spherical space form

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _spherical space form_ is a [[quotient space]] $S^n/G$ of a [[round sphere|round]] [[Riemannian manifold|Riemannian]] [[n-sphere]] ($n \geq 2$) by a [[subgroup]] $G$ of its [[isometry group]], which [[action|acts]] [[free action|freely]] and [[properly discontinuous action|properly discontinuously]].

Equivalently, a spherical space form is a [[Riemannian manifold]] of [[constant function|constant]] [[positive number|positive]] [[sectional curvature]] (an [[elliptic geometry]]) which is [[connected topological space|connected]] and [[geodesically complete Riemannian manifold|geodesically complete]] (see e.g. [Gadhia 07, Lemma 5](#Gadhia07)).

The spherical space forms are one of three classical examples of _[[Clifford-Klein space forms]]_.

Their classification was raised as an open problem by [Killing 1891](#Killing1891) and a complete solution was finally compiled by ([Wolf 74](#Wolf74)) (reviewed in [Gadhia 07, section 2.2](#Gadhia07)). A re-proof of the classification is claimed in ([Allock 15](#Allock15)).

Notice that free group actions by [[isometries]] of [[n-spheres]] are a special case of more general free actions by any [[homeomorphisms]], see at *[[group actions on n-spheres]]* for more.

## Examples

### 7-dim spherical space forms with $Spin$-structure
 {#7DSphericalSpaceFormsWithSpinStructure}

For $n =7$ all the groups $G$ in [Wolf's classification](#Wolf74) [[action|act]] as [[subgroups]] of [[special orthogonal group|SO(8)]], the latter equipped with its defining action on $\mathbb{R}^8$ [[restriction|restricted]] to the action on the [[7-sphere]] $S^7 = S(\mathbb{R}^8) \subset \mathbb{R}^8$.

Therefore one may consider the [[lift]] $\widehat{G}$ of these [[subgroups]] to subgroups of the [[spin group]] $Spin(8) \to SO(8)$ through the [[double cover]]-projection. Such a lift corresponds to a choice of [[spin structure]] on the [[spherical space form]] $S^7/G$.  These Spin-lifts $\widehat{G}$, have been classified in ([Gadhia 07](#Gadhia07)).

Given any such lift $\widehat{G} \subset Spin(8)$, one may consider its [[action|action]] on the two [[irreducible representation|irreducible]] [[real spin representations]] $\mathbf{8}_\pm$ of $Spin(8)$. Write

$$
  N_\pm \;\coloneqq\; dim_{\mathbb{R}}\left( \left(\mathbf{8}_{\pm}\right)^{\widehat{G}} \right)
$$

for the [[dimension]] of the subspace of [[spinors]] that are [[fixed point|fixed]] by the action of $\widehat{G}$. For $\widehat{G}$ non-[[trivial group|trivial]], we have

$$
  (N_+, N_-) = (N,0)
  \phantom{AA} \text{or} \phantom{AA}
  (N_+, N_-) = (0,N)
$$

and hence up to a choice of [[orientation]] there is a unique

$$
  N \in \{0,1,\cdots, 8\}
$$

associated with each 7-dimensional [[spherical space form]] equipped with [[spin structure]].

Hence this allows to stratify [Wolf's classification](#Wolf74) of 7-dimensional spherical space forms, first into the cases that do and that do not admit any [[spin structure]], and then the former further into the dimension $N$ of the space of [[Killing spinors]] that they carry.

In the case  $N \geq 4$
It turns out that the resulting sub-classification demands $\widehat{G}$ to be a [[finite subgroup of SU(2)]]; hence this is an [[ADE classification]] ([MFFGME 09, Sections 3-7](#MFFGME09)):

* $N = 8$: here $\widehat{G} = \mathbb{Z}/2$, the [[cyclic group of order 2]];

* $N = 7$: does not occur;

* $N = 6$: here $\widehat{G} = \mathbb{Z}/k$ (for $k \gt 2)$, a [[cyclic group]];

* $N = 5$: here $\widehat{G} = $ a non-[[cyclic group|cyclic]] [[finite subgroup of SU(2)]], hence a [[binary dihedral group]] or the [[binary tetrahedral group]] or [[binary octahedral group]] or [[binary icosahedral group]], [[diagonal action|acting diagonally]] on $\mathbb{R}^8 \simeq \mathbb{H} \oplus \mathbb{H}$;

* $N = 4$: here $\widehat{G} = $ any [[finite subgroup of SU(2)]] _except_ the [[binary tetrahedral groups]], but acting via the [[diagonal action]] composed with a suitable non-trivial [[outer automorphism]] of $\widehat{G}$ on one of the two sides.

(In the last case, while there is one nontrivial outer automorphism of the binary tetrahedral group, its twisted action yields $N =5$, hence is equivalent to one of the previous cases ([MFFGME 09, section 7.3](#MFFGME09)).)

[[!include 7d spherical space forms -- table]]

This analysis applies to the classification of the [[near horizon geometry]] of smooth (i.e. non-[[orbifold]]) $\geq \tfrac{1}{2}$ [[BPS state|BPS]] [[black brane|black]] [[M2-brane]]-solutions of the [[equations of motion]] of [[11-dimensional supergravity]]:

These are the [[Cartesian product]] $AdS_4 \times (S^7/G)$ of 4-[[dimension|dimensional]] [[anti de Sitter spacetime]] with a 7-[[dimension|dimensional]] [[spherical space form]] $S^7/G$ with [[spin structure]] and $N \geq 4$, as above ([MFFGME 09](#MFFGME09)).

## Properties

### Concordances of higher gerbes over spherical space forms
 {#HigherGerbesOnSphericalSpaceForms}

We discuss the [[concordance]] [[infinity-groupoid|$\infty$-groupoid]] of $\Gamma$-[[principal bundles]] on [[spherical space forms]] $S^{n+2}/G$ (Def. \ref{ShapesOfMappingStacks} below)
in the case that the [[topological group]] $\Gamma$ is a [[braided infinity-group|braided]] [[homotopy n-type]] (Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms} below).

The claim of Theorem \ref{ComparisonMapOfConcordanceInfinityGroupoidsIsEquivalence} below is that this is equivalently the [[hom infinity-groupoid|hom $\infty$-groupoid]] of maps of [[classifying spaces]] $B G \to B \Gamma$.

In the special case that $\Gamma \,=\, $ [[PU(ℋ)]], this result recovers (Example \ref{EquivariantClassifyingSpaceOfPUHBundles} below) the [[equivariant homotopy groups]] of the [[equivariant classifying space]] for [[equivariant principal bundles|equivariant $PU(\mathcal{H})$-principal bundles]] (originally due to [Uribe et al., 2014](twisted+equivariant+K-theory#BEJU14)) -- a statement which falls out of the scope of most traditional theory of [[equivariant principal bundles]] since [[PU(ℋ)]] is not a [[compact Lie group]].

Conversely, Thm. \ref{ComparisonMapOfConcordanceInfinityGroupoidsIsEquivalence} generalizes this classification of equivariant $PU(\mathcal{H})$-principal bundles to any [[braided infinity-group|braided]] [[homotopy n-type|$n$-truncated]] [[structure group]], at the (small) cost of requiring that the [[equivariance group]] corresponds to some $\geq n+2$-dimensional [[spherical space form]]  -- in which case it is essentially an incarnation (via the proof of Prop. \ref{ConcordancesOnSphericalSpaceFormIsMapsOfClassifyingSpaces} below) of the [[smooth Oka principle]] over that spherical space form.

\linebreak

* *[Preliminaries](#Preliminaries)*

* *[The classification theorem](#TheClassificationTheorem)*

* *[Lemmas and proofs](#LemmasAndProofs)*

\linebreak

#### Preliminaries
 {#Preliminaries}

Concretely, we consider the following situation:

\begin{definition}
\label{AssumptionForHigherGerbesOnSphericalSpaceForms}
**Assumption.**
\linebreak
In the following, let

1. $G$ be a [[finite group]] satisfying the following equivalent conditions (equivalent by the [Madsen-Thomas-Wall theorem](group+actions+on+spheres#MadsenThomasWallTheorem)):

   1. for all [[prime numbers]] $p$, if $H \subset G$ is a [[subgroup]] of [[order of a group|order]] $2 p$ or $p^2$, then $H$ is [[isomorphism|isomorphic]] to a [[cyclic group]];

   1. $G$ has a [[G-space|continuous]] [[free action]] on the topological [[d-sphere]] for some $d \in \mathbb{N}$;

   1. $G$ has a [[G-manifold|smooth]] [[free action]] on the [[d-sphere]] for some $d \in \mathbb{N}$ and for some [[smooth structure]] (possibly [[exotic smooth structure|exotic]]);

   1. for each $n \in \mathbb{N}$ there exists $d \geq  n + 2$ such that $G$ has a [[G-manifold|smooth]] [[free action]] on the [[d-sphere]] equipped with some [[smooth structure]] (possibly [[exotic smooth structure|exotic]]):

      \[
        \label{ExistenceOfHighDimensionalSmoothFreeGActionsOnSpheres}
        \underset{n \in \mathbb{N}}{\forall}
        \;\;\;\;
       \underset{ d \geq n+2 }{\exists}
        \;\;\;\;
        \underset{
           {diff \in \mathclap{\phantom{\vert^{\vert}}}}
           \atop
           {SmthStruc(S^d)}
        }{\exists}
        \;\;\;\;
        \underset{
          {
            \rho \curvearrowright S^{d}_{diff}
            \mathclap{\phantom{\vert^{\vert}}}
          }
          \atop
          {
            \in G Act(SmthMfd)
          }
        }{\exists}
        \;\;\;\;
        \rho\;\text{is free}
        \,;
      \]

1. $\Gamma \,\in\, Grp(TopSp)$ be a [[topological group]] whose [[underlying]] [[homotopy type]] [[infinity-group|$\infty$-group]] $Shp(\Gamma) \,\in\, Grp\big( Grpd_\infty \big)$ is

