bundles, is actually an example of Def. \ref{EquivariantPrincipalBundles}, because for the empty bundle the shear map is an isomorphism:

      $$
        \Gamma \times \varnothing
        \overset{\simeq}{\longrightarrow}
        \varnothing \times_X \varnothing
      $$

      (since both [[domain]] and [[codomain]] are isomorphic to the [[empty space]]):

      *Empty bundles are principal! (and locally trivial!), in the above sense.*

      While this degenerate case is irrelevant for ordinary principal bundles, it is crucial for equivariant principal bundles, since their [[fixed loci]] often have empty total space. In fact, if the action $\alpha$ of $G$ on the [[structure group]] $\Gamma$ is trivial (as is often the case in applications) then it follows at once that any fixed locus of an equivariant principal bundle can have only one of two typical fibers, $\Gamma$ or $\varnothing$, while the structure group is $\Gamma$, in both cases.

\end{remark}

## Properties

### As $(G,\alpha,\Gamma)$-principal bundles

\begin{prop}\label{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles}
(**principal bundles internal to $G$-spaces are equivalent to tom Dieck-bundles**)
\linebreak
  The
  $G$-equivariant $(\Gamma,\alpha)$-principal bundles in the [[internalization|internal]] sense of Def. \ref{EquivariantPrincipalBundles} are [[equivalence of categories|equivalent]] to the $(G,\alpha,\Gamma)$-principal bundles in the sense of [tom Dieck 69](#tomDieck69) (both without any condition of [[local trivializability]], at this point).
\end{prop}
\begin{proof}
  This follows immediately by the fact ([this Prop.](equivariant+group#ActionsOfEquivariantGroupsAsSemidirectProductGroupActions)) that $G$-equivariant actions of [[equivariant groups]] $(G,\alpha)$ are equivalent to plain actions of the [[semidirect product group]] $\Gamma \rtimes_\alpha G$.
\end{proof}

### Restriction to fixed loci
 {#RestrictionToFixedLoci}

The [[internalization|internal]]-formulation of $G$-equivariant principal; bundles (Def. \ref{EquivariantPrincipalBundles}) serves to make manifest that they have good behaviour under passage to [[fixed loci]]:

Notice that for any [[subgroup]] $H \subset G$ we have the fixed point functor ([here](fixed+point+space#eq:TopologicalHFixedLociAsFunctorToWeylGroupSpaces))

\[
  \label{TopologicalHFixedLociAsFunctorToWeylGroupSpaces}
  (-)^H
  \;\colon\;
  Topological G Spaces
   \overset{\;\;\;\;\;}{\longrightarrow}
  Topological N(H)/H Spaces
  \,.
\]

\begin{prop}
  For $(\Gamma,\alpha)$ an $G$-equivariant topological group (Def. \ref{EquivariantTopologicalGroup}), $P \to X$ a $G$-equivariant $(\Gamma,\alpha)$-principal bundle (Def. \ref{EquivariantPrincipalBundles}) and $H \subset G$ any [[subgroup]], the $H$-fixed locus (eq:TopologicalHFixedLociAsFunctorToWeylGroupSpaces) $P^H \to X^H$ is canonically an $N(H)/H$-equivariant $(\Gamma^H, \alpha^H)$-principal bundle.
\end{prop}
\begin{proof}
  The point is that passage to $H$-fixed loci (eq:TopologicalHFixedLociAsFunctorToWeylGroupSpaces) is a [[right adjoint]] functor ([this Prop.](fixed+point+space#PassageToFixedLociIsRightAdjoint)) and therefore [[right adjoints preserve limits|preserves all limits]], and so in particular [[preserved limit|preserves]] [[Cartesian products]] and [[fiber products]], hence preserves [[internalization|internal]] [[group objects]] as well as the [principality condition](#EquivariantPrincipality).
\end{proof}

### Over coset spaces
 {#OverCosetSpaces}

\begin{prop}\label{EquivariantCosetSpaceBundles}
**(semidirect product coset space bundles are locally trivial)**
  Let $H \subset G$ be a [[closed subgroup]],
  and let
  ${\widehat H}$ be a lift of $H$ to a [[closed subgroup]] of the [[semidirect product group]], as shown on the left here:

\[
  \label{EquivariantBundleOfCosetSpaces}
  \array{
    {\widehat H} &\subset& \Gamma \rtimes_\alpha G
    &{\phantom{A}}&
    \big(
      \Gamma \rtimes_\alpha G
    \big)
    \big/
    {\widehat H}
    \\
    {}^{\mathllap{\simeq}}
    \big\downarrow
    &&
    \big\downarrow
    {}^{\mathrlap{
      pr_2
    }}
    &&
    \big\downarrow
    {}^{ \mathrlap{p} }
    \\
    H &\subset& G
    &&
    G/H
    \mathrlap{
    \,.
    }
  }
\]

Then the induced [[bundle]] of [[coset spaces]], shown on the right of (eq:EquivariantBundleOfCosetSpaces):

1. is a $(\Gamma,\alpha)$-principal bundle (under Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles})

1. is [[locally trivial]] as an ordinary $\Gamma$-[[fiber bundle]] in [[TopologicalSpaces]] as soon as the the [[coset space]] [[coprojection]] $G \to G/H$ [[coset space coprojection admitting local sections|admits local sections]].

\end{prop}

This is [tom Dieck 69, Lemma in Sec. 2.1](#tomDieck69)

\begin{proof}

The first statement is evident.

For the second statement, let $\sigma$ denote a local section over an open subset $U \subset G/H$:

$$
  \array{
    &&
    G_{\vert \mathcal{U}}
    &\longrightarrow&
    G
    \\
    &
    {}^{\mathllap{
      \sigma
    }}
    \nearrow
    &
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    U
    &=&
    U
    &\longrightarrow&
    G/H
    \mathrlap{\,,}
  }
$$

We claim that then

$$
  \array{
    \Gamma \times U
    &&
    \underoverset
     {\simeq}
     {
       (\gamma,u)
       \mapsto
       [\gamma,\sigma(u)]
     }
     {
      \longrightarrow
    }
    &&
    \Big(
      \big(
        \Gamma \rtimes_\alpha G
      \big) / {\widehat H}
    \Big)_{\vert U}
    \\
    & \searrow && \swarrow
    \\
    && U
  }
$$

is an [[isomorphism]], of $\Gamma$-[[principal bundles]] over $U \subset G/H$.

* It is clear by construction that the map is

  1. [[continuous function|continuous]].

  1. $\Gamma$-equivariant.

* To see that map is an isomorphism it is sufficient to see that it is so over each $u \in U$:

  * Injectivity here follows from the fact that ${\widehat H}$ is a lift of $H$, so that the unique element lifting the [[neutral element]] is the neutral element.

  * To see that the map is surjective over each $u \in U$:

    That $[\gamma, g] \in \big( \Gamma \rtimes_\alpha G \big)/{\widehat H}$ is in the fiber over $u$ means that $ g^{-1} \sigma(u) \in H \subset G$. Then for $\hat h \in \widehat H$ the unique element with $pr_2(\hat h) = g^{-1} \sigma(u)$ we see that $[\gamma, g] = \big[ (\gamma,g) \cdot q \big] = [\cdots, \sigma(u)]$, which is manifestly in the image of our map.

    (Filling in the ellipses here, which is straightforward but unenlightning, actually gives the continuous inverse map.)

\end{proof}

In fact every equivariant principal bundle over a coset space is of this form:

\begin{prop}\label{EquivariantPrincipalBundlesOverCosetSpacesAreSemidirectProductCosetsSpaceBundles}
**(equivariant principal bundle over coset spaces are semidirect product coset space bundles)**
  Let $H \subset G$ be a [[closed subgroup]]. Then every [[Hausdorff space]] $(\Gamma,\alpha)$-principal bundle (Def. \ref{EquivariantPrincipalBundles}) over the [[coset space]]

$$
  P \overset{p}{\longrightarrow} G/H
  \,,
$$

which is [[locally trivial]] as an ordinary $\Gamma$-[[principal bundle]], is [[isomorphism|isomorphic]]

\[
  \label{TrivializingBundleIosmorphismOverCosetSpace}
  \array{
    (\Gamma \rtimes_\alpha G)/{\widehat H}
    &&
    \underoverset
      {\simeq}
      { [\gamma,g] \mapsto (\gamma,g)\cdot p }
      {\longrightarrow}
    &&
    P
    \\
    &
    \searrow
    &&
    \swarrow
    \\
    &&
    G/H
  }
\]

to a [[coset space]]-bundle $(\Gamma \rtimes_\alpha G)/{\widehat H} \longrightarrow G/H$ from Prop. \ref{EquivariantCosetSpaceBundles}, for ${\widehat H}$ a lift of $H$ to a [[closed subgroup]] of the [[semidirect product group]]:

$$
  \array{
    {\widehat H} &\subset& \Gamma \rtimes_\alpha G
    \\
    {}^{\mathllap{\simeq}}
    \big\downarrow
    &&
    \big\downarrow
    {}^{\mathrlap{
      pr_2
    }}
    \\
    H &\subset& G
    \mathrlap{
    \,.
    }
  }
$$

\end{prop}

\begin{proof}

Regard $P$ as equipped with the [[action]] of the [[semidirect product group]] $\Gamma \rtimes_\alpha G$, by Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles}.

Let $e \in P_{[H]}$ be any point in the [[fiber]] over $[H] \in G/H$, and take

$$
  {\widehat H} \;\coloneqq\; Stab_{\Gamma \rtimes_\alpha G}(e)
$$

to be its [[stabilizer subgroup]] under the [[semidirect product group]]-action. We observe that this has the stated properties:

* ${\widehat H}$ is a [[closed subgroup]] because [stabilizer subgroups are closed subgroups](closed+subgroup#StabilizerSubgroupsOfContinuousActionsOnT1SpacesAreClosed) under the Hausdorff-ness assumption;

* under $\Gamma \rtimes_\alpha G \overset{pr_2}{\longrightarrow} G$ we have ${\widehat H} \to H$, since  for all $\hat h \in \widehat H$:

  $$
    \begin{aligned}
      [H]
      &= p(\hat h \cdot e)
      \\
      & = pr_2(\hat h) \cdot p(e)
      \\
      & = pr_2(\hat h) \cdot [H]
    \end{aligned}
  $$

* ${\widehat H} \to H$ is

  * a [[surjection]] because the action of $\Gamma$ on the [[fibers]] of $P$ is a [[transitive action]],

  * an [[injection]] because the action of $\Gamma$ on the [[fibers]] of $P$ is a [[free action]]

  (by [principality](#EquivariantPrincipality)).

Moreover, it is clear by construction that (eq:TrivializingBundleIosmorphismOverCosetSpace) is a $(\Gamma \rtimes_\alpha G)$-[[equivariant function|equivariant]] [[continuous function]] over $G/H$.

Hence to see that (eq:TrivializingBundleIosmorphismOverCosetSpace) is an [[isomorphism]] (a [[homeomorphism]] of underlying [[topological spaces]]) it is sufficient to see that after [[forgetful functor|forgetting]] the $G$-[[action]] we have a morphism between ordinary $\Gamma$-[[principal bundles]] over their common base space, because any such is an [[isomorphism]], as is manifest from its restriction to any common [[local trivialization]].

Therefore it is now sufficient to see that the coset bundle $(\Gamma \rtimes_\alpha G)/{\widehat H} \longrightarrow G/H$ is [[locally trivial]] as a $\Gamma$-[[principal bundle]]. But this is the statement of Prop. \ref{EquivariantCosetSpaceBundles}.

\end{proof}

### Notions of equivariant local triviality
 {#NotionsOfEquivariantLocalTriviality}

The literature considers various different notions of [[local triviality]] of equivariant bundles. Some of these look quite different, but are indeed all equivalent, under mild conditions ([SS 2021, Thm. 2.1.2](#SatiSchreiber21)).