   1. [[braided infinity-group|braided]], in that its [[delooping]] still has [[infinity-group|$\infty$-group]]-structure itself

      \[
        \label{DeloopingOfShapeIsInfinityGroup}
        B Shp(\Gamma) \,\in\, Grp\big( Grpd_\infty \big)
        \,,
      \]

   1. [[n-truncated object of an (infinity,1)-category|truncated]], hence a [[homotopy n-type]] for some $n \in \mathbb{N}$:

   \[
     \label{GammaIsNTruncated}
     \underset{ n \in \mathbb{N} }{\exists}
     \;\;\;\;\;\;
     \tau_n Shp(\Gamma) \,\simeq\, Shp(\Gamma)
     \;\;\;
     \in
     \;
     Grp_\infty
     \,.
   \]

Given $n$ as in (eq:GammaIsNTruncated), possibly increased a little if necessary, we have by (eq:ExistenceOfHighDimensionalSmoothFreeGActionsOnSpheres), a smooth [[free action|free]] [[group action on n-spheres|group action on $S^{n+2}$]], whose [[topological quotient space|quotient space]] is a [[smooth manifold]]:

\[
  \label{CoprojectionInDiscussionOfGerbesOnSphericalSpaceForms}
  S^{n+2}
  \overset{\phantom{-} q \phantom{-}}{\twoheadrightarrow}
  S^{n+2}/G
  \;\;\;
  \in
  \;
  SmthMfd
  \,.
\]
\end{definition}

\begin{example}
The archetypical examples of [[topological groups]] which satisfy Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms} are:

1. The [[circle group]], $\Gamma = $ [[U(1)]], whose underlying homotopy type is that of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},1)$, which is [[homotopy n-type|n-truncated]] for $n \geq 1$
and which is not just [[braided infinity-group|braided]] but even [[abelian infinity-group|abelian]] (i.e. [[E-infinity space|$E_\infty$]]);
with [[delooping]] $K(\mathbb{Z},2)$;

1. the [[projective unitary group]] on a [[complex vector space|complex]] [[separable Hilbert space]],  $\Gamma \,=\,$ [[PU(ℋ)]], whose underlying homotopy type is that of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$,  which is [[homotopy n-type|n-truncated]] for $n \geq 2$,
which is not just [[braided infinity-group|braided]] but even [[abelian infinity-group|abelian]] (i.e. [[E-infinity space|$E_\infty$]]) with [[delooping]]  $K(\mathbb{Z},3)$.

Hence for both of these [[structure groups]], the 7-dimensional spherical space forms discussed [above](#7DSphericalSpaceFormsWithSpinStructure) satisfy Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms}.
\end{example}

\begin{definition}
**([[topological groupoids]] and their underlying [[D-topological space|D-]][[topological stacks]])**
\linebreak
We will write:

\[
  \label{TheTopologicalDeloopingGroupoidOfGamma}
  \array{
    Grp(TopSp)
    &
    \xrightarrow{\; Dtoplg \;}
    &
    Grp(DiffSp)
    &
    \xrightarrow{\;}
    &
    Grpd(SmthGrp_{\infty})
    &
    \xrightarrow{\;}
    &
    SmthGrp_\infty
    \\
    \Gamma \rightrightarrows \ast
    &&
    &\mapsto&
    &&
    \mathbf{B}\Gamma
    \\
    G \rightrightarrows \ast
    &&
    &\mapsto&
    &&
    \mathbf{B}G
    \\
    S^{n+2} \times G \rightrightarrows S^{n+2}
    &&
    &\mapsto&
    &&
    S^{n+2}/G
  }
\]

for

1. the [[topological groupoids]] which are

   1. the [[delooping groupoids]] of $G$ and of $\Gamma$, respectively,

   1. the [[action groupoid]] $S^{n+2} \times G \rightrightarrows S^{n+2}$ of the given $G$-[[group action|action]] on $S^{n+2}$

1. their associated [[D-topological space|D-]][[topological stacks]], regarded as objects in [[smooth infinity-groupoid|$SmthGrpd_\infty$]].

\end{definition}
Notice, as indicated in the last row of (eq:TheTopologicalDeloopingGroupoidOfGamma), that the [[smooth stack]] correspinding to the [[action groupoid]] of $G$ acting on $S^{n+2}$ is [[equivalence in an (infinity,1)-category|equivalent]] to the quotient [[smooth manifold]] $S^{n+2}/G$ itself, due to the condiiton that the $G$-action is [[free action|free]].

\begin{lemma}
\label{CanonicalCechGroupoidOfGammaPrincipalBundlesOnSphericalSpaceForm}
**(canonical [[Cech groupoid]] of $\Gamma$-[[principal bundles]] on [[spherical space form]])**
\linebreak
  Under the truncation condition (eq:GammaIsNTruncated),
  the [[groupoid]] of $\Gamma$-[[principal bundles]]  over $S^{n+2}/G$ ([[internalization|internal to]] [[TopSp]]) is [[equivalence of groupoids|equivalent]] to the groupoid of [[topological functors]] out of the [[action groupoid]] into the [[delooping groupoid]] (eq:TheTopologicalDeloopingGroupoidOfGamma), with [[continuous functions|continuous]] [[natural transformations]] between them:
$$
  \Gamma PrnBdl(TopSp)_{S^{n+2}/G}
  \;\;
  \simeq
  \;\;
  TopFunc
  \big(
    ( S^{n+2} \times G \rightrightarrows S^{n+2} )
    ,\,
    ( \Gamma \rightrightarrows \ast )
  \big)
  \,.
$$
Moreover, the analogous statement holds for bundles over the [[product topological space]] $S^{n+2}/G \times \Delta^k$ with the [[topological k-simplex]] for all $k \in \mathbb{N}$:
$$
  \Gamma PrnBdl(TopSp)_{S^{n+2}/G \times \Delta^k}
  \;\;
  \simeq
  \;\;
  TopFunc
  \big(
    ( S^{n+2} \times G \rightrightarrows S^{n+2} )
    \times \Delta^k
    ,\,
    ( \Gamma \rightrightarrows \ast )
  \big)
  \,.
$$
\end{lemma}
\begin{proof}
The $n$-truncation condition (eq:GammaIsNTruncated) on $\Gamma$ implies that its [[classifying space]] $B \Gamma \,\simeq\, Shp( \left\vert \Gamma \rightrightarrows \ast \right\vert  )$ is an $(n+1)$-type

$$
  \tau_{n+1} B \Gamma \,\simeq\, B \Gamma
  \;\;\;
  \in
  \;
  Grp_\infty
$$

and hence, by [[classifying space|classifying theory]], that every $\Gamma$-principal bundle on $S^{n+2}$ is [[isomorphism|isomorphic]] to the [[trivial bundle]]:

\[
  \label{UniqueGerbesOnNSphere}
  \tau_0
  \big(
     \Gamma PrnBdl(TopSp)_{S^{n+2}}
  \big)
  \;\;
  \simeq
  \;\;
  \tau_0
  \,
  Maps
  \big(
    Shp(S^{n+2})
    ,\,
    B \Gamma
  \big)
  \;\;
  \simeq
  \;\;
  \ast
  \,.
\]

Therefore, every $\Gamma$-[[principal bundle]] on the spherical space form $S^n/G$ trivializes after being [[pullback bundle|pulled back]] along the coprojection $q$ (eq:CoprojectionInDiscussionOfGerbesOnSphericalSpaceForms). The corresponding [[Cech cohomology|Cech cocycle]] is a [[topological functor]]

$$
  \big(
    S^{n+2} \times G^{op} \rightrightarrows S^{n+2}
  \big)
  \xrightarrow{\;\;
    (c_1 \rightrightarrows \ast)
  \;\;}
  \big(
    \ast \times \Gamma^{op} \rightrightarrows \ast
  \big)
  =
  \big(
    \Gamma \rightrightarrows \ast
  \big)
$$

out of the [[action groupoid]] $S^{n+2} \times G^{op} \rightrightarrows X$ of $G$ acting on $S^n$ into the [[delooping groupoid]] $\Gamma \rightrightarrows \ast$:

\begin{tikzcd}
    {}
    \ar[d,-, dotted]
    &&
    {}
    \ar[d,-, dotted]
    \\
    S^{n+2} \times  G \times G
    \ar[rr, "{c_2}"{above}]
    \ar[d, shift left=12pt]
    \ar[d]
    \ar[d, shift right=12pt]
    &&
    \Gamma \times \Gamma
    \ar[d, shift left=12pt]
    \ar[d]
    \ar[d, shift right=12pt]
    \\
    S^{n+2} \times G
    \ar[rr, "c_1"{above}]
    \ar[d, shift left=6pt]
    \ar[d, shift right=6pt]
    \ar[u, shift right=6pt]
    \ar[u, shift left=6pt]
    &&
    \Gamma
    \ar[d, shift left=6pt]
    \ar[d, shift right=6pt]
    \ar[u, shift right=6pt]
    \ar[u, shift left=6pt]
    \\
    S^{n+2}
    \ar[rr, "{ }"{below, name=s, pos=.4}]
    \ar[d, ->>, "{ }"{right, name=t, pos=.7 }]
    \ar[u]
    &&
    \ast
    \ar[u]
    \ar[d, ->>]
    \\
    S^{n+2}/G
    \ar[rr, "c"{below}]
    &&
    \mathbf{B}\Gamma
    %
    \ar[
      from=s,
      to=t,
      Rightarrow,
      "\sim"{sloped, below}
    ]
\end{tikzcd}
With every $\Gamma$-principal bundle trivialized over the covering by $S^{n+2}$ this way, morphisms between principal bundles correspond to Cech [[coboundaries]] between these Cech cocycles, which are canonically identified with continuous [[natural transformations]] between these functors $(c_1 \rightrightarrows \ast)$.
\end{proof}

\begin{definition}
\label{ShapesOfMappingStacks}
**($\infty$-groupoids of concordances of principal bundles on spherical space forms)**
\linebreak
Consider the following two [[infinity-groupoid|$\infty$-groupoids]]:

The [[shape modality|cohesive shape]] of the [[mapping stack]] from $\mathbf{B}G$ to $\mathbf{B}\Gamma$

\[
  \label{ShapeOfMappingStackFromBGToBGamma}
  \begin{aligned}
  &
  \esh
  \,
  Map
  \Big(
    \mathbf{B}G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \\
  &
  \;\simeq\;
  \esh
  \,
  Map
  \Big(
    \big(
      G \rightrightarrows \ast
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \\
  &
  \;\simeq\;
  \underset{\longrightarrow}{\lim}
  N
  TopFunc
  \Big(
    \big(
      G \rightrightarrows \ast
    \big) \times \Delta^{\bullet}
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \;\simeq\;
  \end{aligned}
  \!\!\!\!\!\!\!\!\!
  \left(
    \array{
      \vdots
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          G \rightrightarrows \ast
        \big)
        \times \Delta^2
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_2
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          G \rightrightarrows \ast
        \big)
        \times \Delta^1
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_1
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          G \rightrightarrows \ast
        \big)
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_0
    }
  \right)
\]

and the [[shape modality|cohesive shape]] of the [[mapping stack]] from the [[spherical space form]] $S^{n+2}/G$ to $\mathbf{B}\Gamma$

\[
  \label{ShapeOfMappingStackFromSphericalSpaceFormToBGamma}
  \begin{aligned}
  &
  \esh
  \,
  Map
  \Big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \\
  & \;\simeq\;
  \esh
  \,
  Map
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \\
  &
  \;\simeq\;
  \underset{\longrightarrow}{\lim}
  N
  TopFunc
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big) \times \Delta^{\bullet}
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \;\simeq\;
  \end{aligned}
  \!\!\!\!\!\!\!\!\!
  \left(
    \array{
      \vdots
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          S^{n+2} \times G \rightrightarrows S^{n+2}
        \big)
        \times \Delta^2
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_2
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          S^{n+2} \times G \rightrightarrows S^{n+2}
        \big)
        \times \Delta^1
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_1
      \\
      \big\downarrow
      \big\uparrow
      \big\downarrow
      \\
      TopFunc
      \Big(
        \big(
          S^{n+2} \times G \rightrightarrows S^{n+2}
        \big)
        ,\,
        \big(
          \Gamma \rightrightarrows \ast
        \big)
      \Big)_0
    }
  \right)
\]

In both cases we are showing on the right the canonical [[simplicial sets]] which present these $\infty$-groupoids, obtained by using

1. the expression of [[shape via cohesive path ∞-groupoid|snooth shape via the smooth path $\infty$-groupoid]] (by [this Prop.](shape+via+cohesive+path+∞-groupoid#SmoothShapeModelityGivenBySmoothPathInfinityGroupoid)),

1. the canonical [[Cech groupoid]]-presentation of groupoids of principal bundles, from Lem. \ref{CanonicalCechGroupoidOfGammaPrincipalBundlesOnSphericalSpaceForm},

1. the fact that the [[simplex category|simplicial]] [[homotopy colimit]] of $\infty$-groupoids presented by [[simplicial sets]] is given by the [[diagonal of a bisimplicial set|diagonal]] of the corresponding [[bisimplicial set]] (by [this Prop.](bisimplicial+set#DiagonalAsSimplicialHomotopyColimit)).

\end{definition}

#### The classification theorem
 {#TheClassificationTheorem}

\begin{proposition}
\label{ConcordancesOnSphericalSpaceFormIsMapsOfClassifyingSpaces}
  Under Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms},
  the $\infty$-groupoid of [[concordances]] of $\Gamma$-principal bundles on $S^{n+2}/G$ is [[equivalence of infinity-groupoids|equivalent]] to the [[hom infinity-groupoid|hom $\infty$-groupoid]] from $B G$ to $B \Gamma$:
$$
  \esh
  \,
  Maps
  \big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \big)
  \;\;
  \simeq
  \;\;
  Maps
  \big(
     B G
     ,\,
     B \Gamma
  \big)
  \,.
$$
\end{proposition}
\begin{proof}
  Since $S^{n+2}/G$ is a [[smooth manifold]], the [[smooth Oka principle]] ([here](shape+via+cohesive+path+∞-groupoid#ConsequenceSmoothOkaPrinciple)) gives that
$$
  \begin{aligned}
  \esh
  \,
  Maps
  \big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \big)
  &
  \;\simeq\;
  Maps
  \big(
    \esh
    \,
    S^{n+2}/G
    ,\,
    \esh
    \,
    \mathbf{B}\Gamma
  \big)
  &
  \;=\;
  Maps
  \big(
    Shp
    (
      S^{n+2}/G
    )
    ,\,
    B \Gamma
  \big)
  \,.
  \end{aligned}
$$
From here, the claim follows by the following sequence of [[natural equivalences]] [[equivalence in an (infinity,1)-category|in]] [[Infinity-Grpd|$Grpd_\infty$]]:

\[
  \label{MapsIntoTruncatedBGammaSeeSphericalSpaceFormAsClassifyinSpace}
  \begin{array}{lll}
    Maps
    \big(
      Shp(S^{n+2}/G)
      ,\,
      B \Gamma
    \big)
    &
    \;\simeq\;
    Maps
    \big(
      Shp(S^{n+2} \sslash G)
      ,\,
      B \Gamma
    \big)
    \\
    &
    \;\simeq\;
    Maps
    \big(
      Shp(S^{n+2}) \sslash G
      ,\,
      B \Gamma
    \big)
    \\
    &
    \;\simeq\;
    Maps
    \big(
      \underset{\longrightarrow}{\lim}
      \,
      Shp(S^{n+2}) \times G^{\times_\bullet}
      ,\,
      B \Gamma
    \big)
    \\
    & \;\simeq\;
    \underset{\longleftarrow}{\lim}
    \,
    Maps
    \big(
      Shp(S^{n+2}) \times G^{\times_\bullet}
      ,\,
      B \Gamma
    \big)
    \\
    & \;\simeq\;
    \underset{\longleftarrow}{\lim}
    \,
    Maps
    \big(
      G^{\times_\bullet}
      ,\,
      B \Gamma
    \big)
    &
    \text{ by (eq:GammaIsNTruncated) }
    \\
    & \;\simeq\;
    Maps
    \big(
      \underset{\longrightarrow}{\lim}
      \,
      G^{\times_\bullet}
      ,\,
      B \Gamma
    \big)
    \\
    & \;\simeq\;
    Maps
    \big(
      B G
      ,\,
      B \Gamma
    \big)
  \end{array}
\]
\end{proof}

Moreover:

\begin{definition}
\label{ComparisonMorphismBetweenShapesOfMappingStacks}
There is a canonical comparison morphism between the shapes of mapping stacks in Def. \ref{ShapesOfMappingStacks},
given by [[precomposition]] with the [[terminal object|terminal morphism]] $S^{n+2} \xrightarrow{\; p\;} \ast$:

\[
  \label{ComparisonMorphismBetweenShapesOfMappingStacks}
  \esh
  \,
  Map
  \Big(
    \mathbf{B}G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \xrightarrow
  {\;\;\;
     \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma)
  \;\;\;}
  \esh
  \,
  Map
  \Big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \,
  \,.
\]
\end{definition}

\begin{theorem}
\label{ComparisonMapOfConcordanceInfinityGroupoidsIsEquivalence}
  Under Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms}, the comparison morphism (eq:ComparisonMorphismBetweenShapesOfMappingStacks) is an [[equivalence of infinity-groupoids|equivalence of $\infty$-groupoids]].
\end{theorem}
\begin{proof}
  The claim means that the comparison morphism is a [[weak homotopy equivalence]], hence that it induces [[isomorphisms]] on all [[homotopy groups]] $\pi_n$. For $\pi_0$ this is Prop. \ref{ComparisonMorphismBijectiveOnConnectedComponents} below, while for $\pi_1$ this is Prop. \ref{ComparisonMorphismIsoOnFundamentalGroups} below, whose proof has an evident generalization to all $n$.
\end{proof}

\begin{corollary}
 \label{SmoothOkaForMapsOutOfBGIntoBGammaIfGammaIsBraidedTruncated}
  Under Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms}, there is an [[equivalence of infinity-groupoids|equivalence of $\infty$-groupoids]] of the form:
\[
  \label{TheResultingSmoothOkaLikeEquivalence}
  \esh
  \,
  Map
  \big(
    \mathbf{B}G
    ,\,
    \mathbf{B}\Gamma
  \big)
  \;\;
  \simeq
  \;\;
  Map
  \big(
    \esh
    \,
    \mathbf{B}G
    ,\,
    \esh
    \,
    \mathbf{B}\Gamma
  \big)
  \;=\;
  Map
  \big(
    B G
    ,\,
    B \Gamma
  \big)
\]
\end{corollary}
\begin{proof}
 This is the composite of
Prop. \ref{ConcordancesOnSphericalSpaceFormIsMapsOfClassifyingSpaces}
with
Thm. \ref{ComparisonMapOfConcordanceInfinityGroupoidsIsEquivalence}.
\end{proof}
\begin{remark}
Cor. \ref{SmoothOkaForMapsOutOfBGIntoBGammaIfGammaIsBraidedTruncated} is of the form of the [[smooth Oka principle]] but for [[domain]] not a [[smooth manifold]] but the [[delooping groupoid]] of the [[finite group]] $G$.

In contrast to the case where the domain is a smooth manifold, equivalences of the form (eq:TheResultingSmoothOkaLikeEquivalence) fail in general, unless some extra condition like Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms} is imposed.