> under construction, for details see [SS 2021, Sec. 2.2](#SatiSchreiber21)

\begin{definition}\label{tomDieckEquivariantLocalTrivialization}
**(tom Dieck's equivariant local triviality condition -- [tom Dieck 69, Def. 2.3](#tomDieck69), [tom Dieck 87, p. 58](#tomDieck87))**\linebreak
  An equivariant principal bundle
  (Def. \ref{EquivariantPrincipalBundles}, Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles})
  $$
    P \overset{p}{\longrightarrow} B
    \;\;
    \in
    \;
    (\Gamma,\alpha)PrincipalBundles
  $$
  is **[[locally trivial]]** if there exists

1. an index-[[set]] $I$,

1. an $I$-[[indexed set]] of [[topological subspace|sub]]-[[topological G-space|G-spaces]]

   $$
     U_i \;\subset\; B
     \;\;
     \in
     G Actions
     \big(
      TopologicalSpaces
     \big)
   $$

1. an $I$-[[indexed set]] of [[closed subgroup]] $H_i \subset G$;

1. an $I$-[[indexed set]] of $E_i \overset{p_i}{\to} G/H_i \;\in\; (\Gamma,\alpha)PrincipalBundles$ over their [[coset spaces]];

such that

1. $\big\{ U_i \hookrightarrow B \big\}_{i \in I}$ is an [[open cover]] in [[TopologicalSpaces]];

1. for each $i \in I$ there is a [[homomorphism]] of $(\Gamma,\alpha)PrincipalBundles$

   \[
     \label{tomDieckLocalTrivalityHomomorphism}
     \array{
       E_{\vert U_i}
       &
         \overset{}{\longrightarrow}
       &
       E_i
       \\
       \big\downarrow
       &
         {}^{{}_{}}
       &
       \big\downarrow {}^{\mathrlap{p}}
       \\
       U_i
       &\longrightarrow&
       G/H
     }
   \]

   from the restriction of $E$ to $U_i$ and the given  equivariant bundle over the coset space (as [above](#OverCosetSpaces)).

\end{definition}

\begin{definition}\label{BierstoneEquivariantLocalTrivialization}
**(Bierstone's equivariant local triviality condition -- [Bierstone 78, Sec. 4, p. 619-620](#Bierstone78))**\linebreak
  An equivariant principal bundle
  (Def. \ref{EquivariantPrincipalBundles}, Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles})

  $$
    P \overset{p}{\longrightarrow} X
    \;\;
    \in
    \;
    (\Gamma,\alpha)PrincipalBundles
  $$

  is **[[locally trivial]]** if for each point $x \in X$ (with [[isotropy
group]]/[[stabilizer group]] denoted $G_x \coloneqq Stab_G(x) \subset X$) there exists

1. an [[open neighbourhood]]

   $$
     U_x
       \;\subset\;
     X \;\; \in \; G_x Actions(TopologicalSpaces)
   $$

1. a $G_x$-[[equivariant function|equivariant]] [[homomorphism]] of $\Gamma PrincipalBundles$

   $$
     \array{
       P_{\vert U_x}
       &
         \longrightarrow
       &
       U_x \times p^{-1}(\{x\})
       \\
       {}^{\mathllap{ p_{\vert U_x} }}
       \big\downarrow
         &&
       \big\downarrow
       {}^{
         \mathrlap{ pr_1 }
       }
       \\
       U_x &=& U_x
     }
   $$

   from the restriction of $P$ to $U_x$ to the [[Cartesian product]] of $U_x$ with the [[fiber]] of $P$ over $x$.

\end{definition}

\begin{definition}\label{LashofEquivariantLocalTrivialization}
**(Lashof's equivariant local triviality condition --  [Lashof 82, p. 258](#Lashof82), [Lashof-May 86, p. 267](#LashofMay86))**\linebreak
  An equivariant principal bundle
  (Def. \ref{EquivariantPrincipalBundles}, Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles})

  $$
    P \overset{p}{\longrightarrow} X
    \;\;
    \in
    \;
    (\Gamma,\alpha)PrincipalBundles
  $$

  is **[[locally trivial]]** if for each point $x \in X$ (with [[isotropy
group]]/[[stabilizer group]] denoted $G_x \coloneqq Stab_G(x) \subset X$) there exists

1. an index-[[set]] $I$;

1. an $I$-[[indexed set]] of [[closed subgroups]] $H_i \subset G$;

1. an $I$-[[indexed set]] of $H_i$ [[slice theorem|slices]] $S_i \subset X$;

1. an $I$-[[indexed set]] of $\Gamma_i \;\in\; H_i Actions(TopologicalSpaces)$ lifting $\Gamma \in TopologicalSpaces$;

such that

1. the [[orbits]] of the [[slice theorem|slices]] $\big\{ G\cdot S_i \subset X \big\}_{i \in I}$ form an [[open cover]] over $X$;

1. for each $i \in I$ there is a $G$-[[equivariant function|equivariant]] [[homomorphism]] of $\Gamma PrincipalBundles$

   \[
     \label{LashofLocalTrivalityHomomorphism}
     \array{
       P_{\vert G\cdot S_i}
       &\longrightarrow&
       G
         \times_{H_i}
       \big(
         S_i \times \Gamma_i
       \big)
       \\
       \big\downarrow
       &&
       \big\downarrow
       \\
       G \cdot S_i
       &=&
       G \times_H S_i
     }
   \]

   from the restriction of $P$ over the orbit of the $i$th slice to ...

\end{definition}

* [Lashof-Rothenberg 78, p. 216](#LashofRothenberg78)

(...)

\begin{prop}

(...) Lashof82 $\leftrightarrow$ Bierstone78  (...)

\end{prop}

([Lashof 82, Lemmas 1.1, 1.3](#Lashof82))

\begin{lemma}\label{LashofLocalModelsAreLocallyTrivialAsOrdinaryFiberBundles}
**(Lashof's local models are locally trivial as ordinary fiber bundles)
  If the [[equivariance group]] $G$ is a [[compact Lie group]]
  then for every [[closed subgroup]] $H \subset G$ and [[topological G-space|topological H-spaces]] $S, F$ the canonical projection

$$
  \array{
    G \times_H
    \big(
      S \times F
    \big)
    \\
    \big\downarrow
    \\
    G \times_H S
  }
$$

is a [[locally trivial]] $F$-[[fiber bundle]].

\end{lemma}

([Lashof 82, Cor. 1.2](#Lashof82))

\begin{proof}

Since $G$ is assumed to be a [[compact Lie group]], it admits a [[bi-invariant Riemannian metric]] ([this Prop.](compact+Lie+group#CompactLieGroupsAdmitBiinvariantMetrics)). With respect to this metric, consider a small open [[normal vector|normal]] $\epsilon$-neighbourhood to $H$ at $e$ in $G$

$$
  D_e
  \,\coloneqq\,
  D_\epsilon N_e H
  \,\subset \,
  N H
  \;
    \underoverset{\simeq}{\exp}{\longrightarrow}
  \;
  T H
$$

i.e. of points that with respect to some choice of [[tubular neighbourhood]] of $H \subset G$ are a normal distance $\lt \epsilon$ from $H$.

Then the multiplication action

$$
  \array{
    D_e \times H
    &
    \underoverset{ \simeq }{ (-)\cdot(-) }{\longrightarrow}
    &
    D_e \cdot H
    \\
    (d,h) &\mapsto& d \cdot h
  }
$$

is a [[diffeomorphism]], because, by right-invariance of the chosen metric, the operation

$$
  \array{
    D_e &\overset{}{\longrightarrow}& D_h
    \\
    d &\mapsto& d \cdot h
  }
$$

is even an [[isometry]], for each $h \in H$.

It follows

1) by dimension reasons that

$$
  D \cdot H \;\subset\; G
$$

is an [[open neighbourhood]] of $H$ in $G$, and hence that

$$
  \big(D \cdot H\big) \times_H S
  \hookrightarrow
  G \times_H S
$$

is an [[open neighbourhood]] of $S \subset G_H \times S$.

2) that the restriction of the projection to this neighbourhood is isomorphic to the trivial $F$-fiber bundle:

$$
  \array{
    D \times S \times F
    &
    \simeq
    &
    (D \times H) \times_H (S \times F)
    &
    \overset{\simeq}{\longrightarrow}
    &
    (D \cdot H)
    \times_H
    (S \times F)
    &
    \longrightarrow
    &
    G \times_H ( S \times F )
    \\
    \big\downarrow
    && &&
    \big\downarrow
    &
      {}^{{}_{(pb)}}
    &
    \big\downarrow
    \\
    D \times S
    &\simeq&
    (D \times H) \times_H S
    &
    \underoverset
      { m \times_H id }
      {\simeq}
      {\longrightarrow}
    &
    (D \cdot H) \times_H S
    &\hookrightarrow&
    G \times_H S
    \,.
  }
$$

Finally, since $G \times_H S$ is covered by left $G$-translates of the open subset $(D \cdot H) \times_H S$, and since the same argument applies to each of theses, by left-invariance of the metric, the claim follows.

\end{proof}

\begin{prop}\label{LashofLocalTrivializabilityImpliesTopDieckForTrivialAlpha}
**(Lashof's local trivializability implies tom Dieck's for $\alpha = 1$)**
  In the case that $\alpha = 1$ (in Def. \ref{EquivariantTopologicalGroup}), equivariant local trivializability in the sense of Lashof (Def. \ref{LashofEquivariantLocalTrivialization}) implies local trivializability in the sense of tom Dieck (Def. \ref{tomDieckEquivariantLocalTrivialization}).

\end{prop}

\begin{proof}
  It is sufficient to see that Lashof's local model bundles in (eq:LashofLocalTrivalityHomomorphism) are examples of tom Dieck's local model bundles in (eq:tomDieckLocalTrivalityHomomorphism).

So let $H \subset G$ be a [[closed subgroup]], $S \subset X$ be an $H$-[[slice theorem|slice]] and

\[
  \label{LashofActionOfHOnGammaInProofLashofImpliestoDieck}
  \phi \colon H \to \Gamma
\]

be the [[homomorphism]] through which $H$ acts on $\Gamma$ in Lashof's model for the equivariant bundle over the orbit of the slice.

$$
  \array{
    G \times_H
    (S \times \Gamma)
    \\
    \big\downarrow
    \\
    G \times_H S
    \,.
  }
$$

By Prop. \ref{EquivariantPrincipalBundlesOverCosetSpacesAreSemidirectProductCosetsSpaceBundles} and using that $\alpha = 1$ we obtain from $\phi$ an equivariant principal bundle over $G/H$ by taking the subgroup ${\widehat H}$ in (eq:EquivariantBundleOfCosetSpaces) to be the [[graph of a function|graph]] of $\phi$ (eq:LashofActionOfHOnGammaInProofLashofImpliestoDieck)

$$
  {\widehat H}
  \;\coloneqq\;
  graph(\phi)
  \;\subset\;
  \Gamma \times G
  \,.
$$

Observing that the [[coset space]] by a subgroup of this form coincindes with the quotient by the [[diagonal action]]

$$
  (\Gamma \times G)/ graph(\phi)
  \;\simeq\;
  G \times_H \Gamma
  \,,
$$

makes it evident that tom Dieck's condition (eq:tomDieckLocalTrivalityHomomorphism) is satisfied for Lashof's local model in that the following square is manifestly a [[pullback]]:

$$
  \array{
    G
    \times_H
    (S \times \Gamma)
    &
    \longrightarrow
    &
    G \times_H \Gamma
    &
    =
    &
    (\Gamma \times G)
    \big/
    ( graph(H \overset{\phi}{\to} \Gamma) )
    \\
    \big\downarrow
    &&
    \big\downarrow
    &
    \swarrow
    \\
    G \times_H S
    &
    \underoverset
      {}
      {}
      {\longrightarrow}
    &
    G \times_H H
  }
$$

\end{proof}

(...)

\begin{remark}\label{SufficientConditionForEquivariantEnhancementOfLocalTrivialization}
  **(sufficient condition for plain local trivialization to have equivariant enhancement)**
  Sufficient conditions for existence of plain local trivialization to imply a $G$-equivariant local trivialization, and hence for the inclusion in Prop. \ref{PrincipalBundlesInternalToGSpacesEquivalentTotomDieckBundles} to be an actual [[equivalence of categories]], are:

1. $G$ is a [[finite group]]; or

1. $X$ is a [[locally compact topological space|locally compact]] [[separable metric space]] of [[finite number|finite]] [[dimension of a separable metric space|dimension]] and with a [[finite number]] of $G$-[[orbit types]]; or

1. $\Gamma$ is a [[locally compact topological space|locally compact]] [[separable metric space]] such that for every $G$-[[orbit type]] $G/H \subset X$ the [[fixed locus]] $\Gamma^H$ is an [[absolute neighbourhood retract]] (such as a [[finite number|finite]]-[[dimension of a manifold|dimensional]] [[topological manifold]] or a [[finite number|finite]]-[[dimension of a cell complex|dimensional]] and [[locally finite CW-complex|locally finite]] [[CW-complex]], by the discussion [there](absolute+retract#Examples)).

Because ([Atiyah 66, p. 374](KR+cohomology+theory#Atiyah66)) then
for every plain local trivialization around any [[orbit]] the [[equivariant Tietze extension theorem]] implies the existence of an [[equivariant function]] to $\Gamma$ on an [[open neighbourhood]] of that orbit, which thus constitutes a $G$-equivariant local trivialization.