For example, in the case that both $G$ and $\Gamma$ are [[compact Lie groups]], the $\infty$-groupoids on the left of (eq:TheResultingSmoothOkaLikeEquivalence) are the [[hom infinity-groupoid|hom $\infty$-groupoids]] of the [[global orbit category]] for compact Lie [[equivariance groups]].
\end{remark}

\begin{example}
\label{EquivariantClassifyingSpaceOfPUHBundles}
**(equivariant classifying space of $PU(\mathcal{H})$-bundles)**
\linebreak
In the case that

* the [[structure group]] is $\Gamma = $ [[PU(ℋ)]]

* the [[equivariance group]] $G$ is a [[finite group]] which has any [[free action]] on any [[n-sphere]] for $n \geq 4$

Cor. \ref{SmoothOkaForMapsOutOfBGIntoBGammaIfGammaIsBraidedTruncated} immediately implies that the [[homotopy groups]]

$$
  \pi_k
  \Big(
    Map
    \big(
      \mathbf{B}G
      ,\,
      \mathbf{B}PU(\mathcal{H})
    \big)
  \Big)
  \;\;
  \simeq
  \;\;
  \pi_k
  \Big(
    Map
    \big(
      B G
      ,\,
      B^3 \mathbb{Z}
    \big)
  \Big)
  \;\;
  =
  \;\;
  H^{3-k}_{Grp}
  (
    G
    ;\,
    \mathbb{Z}
  )
$$

are given by the [[group cohomology]] of $G$ with [[coefficients]] in the [[integers]], as shown.

This recovers the statement of [Uribe & al. 2014, Thm. 1.10](twisted+equivariant+K-theory#BEJU14), see also [Uribe & Lück 2014, Thm. 15.17](twisted+equivariant+K-theory#UribeLueck14).
\end{example}

\linebreak

#### Lemmas and proofs
 {#LemmasAndProofs}

\begin{example}
\label{OneMorphismInShapeOfMappingStackFromSphericalSpaceFormToBGamma}
  A [[1-morphism]] in (eq:ShapeOfMappingStackFromSphericalSpaceFormToBGamma)
from a bundle $P_0\vert_0$ to a bundle $P_1\vert_1$ over $S^{n+2}/G$ is a diagram of [[homomorphisms]] of [[principal bundles]] of this form:

\begin{tikzcd}
    P_0\vert_0
    \ar[r]
    \ar[d]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    P_0
    \ar[d, "p_0"]
    \ar[r, "\phi"{above}, "\sim"{below}]
    &
    P_1
    \ar[d, "p_1"]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    P_1\vert_1
    \ar[d]
    \ar[l]
    \\
    S^{n+2}/G \times \{0\}
    \ar[r, hook]
    &
    S^{n+2}/G \times [0,1]
    \ar[r,-,shift left=1pt]
    \ar[r,-,shift right=1pt]
    &
    S^{n+2}/G \times [0,1]
    &
    S^{n+2}/G \times \{1\}
    \ar[l, hook']
\end{tikzcd}
\end{example}

\begin{lemma}
\label{ThePresentationsForShapesOfMappingStacksAreKanComplexes}
  The simplicial set presentations for shapes of mapping stacks in Def. \ref{ShapesOfMappingStacks} have at least all 2-[[horn]] fillers.
\end{lemma}
\begin{proof}
It is useful to denote a 1-morphism
$$
  \big(
    c(-)
    \xrightarrow{\; \gamma(-) \;}
    c'(-)
  \big)
  \;\;\;
  \in
  \;
  TopFunc
  \Big(
    (S^{n+2} \times G \rightrightarrows S^{n+2})
    \times [0,1]
    ,\,
    (\Gamma \rightrightarrows \ast)
  \Big)
  \,,
$$
namely a $[0,1]$-parameterized Cech coboundary
$$
  \gamma(-)
    \colon
  [0,1]
    \xrightarrow{\;}
  Maps
  \big(
    S^{n+2}
    ,\,
    \Gamma
  \big)
  \;\;\;
  \text{s.t.}
  \;\;\;
  \underset{ (t,x) \in [0,1] \times S^{n+2}}{\forall}
  \left(
  \begin{aligned}
    c'(t)(x,\, g )
    &
      \;=\;
    c(t)(x,g)^{\gamma(t)(x)}
    \\
    &
    \,\coloneqq\,
    (\gamma(t)(x))^{-1}
      \cdot
    c(t)(x, g)
     \cdot
    \gamma(t)(x)
  \end{aligned}
  \right)
  \,.
$$
as follows:

\begin{tikzcd}[row sep={between origins, 60pt}, column sep={between origins, 60pt}]
    c(0)
    \ar[rr, "{\gamma(0)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(0)^{\gamma(0)}
    \ar[d,-, dotted]
    \\
    c(t)
    \ar[rr, "{\gamma(t)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(t)^{\gamma(t)}
    \ar[d,-, dotted]
    \\
    c(1)
    \ar[rr, "{\gamma(1)}"{description, pos=.6}]
    &{}&
    c(1)^{\gamma(1)}
\end{tikzcd}

Then a pair of composable 1-morphisms, hence a $\Lambda^2_1$-[[horn]], looks like this:

\begin{tikzcd}[row sep={between origins, 60pt}, column sep={between origins, 60pt}]
    c(0)
    \ar[rr, "{\gamma(0)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(0)^{\gamma(0)}
    \ar[d,-, dotted]
    \\
    c(t)
    \ar[rr, "{\gamma(t)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(t)^{\gamma(t)}
    \ar[d,-, dotted]
    \\
    c(1)
    \ar[rr, "{\gamma(1)}"{description, pos=.6}]
    &{}&
    c(1)^{\gamma(1)}
    \ar[r, -, shift left=1pt]
    \ar[r, -, shift right=1pt]
    &[-20pt]
    c'(0)
    \ar[rr, "{\gamma'(0)}"{description, pos=.6}]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(0)^{\gamma'(0)}
    \ar[d, -, dotted]
    \\
    &&&
    c'(t')
    \ar[rr, "{\gamma'(t')}"{description, pos=.6}]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(t')^{\gamma'(t')}
    \ar[d, -, dotted]
    \\
    &&&
    c'(1)
    \ar[rr, "{\gamma'(1)}"{description, pos=.6}]
    &
    {}
    &
    c'(1)^{\gamma'(1)}
\end{tikzcd}

For any such we may form the following composable pair of 1-morphisms of cocycles over $S^{n+2}/G \times \big( \underset{\ni t}{[0,1]} \underset{\ast}{\sqcup} \underset{\ni t'}{[0,1]}  \big)$:

\begin{tikzcd}[row sep={between origins, 60pt}, column sep={between origins, 60pt}]
    c(0)
    \ar[rr, "{\gamma(0)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(0)^{\gamma(0)}
    \ar[d,-, dotted]
    \ar[rr, "\gamma'(0)"{description, pos=.6}]
    &
    {}
    \ar[d, -, shift left=1pt]
    \ar[d, -, shift right=1pt]
    &
    c(0)^{ \gamma(0) \mathrlap{\gamma'(0)} }
    \ar[d, -, dotted]
    \\
    c(t)
    \ar[rr, "{\gamma(t)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(t)^{\gamma(t)}
    \ar[d,-, dotted]
    \ar[rr, "\gamma'(0)"{description, pos=.6}]
    &
    {}
    \ar[d, -, shift left=1pt]
    \ar[d, -, shift right=1pt]
    &
    c(t)^{ \gamma(t) \mathrlap{\gamma'(0)} }
    \ar[d, -, dotted]
    \\
    c(1)
    \ar[rr, "{\gamma(1)}"{description, pos=.6}]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, shift left=1pt]
    \ar[d, -, shift right=1pt]
    &
    c'(0)
    \ar[rr, "{\gamma'(0)}"{description, pos=.6}]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(0)^{\gamma'(0)}
    \ar[d, -, dotted]
    \\
    c'(t')^{\gamma(1)^{-1}}
    \ar[
      rr,
      "{\gamma(1)}"{description, pos=.4}
    ]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, shift left=1pt]
    \ar[d, -, shift right=1pt]
    &
    c'(t')
    \ar[rr, "{\gamma'(t')}"{description, pos=.6}]
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(t')^{\gamma'(t')}
    \ar[d, -, dotted]
    \\
    c'(1)^{\gamma(1)^{-1}}
    \ar[
      rr,
      "{\gamma(1)}"{description, pos=.4}
    ]
    &
    {}
    &
    c'(1)
    \ar[rr, "{\gamma'(1)}"{description, pos=.6}]
    &
    {}
    &
    c'(1)^{\gamma'(1)}
\end{tikzcd}

When pulled back along the canonical topological horn filler retraction $\Delta^2 \to \Lambda^2_{1}$ this yields a 2-cell which fills the original 2-horn.