\end{remark}

\linebreak

### Universal equivariant principal bundles
 {#UniversalEquivariantPrincipalBundles}

\begin{defn}\label{calBGammaForDiscreteG}
**(Murayama-Shimakawa groupoid for discrete $G$)** \linebreak
  For $G$ a [[discrete group]] and $(\Gamma,\alpha)$ a $G$-[[equivariant group|equivariant]] [[topological group]] (hence a [[topological group]] equipped with an [[action]] $\alpha$ of $G$ by continuous group [[automorphisms]]), consider the [[topological groupoid]] which is the [[functor groupoid]] from the [[pair groupoid]] of $G$ to the [[delooping groupoid]] of $\Gamma$:

\[
  \label{HomGroupoidFromPairGroupoidOfGToDeloopingGroupoidOfGamma}
  \mathcal{B}\Gamma
  \;\coloneqq\;
  Groupoids(TopSpaces)
  \left(
    G \times G
      \underoverset{pr_2}{pr_1}{\rightrightarrows}
    G,
    \;
    \Gamma \rightrightarrows \ast
  \right)
\]

but regarded as an [[internal groupoid]] in [[topological G-spaces]]

$$
  \mathcal{B}\Gamma
  \;\in\;
  Groupoids
  \big(
    G Actions(TopologicalSpaces)
  \big)
$$

by equipping it with the $G$-[[group action|action]] given on [[functors]] $F$ and [[natural transformations]] $\eta$ in (eq:HomGroupoidFromPairGroupoidOfGToDeloopingGroupoidOfGamma) by:

$$
  (g \cdot F)
  (g_1, g_2)
  \;\coloneqq\;
  \alpha(g)
  \big(
    F(g_1 g, g_2 g)
  \big)
  \,,
  {\phantom{AAA}}
  (g \cdot \eta)
  (g_1)
  \;\coloneqq\;
  \alpha(g)(\eta(g_1))
  \,.
$$

\end{defn}

> add word on choice of topology on the mapping spaces, see [Murayama-Shimakawa 95, p. 1293 (5 of 7)](#MurayamaShimakawa95)

\begin{remark}\label{FixedPointsOfEquivariantClassifyingSpaceAreHomotopyFixedPoints}
  This means that for any $H \subset G$ the $H$-[[fixed point space]]
 of $\mathcal{B}\Gamma$ is the $H$-[[homotopy fixed point]]-space of $B \Gamma$ ([Thomason 83, (3.2)](homotopy+fixed+point#Thomason83)).
\end{remark}

\begin{prop}\label{RealizationOfMurayamaShimakawaGroupoidForDiscreteGIsClassifyingSpace}
  The [[fat geometric realization]] of the [[nerve]] of the $G$-topological groupoid from Def. \ref{HomGroupoidFromPairGroupoidOfGToDeloopingGroupoidOfGamma}
$$
  \left\Vert
    \mathcal{B}\Gamma
  \right\Vert
  \;\;
  \in
  G Actions(TopSpaces)
$$
is a [[classifying space]] for $G$-equivariant $(\Gamma,\alpha)$-principal bundles.
\end{prop}
\begin{proof}
  This is Theorem 3.1 in [Murayama-Shimakawa 95](#MurayamaShimakawa95),
  using the remark on the bottom of p. 1294 (6 of 7) that for discrete group $G$ the construction in Theorem 3.1 may be simplified.

For $G$ discrete and $\Gamma$ [[discrete group|discrete]] or [[compact Lie group|compact Lie]] the same statement appears as [Guillou, May & Merling 17, Thm. 3.11](#GuillouMayMerling17). Notice that  Scholium 3.12 there doubts that [Murayama-Shimakawa 95](#MurayamaShimakawa95)'s result holds for non-discrete $G$.
\end{proof}

\begin{prop}\label{FixedLociOfcalBGammaForDiscreteG}
  Let $H \subset G$ any [[subgroup]], there is an [[equivalence]] of [[topological groupoids]] between the $H$-[[fixed locus]] of the $G$-groupoid from Def. \ref{calBGammaForDiscreteG} and

1. for trivial $\alpha$: the [[functor groupoid]] between the [[delooping groupoids]] of $G$ and $\Gamma$:

   $$
     \big(
       \mathcal{B}\Gamma
     \big)^H
     \;\;
     \simeq
     \;\;
     Groupoids(TopSpaces)
     \big(
       H \rightrightarrows \ast
       \,,
       \Gamma \rightrightarrows \ast
     \big)
   $$

1. for general $\alpha$: the [[action groupoid]] of the [[conjugation action]] of $\Gamma \overset{\gamma \mapsto (\gamma,e)}{\hookrightarrow} \Gamma \rtimes_\alpha G$ on [[group homomorphisms]] $\phi \colon G \to \Gamma \rtimes_\alpha G$ which are [[sections]] ($pr_2 \circ \phi = id_G$) of $pr_2 \colon \Gamma \rtimes_\alpha G \to G$:

   $$
     \big(
       \mathcal{B}\Gamma
     \big)^H
     \;\;
     \simeq
     \;\;
     \Big(
       Groups(TopSpaces)_{/G}
       \big(
         H, \, \Gamma \rtimes_\alpha G
       \big)
     \Big)
     \sslash
     \Gamma
     \,.
   $$

\end{prop}
\begin{proof}
  The first statement is the evident specialization of the second. It may help to go through the proof first in this special case of trivial $\alpha$, as the presence of $\alpha$ can tend to obscure the simple logic behind it.

  First consider the case $H = G$.
  Here the point to notice is that $G$-invariance of a functor
  $
    F
    \colon
    G \times G
    \to
    \Gamma
  $
  means that its value depends only on the difference of its arguments
  $$
    F(g_1, g_2) = F'( g_1 g_2^{-1} )
  $$
  and functoriality then means equivalently that $g \mapsto (F'(g), g)$
  is a group homomorphism to $\Gamma \rtimes_\alpha$.

  Similarly, $G$-invariance of a naturaltrans formation
  $\eta \colon G \to \Gamma$ means that its components are fixed by
  its values on $e \in G$ via

  $$
    \eta( g^{-1} ) \;=\; \alpha(g)(\eta(e))
    \,.
  $$

  With this, the naturality square for $\eta$ commutes precisely
  if two sections $g \mapsto (F'(g), g)$, as above,
  are related by conjugation with $\gamma_e \in \Gamma \rtimes_\alpha G$.

  This proves the equivalence in the case $H = G$.

  Next to see the claim for general $H \subset G$, notice that
  restriction along
  $(H \times H \rightrightarrows H) \hookrightarrow (G \times G \rightrightarrows G)$ followed by the above equivalence (now for $G$ replaced by $H$) gives a canonical comparison functor $p$. We claim
that a homotopy-inverse is given by the evident inclusion functor $i$ that fills up unspecified data by neutral elements. It is clear that $p \circ i = id$ and one checks that there is a homotopy $(id \Rightarrow i \circ p )$ (which is straightforward if one chooses good diagrammatic notation...).

\end{proof}

\begin{remark}
  To the extent that passage to fixed loci commutes with
  realization (this would be guaranteed if we could use ordinary
  geometric realization in Prop. \ref{RealizationOfMurayamaShimakawaGroupoidForDiscreteGIsClassifyingSpace} which works as soon as $\Gamma$ is compact Lie),
  Prop. \ref{FixedLociOfcalBGammaForDiscreteG} immediately implies
  the behaviour of equivariant classifying spaces under
  fixed loci according to [Lashof 82, Thm. 2.17](#Lashof82) and [Lashof & May 86, Thm. 10](#LashofMay86).

  Related discussion is in [Guillou, May & Merling 17, pp. 15](#GuillouMayMerling17).
\end{remark}

(...)

\begin{proposition}
\label{EquivariantClassifyingSpaceForAbelianCompactLieStructureGroup}
  If $\Gamma$ is a [[compact Lie group]] which is also [[abelian group|abelian]] (hence a [[direct product group]] of an [[n-torus]] with a abelian [[finite group]]) and regarded as carrying trivial $G$-action, then a classifying [[G-space]] for $G$-equivariant $\Gamma$-principal bundles is given by $Map(E G ,\, B\Gamma)$.
\end{proposition}
This is [Lashof, May & Segal 1983, Lem. 1 with Thm. 2](#LashofMaySegal83),
re-stated more explicitly in [May 1990, Thm. 10](#May90).

This implies that for abelian compact Lie structure groups $\Gamma$, the isomorphism classes of $G$-equivariant $\Gamma$-principal bundles over some [[G-CW complex]] $X$ have the same classification as plain $\Gamma$-[[principal bundles]] over its [[Borel construction]] $X_G \,\coloneqq\, (E G \times X)/G$:

$$
  \begin{aligned}
    \big(
       G Equ \Gamma PrnBdl_X
    \big)_{/\sim}
    &
    \;\simeq\;
    \tau_0
    \,
    Map\big( X ,\, \Map(EG ,\, B \Gamma) \big)^G
    \\
    &
    \;\simeq\;
    \tau_0
    \,
    Map\big( E G \times X ,\, B \Gamma \big)^G
    \\
    &
    \;\simeq\;
    \tau_0
    \,
    Map\big( (E G \times X)/G ,\, B \Gamma \big)
    \\
    &
    \;\simeq\;
    \big(
      \Gamma PrnBdl_{X_G}
    \big)_{/\sim}
    \,.
  \end{aligned}
$$
In the form of the resulting composite bijection, this statement appears as [May 1990, Thm. 3](#May90).

\linebreak

## Examples

* [[equivariant tautological line bundle]]

* [[universal equivariant PU(H)-bundle]]

* in [[arithmetic geometry]]: [[shtuka]]

## Related concepts

* [[twisted bundle]]

* [[equivariant differential topology]]

* [[equivariant cohomology]]

* [[equivariant connection]]

* [[equivariant K-theory]]

## References
 {#References}

### Equivariant fiber bundles
 {#ReferencesEquivariantFiberBundles}

On general (topological) equivariant [[fiber bundles]]/[[principal bundles]]:

Precursor discussion:

* T. E. Stewart, *Lifting Group Actions in Fibre Bundles*, Annals of Mathematics Second Series, Vol. 74, No. 1 (1961), pp. 192-198 ([jstor:1970310](https://www.jstor.org/stable/1970310))

  > (for [[structure group]] commuting with the [[equivariance group]], but noticing already that a [[normal subgroup]]-extension could be considered, more generally)

The original definition:

* {#tomDieck69} [[Tammo tom Dieck]], _Faserbündel mit Gruppenoperation_, Arch. Math 20, 136–143 (1969) ([doi:10.1007/BF01899003](https://doi.org/10.1007/BF01899003))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

* {#Bierstone78} [[Edward Bierstone]], _The equivariant covering homotopy property for differentiable $G$-fibre bundles_, J. Differential Geom. 8(4): 615-622 (1973) ([doi:10.4310/jdg/1214431963](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-8/issue-4/The-equivariant-covering-homotopy-property-for-differentiable-G-fibre-bundles/10.4310/jdg/1214431963.full))

  > (for [[structure group]] commuting with the [[equivariance group]], but with [[equivariant differential topology|differentiable structure]])

* [[Goro Nishida]], Section 2 of: _The transfer homomorphism in equivariant generalized cohomology theories_, J. Math. Kyoto Univ. 18(3): 435-451 (1978) ([doi:10.1215/kjm/1250522505](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-18/issue-3/The-transfer-homomorphism-in-equivariant-generalized-cohomology-theories/10.1215/kjm/1250522505.full))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

* {#LashofRothenberg78} [[Richard Lashof]], [[Melvin Rothenberg]], Section 1 of: _$G$-Smoothing theory_, p. 211-266 in: _Algebraic and Geometric Topology_, Proceedings of Symposia in Pure Mathematics, Vol. XXXII, Part 1, American Mathematical Society 1978 ([doi:10.1090/pspum/032.1](https://doi.org/10.1090/pspum/032.1), [[LashofRothenbergGSmoothingTheory.pdf:file]])

  > (for [[structure group]] commuting with the [[equivariance group]], but with [[equivariant differential topology|differentiable structure]])

* {#Lashof82} [[Richard Lashof]], _Equivariant bundles_, Illinois J. Math. 26(2): 257-271, 1982 ([doi:10.1215/ijm/1256046796](https://doi.org/10.1215/ijm/1256046796))

  > (for [[structure group]] commuting with the [[equivariance group]])

* {#LashofMay86} [[Richard Lashof]], [[Peter May]], _Generalized equivariant bundles_, Bull. Soc. Math. Belg. Sér. A 38, 265-271, 1986 ([pdf](http://math.uchicago.edu/~may/PAPERS/55.pdf), [[LashofMayGeneralizedEquivariantBundles.pdf:file]])