Similarly, a $\Lambda^2_0$-horn is the following type of data:

\begin{tikzcd}[row sep={between origins, 60pt}, column sep={between origins, 60pt}]
    c(0)
    \ar[rr, "{\gamma(0)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(0)^{\gamma(0)}
    \ar[d,-, dotted]
    \\
    c(t)
    \ar[rr, "{\gamma(t)}"{description, pos=.6}]
    \ar[d,-, dotted]
    &
    {}
    \ar[d,-, dotted]
    &
    c(t)^{\gamma(t)}
    \ar[d,-, dotted]
    \\
    c(1)
    \ar[rr, "{\gamma(1)}"{description, pos=.6}]
    &{}&
    c(1)^{\gamma(1)}
    \ar[r, -, shift left=1pt]
    \ar[r, -, shift right=1pt]
    &[-20pt]
    c'(1)^{\gamma'(1)}
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(1)
    \ar[ll, "{\gamma'(0)}"{description, pos=.6}]
    \ar[d, -, dotted]
    \\
    &&&
    c'(t')^{\gamma'(t')}
    \ar[d, -, dotted]
    &
    {}
    \ar[d, -, dotted]
    &
    c'(t')
    \ar[d, -, dotted]
    \ar[ll, "{\gamma'(t')}"{description, pos=.6}]
    \\
    &&&
    c'(0)^{\gamma'(0)}
    &
    {}
    &
    c'(0)
    \ar[ll, "{\gamma'(0)}"{description, pos=.6}]
\end{tikzcd}

and is filled by the evident directly analogous procedure.
\end{proof}

$\,$

\begin{lemma}
\label{ConjugacyClassesOfGroupHomomorphismsSurjectOnPrincipalBundlesOverSphericalSpaceForm}
  Under the truncation assumption (eq:GammaIsNTruncated),
  the canonical [[function]]
  $$
    Hom(G,\Gamma)_{/\sim_{conj}}
    \xrightarrow{\phantom{--}\sim\phantom{--}}
    \Big(
      \Gamma PrnBdl(TopSp)_{S^{n + 2}/G}
    \Big)_{/\sim_{iso}}
  $$
  is an [[isomorphism]].
\end{lemma}
\begin{proof}
The [[homotopy class]] of the [[classifying space|classifying map]] of a [[principal bundle]] is represented by the [[topological realization]] of any of its [[Cech cohomology|Cech cocycles]]. Therefore, Lemma \ref{CanonicalCechGroupoidOfGammaPrincipalBundlesOnSphericalSpaceForm} implies that in the following [[commuting diagram]] the left and the composite function are [[bijections]]:

\begin{tikzcd}[column sep=small]
  \Gamma \mathrm{PrnBdl}(\mathrm{TopSp})_{ S^{n+2}/G }
  \ar[r, "{\sim}"{yshift=-1pt}]
  \ar[
    rr,
    rounded corners,
    to path={
      -- ([yshift=-11pt]\tikztostart.south)
      -- node[below, yshift=1pt]{ \scalebox{.7}{$\sim$} }
         ([yshift=-08pt]\tikztotarget.south)
      -- (\tikztotarget.south)
    }
  ]
  &
  \tau_0
  \,
  \mathrm{TopFunc}
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \ar[
    r,
    "\color{blue}\sim"{above, yshift=-1pt}
  ]
  &
  \tau_0
  \,
  \mathrm{Maps}
  \Big(
    \mathrm{Shp}
    \big(
      S^{n+2} /\!/ G
    \big)
    ,\,
    B \Gamma
  \Big)
\end{tikzcd}

It follows by [[2-out-of-3]] that also the function on the right is a bijection:
\[
  \label{CocyclesToClassifyingMapsIsBijection}
  \tau_0
  \,
  \mathrm{TopFunc}
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \xrightarrow{\;\; \sim \;\;}
  \tau_0
  \,
  \mathrm{Maps}
  \Big(
    \mathrm{Shp}
    \big(
      S^{n+2} /\!/ G
    \big)
    ,\,
    B \Gamma
  \Big)
\]

Observe that the composite of the [[weak homotopy equivalence|equivalences]] in (eq:MapsIntoTruncatedBGammaSeeSphericalSpaceFormAsClassifyinSpace) is again the morphism induced by pre-composition with

$$
  S^{n+2} \xrightarrow{\; p \;} \ast
  \;\;\;\;
  \text{or}
  \;\;\;\;
  \ast \xrightarrow{\; i \;} S^{n+2}
  \,.
$$

Therefore, by [[functor|functoriality]] of the [[nerve]]-operation followed by [[topological realization of simplicial topological spaces|topological realization]], the following [[commuting diagram|diagram commutes]]:

\begin{tikzcd}[column sep=small]
  {}
  &[60pt]
  \tau_0
  \,
  \mathrm{TopFunc}
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \ar[
    r,
    "\sim"{above}
  ]
  \ar[d, hook, "{ i^\ast }"{left}]
  &
  \tau_0
  \,
  \mathrm{Maps}
  \Big(
    \mathrm{Shp}
    \big(
      S^{n+2} / G
    \big)
    ,\,
    B \Gamma
  \Big)
  \ar[d, , "{ i^\ast }"{left}, "\sim"{sloped, xshift=-6pt, yshift=+4pt}]
  &[150pt]
  {}
  \\
  {}
  &
  \mathllap{
    \mathrm{Hom}(G,\Gamma)_{/\sim_{\mathrm{conj}}}
    \;\simeq\;\;
  }
  \tau_0
  \,
  \mathrm{TopFunc}
  \Big(
    \big(
      G \rightrightarrows \ast
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \ar[d, ->>, "{ p^\ast }"{left}]
  \ar[r, ->>]
  &
  \tau_0
  \,
  \mathrm{Maps}
  \Big(
    \ast /\!/ G
    ,\,
    B \Gamma
  \Big)
  \mathrlap{
    \;\;\simeq\;
    \tau_0
    \,
    \mathrm{Maps}
    \Big(
      B G
      ,\,
      B \Gamma
    \Big)
  }
  \ar[
    d, "{ p^\ast }"{left}, "\sim"{sloped, xshift=-6pt, yshift=+4pt}
  ]
  &
  {}
  \\
  {}
  &
  \tau_0
  \,
  \mathrm{TopFunc}
  \Big(
    \big(
      S^{n+2} \times G \rightrightarrows S^{n+2}
    \big)
    ,\,
    \big(
      \Gamma \rightrightarrows \ast
    \big)
  \Big)
  \ar[
    r,
    "\sim"{above}
  ]
  &
  \tau_0
  \,
  \mathrm{Maps}
  \Big(
    \mathrm{Shp}
    \big(
      S^{n+2} / G
    \big)
    ,\,
    B \Gamma
  \Big)
  \mathrlap{
     \;\;\simeq\;
     \big(
       \Gamma \mathrm{PrnBld}(\mathrm{TopSp})_{S^{n+2}/G}
     \big)_{/\sim_{\mathrm{iso}}}
  }
  &
  {}
\end{tikzcd}

Here

* the top and bottom horizontal functions are bijections by (eq:CocyclesToClassifyingMapsIsBijection),

* the right vertical functions are bijections by (eq:MapsIntoTruncatedBGammaSeeSphericalSpaceFormAsClassifyinSpace).

By commutativity of the total rectangle,
this implies that the left vertical functions exhibit a [[retraction]], as shown, in particular the bottom left function is [[surjective]]. (Moreover, the commutativity of the two squares separately each implies that the middle horizontal function is also a surjection, as shown.)

But the bottom left function is clearly also [[injective]]: If $\phi \colon S^{n+2} \xrightarrow{\;} \Gamma$ is a Cech coboundary between Cech cocycles that are constant along $S^{n+2}$, then $\phi(\ast)$ conjugates the corresponding [[group homomorphisms]] into each other.

Therefore the bottom left morphism is both injective as well as surjective, hence bijective.
\end{proof}

More generally, the analogous conclusion evidently still holds for $\Gamma$-principal bundles on the [[topological product space]] $S^{n+2}/G \times \Delta^k$ with the [[topological k-simplex]]

$$
  \tau_0
  \,
  \Gamma PrnBdl(TopSp)_{S^{n+2}/G \times \Delta^k}
  \;\;
   \simeq
  \;\;
  \tau_0
  \,
  TopFunc
  \big(
    S^{n+2} \times G \times \Delta^k
    \rightrightarrows
    S^{n+2} \times \Delta^k
    ,\,
    \Gamma \rightrightarrows \ast
  \big)
  \,.
$$

\begin{lemma}
\label{OneMorphismsInShapeOfMapsFromSphericalFormToBGammaEquivalentlyHaveConstantUnderlyingBundle}
  Every [[1-morphism]]
  in
  $\esh \,Maps\big( S^{n+2}/G ,\, \mathbf{B}\Gamma \big)$
  (Exp. \ref{OneMorphismInShapeOfMappingStackFromSphericalSpaceFormToBGamma})
  is equivalent (homotopic relative its endpoints) to
  one of the form
  \[
  \label{OneMorphismInShapeOfMappingStacksOutOfAConstantBundle}
  \array{
    P \times \Delta^1
    &\xrightarrow{\phantom{--}}&
    P'
    \\
    \big\downarrow
    {}^{\mathrlap{p \times id_{[0,1]}}}
    &&
    \big\downarrow
    {}^{\mathrlap{p'}}
    \\
    S^{n+2}/G \times \Delta^1
    &=&
    S^{n+2}/G \times \Delta^1
  }
  \]
\end{lemma}
\begin{proof}
  By the [[classifying space|classification theory]] of principal bundles (or, more concretely, by the proof of [this Prop.](concordance#ConcordantTopologicalPrincipalBundlesAreIsomorphic)), every principal bundle on a [[cylinder]] like $S^{n+2}/G  \times [0,1]$ is isomorphic to the constant re-extension of its restriction to one end of the cylinder. With the given 1-morphism denoted as in Exp. \ref{OneMorphismInShapeOfMappingStackFromSphericalSpaceFormToBGamma} we write $\ell$ for such an isomorphism onto its $P_0$-component:

\begin{tikzcd}
    &
    P_0
    \ar[dd, "p_0"{description}]
    \\
    P_0\vert_0 \times \Delta^1
    \ar[dd, "p_0\vert_0 \,\times\, \mathrm{id}_{[0,1]}"{description}]
    \ar[ur, "\ell"{above}, "\sim"{below, sloped}]
    &
    \\
    &
    S^{n+2}/G \times \Delta^1
    \\
    S^{n+2}/G \times \Delta^1
    \mathrlap{\,,}
    \ar[ur, -, shift left=1pt]
    \ar[ur, -, shift right=1pt]
\end{tikzcd}

which is such that the restriction of $\ell$ to $\{0\} \subset [0,1]$ is the [[identity morphism]]

\[
  \label{RestrictionOfEllToLeftEndpointIsIdentity}
  \ell\vert_0 \;\colon\; P_0 \xrightarrow{\; id \;} P_0
  \mathrlap{\,.}
\]