  > (for [[structure group]] [[group extension|extending]] the [[equivariance group]])

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger, Section IV.1 of: _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics Vol.1213. 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778))

  > (for [[structure group]] [[group extension|extending]] the [[equivariance group]])

* {#tomDieck87} [[Tammo tom Dieck]], Section I.8 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

With [[abelian group|abelian]] [[structure group]]:

* {#LashofMaySegal83} [[Richard Lashof]], [[Peter May]], [[Graeme Segal]], _Equivariant bundles with abelian structural group_, Contemporary Mathematics,  Volume 19, 1983 ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019), [pdf](http://www.math.uchicago.edu/~may/PAPERS/45.pdf), [[LashofMaySegalEquivariantBundles.pdf:file]])

  > (for [[structure group]] commuting with the [[equivariance group]])

* [[Andrzej Kozlowski]], _Equivariant Bundles and Cohomology_, Transactions of the American Mathematical Society, Vol. 296, No. 1 (Jul., 1986), pp. 181-190 ([jstor:2000568](https://www.jstor.org/stable/2000568))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

More on [[equivariant principal bundles]] and their [[classifying spaces]]/[[universal principal bundles]]:

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Théorie de l'homotopie, Astérisque, no. 191 (1990), 15 p. ([numdam:AST_1990__191__239_0](http://www.numdam.org/item/?id=AST_1990__191__239_0))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

* {#MurayamaShimakawa95} [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9), [jstor:2160733](https://www.jstor.org/stable/2160733))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

* [[Wolfgang Lück]], *Survey on Classifying Spaces for Families of Subgroups*,  In: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects* Progress in Mathematics, **248** Birkhäuser (2005) ([arXiv:math/0312378](https://arxiv.org/abs/math/0312378), [doi:10.1007/3-7643-7447-0_7](https://doi.org/10.1007/3-7643-7447-0_7))

* {#UribeLück14} [[Bernardo Uribe]], [[Wolfgang Lück]], *Equivariant principal bundles and their classifying spaces*, Algebr. Geom. Topol. **14** (2014) 1925-1995 [[arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925)]

  > (for [[structure group]] commuting with the [[equivariance group]])

* {#Rezk14} [[Charles Rezk]], Sec. 2.3 in: *[[Global Homotopy Theory and Cohesion]]*, 2014 ([pdf](http://www.math.uiuc.edu/~rezk/global-cohesion.pdf), [[Rezk14.pdf:file]])

  > (for [[structure group]] commuting with the [[equivariance group]],
    and in relation to the [[global orbit category]] with an eye towards [[equivariant homotopy theory]])

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]], _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

  > (for [[structure group]] [[semidirect product group|split]]-[[group extension|extending]] the [[equivariance group]])

* {#Rezk16} [[Charles Rezk]], *Classifying spaces for 1-truncated compact Lie groups*, Algebr. Geom. Topol. **18** (2018) 525-546 ([arXiv:1608.02999](https://arxiv.org/abs/1608.02999))

  > (for [[structure group]] a [[1-truncated]] [[compact Lie group]] commuting with the [[equivariance group]])

Review and examples over the [[2-sphere]]:

* [[Eyup Yalcinkaya]], *Equivariant Principal Bundles over the 2-Sphere* ([arXiv:1902.06293](https://arxiv.org/abs/1902.06293))

See also:

* {#Zou20} [[Foling Zou]], _Notes on equivariant bundles_ ([arXiv:2008.01268](https://arxiv.org/abs/2008.01268))

Discussion in the context of [[principal infinity-bundles|principal $\infty$-bundles]]:

* {#SatiSchreiber21} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]* ([arXiv:2112.13654](https://arxiv.org/abs/2112.13654))

### Equivariant vector bundles

On equivariant [[vector bundles]]:

* [tom Dieck 87, Section I.9](#tomDieck87)

In the context of [[equivariant K-theory]]:

* {#Segal68} [[Graeme Segal]], Section 1 of: _Equivariant K-theory_, Inst. Hautes Etudes Sci. Publ. Math.  No. 34 (1968) p. 129-151 ([numdam:PMIHES_1968__34__129_0](http://www.numdam.org/item/PMIHES_1968__34__129_0))

* {#HJJS08} [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Section 13 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

In a context of [[equivariant differential topology]]:

* {#Wasserman69} [[Arthur Wasserman]], section 2 of: _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 (<a href="https://doi.org/10.1016/0040-9383(69)90005-6">doi:10.1016/0040-9383(69)90005-6</a>[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

In the context of [[orientation in generalized cohomology]] generalized to [[equivariant complex oriented cohomology theory]]:

* [[Steven Costenoble]], [[Peter May]], [[Stefan Waner]], _Equivariant orientation theory_, Homology Homotopy Appl.
Volume 3, Number 2 (2001), 265-339 ([euclid:hha/1139840256](https://projecteuclid.org/euclid.hha/1139840256))

### As twists for twisted equivariant K-theory

On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

* [Uribe & Lück 2014, Sec. 15](#UribeLück14)

### For equivariant elliptic cohomology

In a context of [[equivariant elliptic cohomology]]:

* [[Matthew Ando]], [[John Greenlees]], Part I of: _Circle-equivariant classifying spaces and the rational equivariant sigma genus_, Math. Z. 269, 1021–1104 (2011) ([arXiv:0705.2687](https://arxiv.org/abs/0705.2687), [doi:10.1007/s00209-010-0773-7](https://doi.org/10.1007/s00209-010-0773-7))

### For gauge theory on orbifolds

The general notion of equivariant bundles from [tom Dieck 69](#tomDieck69) (with [[action]] of the [[semidirect product]] of the [[gauge group]] with the [[equivariance group]]) gets a brief mentioning in

* {#Horava96} [[Petr Hořava]], Section 5.3 of: _Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions_, J. Geom. Phys. 21:1-33, 1996 ([arXiv:hep-th/9404101](https://arxiv.org/abs/hep-th/9404101))

in a context of [[Chern-Simons theory]] on [[orbifolds]].

[[!redirects equivariant bundles]]

[[!redirects equivariant principal bundle]]
[[!redirects equivariant principal bundles]]

[[!redirects equivariant fiber bundle]]
[[!redirects equivariant fiber bundles]]

[[!redirects equivariant vector bundle]]
[[!redirects equivariant vector bundles]]

[[!redirects equivariant classifying space]]
[[!redirects equivariant classifying spaces]]

[[!redirects universal equivariant bundle]]
[[!redirects universal equivariant bundles]]

[[!redirects universal equivariant principal bundle]]
[[!redirects universal equivariant principal bundles]]


equivariant chain complex

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## Definition

Let $G$ be a [[finite group]].