From this we may construct the following  [[2-morphism]] in (eq:ShapeOfMappingStackFromSphericalSpaceFormToBGamma):

\[\label{A2MorphismInShapeOfMappingStackWithDegeenrateTwoFace}\]
\begin{tikzcd}[column sep={between origins, 90pt}]
    &
    s_0^\ast
    (
      P_0
    )
    \ar[dd]
    \ar[dr, "s_0^\ast\phi"{above}, "\sim"{sloped, below}]
    &[-6pt]
    \\
    s_0^\ast
    (
      P_0\vert_0 \times \Delta^1
    )
    \ar[ur, "s_0^\ast(\ell)"{above}, "\sim"{below, sloped}]
    \ar[rr, crossing over]
    \ar[dd]
    &
    &
    s_0^\ast
    (
      P_1
    )
    \ar[dd]
    \\
    &
    S^{n+2}/G \times \Delta^2
    \ar[dr, -, shift left=1pt]
    \ar[dr, -, shift right=1pt]
    \\
    S^{n+2}/G \times \Delta^2
    \ar[ur, -, shift left=1pt]
    \ar[ur, -, shift right=1pt]
    \ar[rr, -, shift left=1pt]
    \ar[rr, -, shift right=1pt]
    &&
    S^{n+2}/G \times \Delta^2
\end{tikzcd}

Here $\sigma_1 \,\colon\, \Delta^2 \to \Delta^1$ denotes the map of [[topological simplices]] which collapses the 2-[[face]]:

\begin{tikzcd}
    \Delta^1
    \ar[r, "\delta_2"]
    \ar[
      rr,
      rounded corners,
      to path={
        -- ([yshift=-7pt]\tikztostart.south)
        -- node[below]{\scalebox{.7}{$\mathrm{const}_0$}}
          ([yshift=-7pt]\tikztotarget.south)
        -- (\tikztotarget.south)
      }
    ]
    &
    \Delta^2
    \ar[r, "\sigma_1"]
    &
    \Delta^1
    &[10pt]
    \Delta^1
    \ar[r, "\delta_{0,1}"]
    \ar[
      rr,
      rounded corners,
      to path={
        -- ([yshift=-7pt]\tikztostart.south)
        -- node[below]{\scalebox{.7}{$\mathrm{id}$}}
          ([yshift=-7pt]\tikztotarget.south)
        -- (\tikztotarget.south)
      }
    ]
    &
    \Delta^2
    \ar[r, "\sigma_1"]
    &
    \Delta^1
    \mathrlap{\,.}
\end{tikzcd}

Therefore the 2-[[face]] of the above [[2-morphism]] (eq:A2MorphismInShapeOfMappingStackWithDegeenrateTwoFace) is degenerate (where the last step uses (eq:RestrictionOfEllToLeftEndpointIsIdentity)):

$$
  \begin{aligned}
    \delta_2^\ast
    \big(
      \sigma_1^\ast(\ell)
    \big)
    &
    \;=\;
    const_0^\ast(\ell)
    \\
    &
    \;=\;
    \ell\vert_0 \times id_{\Delta^1}
    \\
    &
    \;=\;
    id_{P_0} \times id_{\Delta^1}
    \;=\;
    id_{ P_0 \times \Delta^1 }
    \mathrlap{\,,}
  \end{aligned}
$$

while the 0-[[face]] is the original morphism $\phi$

$$
  \delta_0^\ast
  \big(
    s_0^\ast(\phi)
  \big)
  \;=\;
  \mathrm{id}_{\Delta^1}^\ast(\phi)
  \;=\;
  \phi
  \,,
$$

and the 1-[[face]] is of the claimed form (eq:OneMorphismInShapeOfMappingStacksOutOfAConstantBundle):

$$
  \begin{aligned}
    \delta_1^\ast
    \big(
      s_0^\ast(\phi)
      \circ
      s_0^\ast(\ell)
    \big)
    &
    \;=\;
    \delta_1^\ast
    \big(
      s_0^\ast( \phi \circ \ell )
    \big)
    \\
    &
    \;=\;
    \phi \circ \ell
    \;\colon\;
    P_0\vert_0 \xrightarrow{\;} P_1
    \mathrlap{\,.}
  \end{aligned}
$$
Hence the 2-morphism (eq:A2MorphismInShapeOfMappingStackWithDegeenrateTwoFace) exhibits the claimed homotopy relative endpoints.
\end{proof}

\begin{lemma}\label{ComparisonMorphismIsInjectiveOnConnectedComponents}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is [[injective]] on [[connected components]].
\end{lemma}
\begin{proof}
Given a 1-morphism
$$
  c(-)
    \xrightarrow{\;\gamma(-)\;}
  c'(-)
$$
such that both $c(0)$ and $c'(1)$ are constant along $S^{n+2}$
$$
  \underset{x \in S^{n+2}}{\forall}
  \big(
    c(0)(x,g) \;=\; c(0)(\ast,g)
    \,,
    \;\;\;\;
    c'(1)(x,g) \;=\; c'(1)(\ast,g)
  \big)
$$
we need to show that there is a 1-morphism between $c(0)$ and $c'(1)$ all whose components are constant along $S^{n+2}$.

Now by Lemma \ref{OneMorphismsInShapeOfMapsFromSphericalFormToBGammaEquivalentlyHaveConstantUnderlyingBundle} we know that there is a 1-morphism $c(-) \xrightarrow{\gamma(-)} c'(-)$ such that $c$ is constant along $[0,1] \times S^{n+2}$, i.e. such that
$$
  \underset{
    (t,x) \,\in\, [0,1] \times S^{n+2}
  }{\forall}
  \;\;
  c(t)(x,g) \,=\, c(0)(\ast,g)
  \,.
$$

But the remaining data is then all in $\gamma(-)$. Hence restricting $\gamma$ to $\ast \,\in\, S^{n+2}$ (and then re-extending it as a constant function on this value) yields a 1-morphism of the desired form.
\end{proof}

\begin{lemma}\label{ComparisonMorphismIsSurjectiveOnConnectedComponents}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is [[surjective]] on [[connected components]].
\end{lemma}
\begin{proof}
  We need to show that every cocycle $c(0)$ there exists a cocycle $c'(1)$ which is constant along $S^{n+2}$ and a 1-morphism $c(-) \xrightarrow{\gamma(-)} c'(-)$. But by Lemma \ref{ConjugacyClassesOfGroupHomomorphismsSurjectOnPrincipalBundlesOverSphericalSpaceForm} there is even a Cech coboundary $\gamma(0)$ with $c'(1) = c(0)^{\gamma(0)}$. Hence taking $c(t) \coloneqq c(0)$ and $\gamma(t) \coloneqq \gamma(1)$ gives the required morphism.
\end{proof}

\begin{proposition}\label{ComparisonMorphismBijectiveOnConnectedComponents}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is [[bijective]] on [[connected components]]:
$$
  \pi_0
  \big(
    \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma)
  \big)
  \;\;
  \colon
  \;\;
  \pi_0
  \bigg(
  \esh
  \,
  Map
  \Big(
    \mathbf{B}G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \bigg)
  \xrightarrow
  {\;\;\;
     \sim
  \;\;\;}
  \pi_0
  \bigg(
  \esh
  \,
  Map
  \Big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \bigg)
$$
\end{proposition}
\begin{proof}
  By Lemma \ref{ComparisonMorphismIsInjectiveOnConnectedComponents} it is injective, and by Lemma \ref{ComparisonMorphismIsSurjectiveOnConnectedComponents} it is surjective.
\end{proof}

\begin{lemma}
\label{ComparisonMorphisnInjectiveOnFundamentalGroups}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is [[injective]] on [[fundamental groups]].
\end{lemma}
\begin{proof}
  Since both simplicial sets have all 2-horn fillers, by Lem. \ref{ThePresentationsForShapesOfMappingStacksAreKanComplexes}, it is sufficient to show for $c(-) \xrightarrow{\gamma(-)} c(-)$ a 1-morphism with $c(0) = c(1)$ and all data constant along $S^{n+2}$ that if this is homotopic relative boundary to the identity on $c(0)$ by any 2-cell, then it is so by a 2-cell all whose data is constant along $S^{n+2}$.

Idea: As in Lem. \ref{OneMorphismsInShapeOfMapsFromSphericalFormToBGammaEquivalentlyHaveConstantUnderlyingBundle} we find that the given homotopy is itself equivalent to one whose underlying cocycle is constant along $S^{n+2}$. The remaining data is all in $\gamma(-)$, so that restricting that to $\ast \in S^{n+2}$ (and then re-extending as a constant function) yields the desired homotopy.
\end{proof}

\begin{lemma}\label{DeformationOfCechCoboundaryGivesConcordanceOfCocycles}
  Let $c(-) \xrightarrow{\gamma(-)} c(-)^{\gamma(-)}$ be a 1-morphism such that $c(1)$ is trivial: $\underset{(x,g) \in S^{n+2} \times G}{\forall} \,  c(1)(x,g) = \mathrm{e}$. Then for every continuous function

$$
  \widehat{\gamma} \,\colon\,
  [0,1] \times [0,1] \times S^{n+2} \xrightarrow{\;\;} \Gamma
$$

with

$$
  \widehat{\gamma}(0) \,=\, \gamma
  \,,
  \;\;\;
  \widehat{\gamma}(s)(1,\ast) \,=\, \gamma(1,\ast)
$$

this 1-morphism is homotopic relative boundary to

$$
  c(-)
  \xrightarrow{ \widehat{\gamma}(1)(-) }
  c(-)^{ \widehat{\gamma}(1)(-) }
  \,.
$$
\end{lemma}
\begin{proof}
  There is the obvious way to turn $\widehat \gamma$ into
a 2-morphism of the shape of the simplicial square whose two vertical 1-faces are degenerate (the right one by the assumption that $c(1)$ is trivial, so that $c(1)^{\gamma} \,=\, c(1)$):

\begin{tikzcd}[row sep={between origins, 30pt}, column sep={between origins, 40pt}]
    &&
    c(0)
    \ar[rrrrrrrr, "{\gamma(0)}"{description, pos=.5}]
    \ar[dddd,-, dashed]
    &&&&
    \phantom{AA}
    \ar[dddd,-, dotted, crossing over]
    &&&&
    c(0)^{\gamma(0)}
    \ar[dddd,-, dotted]
    \\
    &
    c(0)
    \ar[ur,-,dashed]
    \ar[dddd,-,dotted]
    \ar[rrrrrrrr, crossing over, "{\widehat{\gamma}(t')(0)}"{description}]
    &&&&
    \phantom{AA}
    \ar[ur,-,dotted]
    \ar[ddd,-,dotted, crossing over]
    &&&&
    c(0)^{\widehat{\gamma}(t')(0)}
    \ar[ur,-,dotted]
    \\
    c(0)
    \ar[ur,-,dashed]
    \ar[rrrrrrrr, crossing over, "{ \widehat{\gamma}(1)(0)}"{description}]
    \ar[dddr, -, dashed]
    &&&&
    \phantom{AA}
    \ar[ur,-,dotted]
    &&&&
    c(0)^{\widehat{\gamma}(1)(0)}
    \ar[ur, -, dotted]
    \\
    &&&&&
    {}
    \\
    &&
    c(t)
    \ar[rrrrrrrr, "{\gamma(t)}"{description, pos=.5}]
    \ar[dddd,-, dashed]
    &&&{}&
    \phantom{AA}
    \ar[dddd,-, dotted]
    &{}&&&
    c(t)^{\gamma(t)}
    \ar[dddd,-, dotted]
    \\
    &
    c(t)
    \ar[ur,-,dotted]
    \ar[dddd,-,dotted]
    \ar[rrrrrrrr, crossing over, "{ \widehat{\gamma}(t')(t) }"{description}]
    \ar[dddr, -, dashed]
    &&&&
    \phantom{AA}
    \ar[ur,-,dotted]
    \ar[uu, -, dotted, crossing over]
    \ar[ddd, -, dotted]
    &&&&
    c(t)^{ \widehat{\gamma}(t')(t) }
    \ar[ur,-,dotted]
    \ar[uuuu,-,dotted, crossing over]
    \\
    c(t)
    \ar[ur, -, dotted]
    \ar[uuuu, -, dashed]
    \ar[rrrrrrrr, crossing over, "\widehat{\gamma}(1)(t)"{description}]
    &&&&
    \phantom{AA}
    \ar[ur, -, dotted]
    \ar[uuuu, -, dotted, crossing over]
    &&&&
    c(t)^{\widehat{\gamma}(1)(t)}
    \ar[ur, -, dotted]
    \ar[uuuu, -, dotted, crossing over]
    \\
    &
    &&&&
    {}
    \\
    &&
    c(1)
    \ar[rrrrrrrr, "{\gamma(1)}"{description, pos=.5}]
    &&&
    {}
    &
    {}
    &&&&
    c(1)^{\gamma(1)}
    \\
    &
    c(1)
    \ar[ur,-,dashed]
    \ar[rrrrrrrr, "{ \widehat{\gamma}(t')(1) }"{description}]
    &&&&
    \phantom{AA}
    \ar[ur,-,dotted]
    \ar[uu,-,crossing over, dotted]
    &&&&
    c(1)^{\widehat{\gamma}(t')(1)}
    \ar[ur,-,dotted]
    \ar[uuuu,-,crossing over, dotted]
    \\
    c(1)
    \ar[ur,-,dashed]
    \ar[uuuu,-,dashed]
    \ar[rrrrrrrr, "{ \widehat{\gamma}(1)(1) }"{description}]
    &&&&
    \phantom{AA}
    \ar[ur, -, dotted]
    \ar[uuuu, -, dotted, crossing over]
    &&&&
    c(1)^{\widehat{\gamma}(1)(1)}
    \ar[ur, -, dotted]
    \ar[uuuu, -, dotted, crossing over]
\end{tikzcd}

\end{proof}

\begin{lemma}
\label{ComparisonMorphismSurjectiveOnFundamentalGroups}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is [[surjective]] on [[fundamental groups]].
\end{lemma}
\begin{proof}
  Since both simplicial sets have all 2-horn fillers, by Lem. \ref{ThePresentationsForShapesOfMappingStacksAreKanComplexes}, it is sufficient to show for $c(-) \xrightarrow{\gamma(-)} c(-)$ any 1-morphism with $c(0) = c(1)$ that it is homotopic relative boundary to one all whose data is constant along $S^{n+2}$, and by Lem. \ref{ComparisonMorphismIsSurjectiveOnConnectedComponents} it is sufficient to assume that the cocycle $c(0)$ is already constant along $S^{n+2}$.

Hence considering this case, Prop. \ref{OneMorphismsInShapeOfMapsFromSphericalFormToBGammaEquivalentlyHaveConstantUnderlyingBundle} give a homotopy relative boundary to a 1-morphism whose underlying cocycle $c(-)$ is constant along $S^{n+2}$. The remaining data is

$$
  \gamma(-)
  \;\colon\;
  [0,1] \times S^{n+2} \xrightarrow \Gamma
  \,.
$$

or equivalently

$$
  \tilde{\gamma}(-)
  \;\colon\;
  S^{n+2} \xrightarrow Maps\big([0,1],  \Gamma\big)
  \,.
$$

We need to show that any such map is pointed-homotopic to the map constant on the basepoint $t \mapsto \gamma(t)(\ast)$. By Lemma \ref{DeformationOfCechCoboundaryGivesConcordanceOfCocycles} this is the case if

$$
  \pi_{n+2}
  \Big(
     Maps\big([0,1],  \Gamma\big)
  \Big)
  \;\simeq\;
  \pi_{n+2}
  (
     \Gamma
  )
  \;\overset{!}{=}\;
  \ast
$$

and this holds by the truncation assumption (eq:GammaIsNTruncated).
\end{proof}

\begin{remark}
\label{ConcordancesOfHigherGerbesOnSphericalSpaceFormHaveGroupStructure}
  With (eq:DeloopingOfShapeIsInfinityGroup) in Assumption \ref{AssumptionForHigherGerbesOnSphericalSpaceForms}, Prop. \ref{ConcordancesOnSphericalSpaceFormIsMapsOfClassifyingSpaces} implies that the space of concordances has [[infinity-group|$\infty$-group]] structure:
$$
  \esh
  \,
  Maps
  \big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \big)
  \;\;\;
  \in
  \;
  Grp(Grpd_\infty)
  \,.
$$
\end{remark}

\begin{proposition}
\label{ComparisonMorphismIsoOnFundamentalGroups}
The comparison morphism (Def. \ref{ComparisonMorphismBetweenShapesOfMappingStacks}) is an [[isomorphism]] on [[fundamental groups]] (for any basepoint):
$$
  \pi_1
  \big(
    \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma)
  \big)
  \;\;
  \colon
  \;\;
  \pi_1
  \bigg(
  \esh
  \,
  Map
  \Big(
    \mathbf{B}G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \bigg)
  \xrightarrow
  {\;\;\;
     \sim
  \;\;\;}
  \pi_1
  \bigg(
  \esh
  \,
  Map
  \Big(
    S^{n+2}/G
    ,\,
    \mathbf{B}\Gamma
  \Big)
  \bigg)
$$
\end{proposition}
\begin{proof}
  By Lemma \ref{ComparisonMorphisnInjectiveOnFundamentalGroups} this function is [[injective]] on fundamental groups.

In order to see surjectivity, we may use that the $\infty$-groupoid of concordances has
[[infinity-group|group structure]] (Rem. \ref{ConcordancesOfHigherGerbesOnSphericalSpaceFormHaveGroupStructure}), which implies that all its [[connected components]] have isomorphic [[homotopy groups]]. Therefore it is sufficient to consider the connected component of the trivial cocycle $c$. Here, Lemma \ref{ComparisonMorphismSurjectiveOnFundamentalGroups} gives the surjectivity.
\end{proof}

(...)

## Literature

See also the references at *[[group actions on spheres]]*.

### Riemannian spherical space forms

Historically and by default, spherical space forms are understood in [[Riemannian geometry]] hence with finite groups acting (freely and) by [[isometries]] on [[n-spheres]].

Historical articles:

* {#Killing1891} [[Wilhelm Killing]], _Ueber die Clifford-Klein’schen Raumformen_, Math. Ann. 39 (1891), 257–278 ([doi:10.1007/BF01206655](https://doi.org/10.1007/BF01206655))

Solution of the classification problem:

* {#Wolf74} [[Joseph Wolf]], _Spaces of constant curvature_, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 ([doi:10.1090/chel/372](https://doi.org/10.1090/chel/372))

Streamlined re-proof:

* {#Allcock15} [[Daniel Allcock]], _Spherical space forms revisited_, Trans. Amer. Math. Soc. 370 (2018), 5561-5582  ([arXiv:1509.00906](https://arxiv.org/abs/1509.00906), [doi:10.1090/tran/7167](https://doi.org/10.1090/tran/7167))

Discussion for the [[7-sphere]] with application to [[near horizon geometries]] of [[M2-brane]] ([[AdS/CFT duality|AdS/CFT dual]] to the [[ABJM model]]):

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* {#Gadhia07} [[Sunil Gadhia]], _Supersymmetric quotients of M-theory and supergravity backgrounds_, PhD thesis, School of Mathematics, University of Edinburgh, 2007 ([spire:1393845](http://inspirehep.net/record/1393845/))

### Topological spherical space form

General quotients of $n$-spheres by finite group actions that are (free and) just required to be continuous are known as *topological spherical space forms*:

* {#MadsenThomasWall76} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *The topological spherical space form problem II: existence of free actions*, Topology Volume 15, Issue 4, 1976, Pages 375-382 (<a href="https://doi.org/10.1016/0040-9383(76)90031-8">doi:10.1016/0040-9383(76)90031-8</a>)

* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*,  Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))

Review:

* [[Ian Hambleton]], _Topological spherical space forms_, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston ([arXiv:1412.8187](https://arxiv.org/abs/1412.8187), [ISBN:9781571463012](https://www.intlpress.com/site/pub/pages/books/items/00000439/index.php))