An _$G$-equivariant chain complex_ is a [[functor]]

$$
  G Orbits \longrightarrow ChainComplexes
$$

from the [[orbit category]] of $G$ to a [[category of chain complexes]].

## Related concepts

* [[model structure on equivariant chain complexes]]

## References

* {#Triantafillou82} [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects equivariant chain complexes]]

[[!redirects equivariant cochain complex]]
[[!redirects equivariant cochain complexes]]


equivariant Chern character

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## Idea

The _equivariant Chern character_ is the generalization of the [[Chern character]] from [[topological K-theory]] to [[equivariant topological K-theory]], equivalently the specialization of the [[equivariant Chern-Dold character]] for [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[equivariant cohomology]] to [[equivariant topological K-theory]].

The equivariant Chern character has a variety of different but equivalent concrete incarnations, depending on the choice of presentation of the [[rational equivariant K-theory]] that it takes values in:

[[!include incarnations of rational equivariant topological K-theory -- table ]]

## Related concepts

* [[twisted Chern character]], [[twisted equivariant Chern character]]

* [[equivariant Chern-Dold character]]

* [[equivariant rationalization]]

* [[delocalized equivariant cohomology]]

* [[Chen-Ruan cohomology]]

## References

* [[Jolanta Słomińska]], *On the Equivariant Chern homomorphism*, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys, Vol. 24 (1976), 909-913.

* [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

* [[Michel Duflo]], [[Michèle Vergne]], *Cohomologie équivariante et descente*, Astérisque, no. 215 (1993) ([numdam:AST_1993__215__5_0](http://www.numdam.org/book-part/AST_1993__215__5_0))

* {#LueckOliver01} [[Wolfgang Lück]], [[Bob Oliver]], _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_, Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15), [pdf](https://www.math.univ-paris13.fr/~bobol/lo2.pdf))

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], Section 5 of: _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))

* [[Varghese Mathai]], [[Danny Stevenson]], *Chern character in twisted K-theory: equivariant and holomorphic cases*, Commun. Math. Phys. **236** 161-186, 2003 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7))

* {#Moerdijk02} [[Ieke Moerdijk]], p. 18 of: _Orbifolds as Groupoids: an Introduction_, in: [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]] (eds.) _[[Orbifolds in Mathematics and Physics]]_, Contemporary Math 310 , AMS (2002), 205–222 ([arXiv:math.DG/0203100](http://arxiv.org/abs/math.DG/0203100))

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], Theorem 6.1 in: _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

* [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], _Inertia and delocalized twisted cohomology_, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) ([arXiv:math/0609576](https://arxiv.org/abs/math/0609576), [doi:10.4310/HHA.2008.v10.n1.a6](https://dx.doi.org/10.4310/HHA.2008.v10.n1.a6))

* Pierre Albin, [[Richard Melrose]], _Delocalized equivariant cohomology and resolution_ ([arXiv:1012.5766](https://arxiv.org/abs/1012.5766))

Review in:

* [[Ulrich Bunke]], Section 3.1 in: _Orbifold index and equivariant K-homology_, Math. Ann. 339, 175–194 (2007) ([arXiv:math/0701768](https://arxiv.org/abs/math/0701768), [doi:10.1007/s00208-007-0111-5](https://doi.org/10.1007/s00208-007-0111-5))

* {#Stefanich} German Stefanich, _Chern Character in Twisted and Equivariant K-Theory_ ([pdf](https://math.berkeley.edu/~germans/Chern2.pdf))


equivariant Chern-Dold character

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## Idea

The _equivariant Chern-Dold character_ is the generalization of the _[[Chern-Dold character]]_ from [[Whitehead-generalized cohomology]] to [[equivariant cohomology]], and the generalization of the [[equivariant Chern character]] on [[equivariant K-theory]] to other equivariant generalized cohomology theories.

## Related concepts

* [[equivariant Chern character]]

* [[equivariant rationalization]]

* [[Chern-Dold character]]

## References

* [[Wolfgang Lück]], _Chern characters for proper equivariant homology theories and applications to K- and L-theory_, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 ([doi:10.1515/crll.2002.015](https://doi.org/10.1515/crll.2002.015), [pdf](https://www.him.uni-bonn.de/lueck/data/eh.pdf))

* [[Wolfgang Lück]], _Equivariant Cohomological Chern Characters_, International Journal of Algebra and ComputationVol. 15, No. 05n06, pp. 1025-1052 (2005) ([arXiv:math/0401047](https://arxiv.org/abs/math/0401047), [doi:10.1142/S0218196705002773](https://doi.org/10.1142/S0218196705002773))

* [[Wolfgang Lück]], _Equivariant Chern characters_, 2006 ([pdf](https://www.him.uni-bonn.de/lueck/data/goettingen_equi_Chern061130trans.pdf), [[LuckEquivariantChernCharacters.pdf:file]])

[[!redirects equivariant Chern-Dold characters]]


equivariant cobordism theory

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## Idea

The generalization of [[cobordism theorem]] to [[equivariant homotopy theory]]/[[equivariant differential topology]].

## Theorems

* [[equivariant Pontrjagin theorem]]

* (...)

## Related concepts

* [[Real cobordism cohomology]]

* [[differential cobordism cohomology]]

* [[equivariant differential topology]]

* [[equivariant homotopy theory]], [[equivariant stable homotopy theory]]

* [[equivariant cohomology]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

For more see at the references at *[[equivariant bordism homology theory]]*.

* Po Hu, [[Igor Kriz]], Yunze Lu, *Coefficients of the $\Sigma_3$-equivariant complex cobordism ring* [[arXiv:2109.00991](https://arxiv.org/abs/2109.00991)]

Review in [[global equivariant homotopy theory]]:

* Ningchuan Zhang, *Equivariant cobordism* (2019) [[pdf](https://iwoat.github.io/2019/notes/Lecture-14.pdf)]

[[!redirects equivariant bordism theory]]

[[!redirects equivariant cobordism]]
[[!redirects equivariant bordism]]

[[!redirects equivariant cobordism cohomology]]
[[!redirects equivariant cobordism cohomology theory]]