### Smooth spherical space forms

If the action by $G$ on the $n$-sphere is (free and) [[smooth function|smooth]], one speaks of *smooth spherical space forms*:

* [Madsen, Thomas and Wall 1983](#MadsenThomasWall83)

* {#Madsen78} [[Ib Madsen]], *Smooth spherical space forms*, Geometric Applications of Homotopy Theory I. Springer 1978. 303-352 ([doi:10.1007/BFb0069224](https://link.springer.com/book/10.1007/BFb0069224),  [pdf](https://link.springer.com/content/pdf/10.1007/BFb0069243.pdf))

[[!redirects spherical space forms]]


spherical T-duality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Spherical T-duality_ ([Bouwknegt-Evslin-Mathai 14a](#BouwknegtEvslinMathai14a)) is the name given to a variation of [[topological T-duality]] where the role of the [[circle]] $S^1$, or the [[circle group]] $U(1)$, is replaced by the [[3-sphere]] $S^3$, or the [[special unitary group]] $SU(2)$. Where topological T-duality relates pairs consisting of total spaces of $U(1)$-[[principal bundles]] equipped with a [[cocycle]] in degree-3 [[ordinary cohomology]], spherical T-duality relates pairs consisting of $SU(2)$-[[principal bundles]] (or just $S^3$-[[fiber bundles]] ([Bouwknegt-Evslin-Mathai 14b](#BouwknegtEvslinMathai14b))) equipped with cocycles in degree-7 cohomology. As for topological T-duality, under suitable conditions spherical T-duality lifts to an isomorphism of [[twisted K-theory]] classes of these bundles with twisting by the 7-class.

In the approximation of [[schreiber:Introduction to Higher Supergeometry|rational super homotopy theory]], topological spherical T-duality has been derived for the [[M5-brane]], not on 11d [[super Minkowski spacetime]] itself, but on its [[M2-brane]]-[[extended super Minkowski spacetime]], and from there on the [[exceptional geometry|exceptional super spacetime]]; see [FSS 18a](#FiorenzaSatiSchreiber18a), reviewed in [FSS 18b](#FiorenzaSatiSchreiber18b).

## Related concepts

* A relation to _[[Poisson-Lie T-duality]]_ for [[special unitary group|SU(2)]] might seem plausible, or not; but this remains open.

## References

The idea is due to

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], _A higher categorical analogue of topological T-duality for sphere bundles_, Annals of K-Theory, Vol. 5 (2020), No. 1, 1–42 ([arXiv:1601.06285](http://arxiv.org/abs/1601.06285), [doi:doi:10.2140/akt.2020.5.1](https://msp.org/akt/2020/5-1/p01.xhtml))

which in the course considers higher [[Snaith theorem|Snaith spectra]] and higher order [[iterated algebraic K-theory]].

A special case of this general story is discussed in some detail in

* {#BouwknegtEvslinMathai14a} [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], _Spherical T-Duality_, Commun. Math. Phys. 337:909-954, 2015 ([arXiv:1405.5844](http://arxiv.org/abs/1405.5844))

* {#BouwknegtEvslinMathai14b} [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], _Spherical T-duality II: An infinity of spherical T-duals for non-principal SU(2)-bundles_, J.Geom.Phys.92:46-54, 2015 ([arXiv:1409.1296](http://arxiv.org/abs/1409.1296))

* [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], _Spherical T-Duality and the spherical Fourier-Mukai transform_ ([arXiv:1502.04444](http://arxiv.org/abs/1502.04444))

See also

* Lachlan Macdonald, [[Varghese Mathai]], Hemanth Saratchandran, _On the Chern character in Higher Twisted K-theory and spherical T-duality_ ([arXiv:2007.02507](https://arxiv.org/abs/2007.02507))

The realization in [[M-theory]] at the level of rational super homotopy theory is derived in

* {#FiorenzaSatiSchreiber18a} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Higher T-duality of super M-branes]]_ ([arXiv:1803.05634](https://arxiv.org/abs/1803.05634))

* {#FiorenzaSatiSchreiber18b} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Higher T-duality in M-theory via local supersymmetry]]_, Physics Letters B Volume 781, 10 June 2018, Pages 694-698 ([arXiv:1805.00233](https://arxiv.org/abs/1805.00233), [doi:10.1016/j.physletb.2018.04.058](https://doi.org/10.1016/j.physletb.2018.04.058))

[[!redirects spherical T-dualities]]


spiking neural network

#Contents#
* table of contents
{:toc}

## Idea

A spiking neural network is a [[neural network]] which behaves in a time dependent manner similar to biological neurons: neurons only send a signal only when a threshold is reached.

## See also

* [[neural network]]

* [[neuromorphic computing]]

## References

* Amirhossein Tavanaei, Masoud Ghodrati, Saeed Reza Kheradpisheh, Timothee Masquelier, Anthony S. Maida, *Deep Learning in Spiking Neural Networks*, Neural Networks, Volume 111, March 2019, Pages 47-63 ([arXiv:1804.08150](https://arxiv.org/abs/1804.08150), [doi:10.1016/j.neunet.2018.12.002](https://doi.org/10.1016/j.neunet.2018.12.002))

* Milad Mozafari, Mohammad Ganjtabesh, Abbas Nowzari-Dalini, Timothée Masquelier, *SpykeTorch: Efficient Simulation of Convolutional Spiking Neural Networks with at most one Spike per Neuron*, Frontiers in Neuroscience, 13:625. ([arXiv:1903.02440](https://arxiv.org/abs/1903.02440), [doi:10.3389/fnins.2019.00625](https://doi.org/10.3389/fnins.2019.00625))

* Benjamin Cramer, Sebastian Billaudelle, Simeon Kanya, Aron Leibfried, Andreas Grübl, Vitali Karasenko, Christian Pehle, Korbinian Schreiber, Yannik Stradmann, Johannes Weis, Johannes Schemmel, Friedemann Zenke, *Surrogate gradients for analog neuromorphic computing*, [[Proceedings of the National Academy of Sciences of the United States of America]], Volume 119, No. 4, January 14, 2022 ([arXiv:2006.07239](https://arxiv.org/abs/2006.07239), [doi:10.1073/pnas.2109194119](https://doi.org/10.1073/pnas.2109194119))

* Shikhar Gupta, Arpan Vyas, Gaurav Trivedi, *FPGA Implementation of Simplified Spiking Neural Network* ([arXiv:2010.01200](https://arxiv.org/abs/2010.01200))

* Youngeun Kim, Priyadarshini Panda, *Revisiting Batch Normalization for Training Low-latency Deep Spiking Neural Networks from Scratch* ([arXiv:2010.01729](https://arxiv.org/abs/2010.01729))

* Youngeun Kim, Priyadarshini Panda, *Visual Explanations from Spiking Neural Networks using Interspike Intervals*, ([arXiv:2103.14441](https://arxiv.org/abs/2103.14441))

* Zhehui Wang, Xiaozhe Gu, Rick Goh, Joey Tianyi Zhou, Tao Luo, *Efficient Spiking Neural Networks with Radix Encoding* ([arXiv:2105.06943](https://arxiv.org/abs/2105.06943))

* Jason K. Eshraghian, Max Ward, Emre Neftci, Xinxin Wang, Gregor Lenz, Girish Dwivedi, Mohammed Bennamoun, Doo Seok Jeong, Wei D. Lu, *Training Spiking Neural Networks Using Lessons From Deep Learning*, ([arXiv:2109.12894](https://arxiv.org/abs/2109.12894))

Wikipedia article: [Spiking neural network](https://en.wikipedia.org/wiki/Spiking_neural_network)


spin

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=--
#### Quantum systems
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The __spin__ of a [[particle]] in [[quantum mechanics]]/[[quantum field theory]] is its intrinsic [[angular momentum]], but subject to the possibility that the particle may be a [[fermion]] so that that its [[angular momentum]] is not determined by the [[representation theory]] of the [[special orthogonal group]], but of the [[spin group]].

## Properties

### Spin-Statistics theorem

By the [[spin-statistics theorem]], the spin of any [[elementary particle]] must be (in magnitude) half of an [[integer]] multiple of [[Planck's constant]] ($\hbar = h/2\pi$), an integer if the particle is a [[boson]] but only a half-integer if the particle is a [[fermion]].

## Related concepts

* [[spin group]],

* [[spin representation]], [[real spin representation]]

* [[Fierz identity]]

* [[proton spin crisis]]

* [[spinor]]

* [[Dirac spinor]], [[Weyl spinor]], [[Majorana spinor]]

* [[Dirac field]]

* [[Ising model]], [[Heisenberg model]]

* [[quantization of the 2-sphere]]

* [[supersymmetry]]

* [[division algebras and supersymmetry]]

* [[geometry of physics -- supersymmetry]]

## References
 {#References}

The concept of intrinsic spin of the [[electron]] (making it a "[[spinor]]") originates with:

* [[Samuel A. Goudsmit]], [[George E. Uhlenbeck]]: *Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons*, Naturwissenschaften **13** 47 (1925) 953 [pdf, [[GoudsmitUhlenbeck-ElektronSpin.pdf:file]]]

Historical recollection:

* [[Samuel A. Goudsmit]], *The discovery of the electron spin*, in: *Foundations of Modern EPR*, World Scientific (1998) [[pdf](https://www.ifsc.usp.br/~laf/quantica1/goudsmit_discovery_of_spin.pdf), [webpage](https://lorentz.leidenuniv.nl/history/spin/goudsmit.html), [doi:10.1142/3624](https://doi.org/10.1142/3624)]

* [[Richard G. Milner]], *A Short History of Spin*, [Proceedings of PSTP2013](https://pos.sissa.it/182/PSTP2013), POS **182** (2014) [[arXiv:1311.5016](https://arxiv.org/abs/1311.5016), [doi:10.22323/1.182.0003 ](https://doi.org/10.22323/1.182.0003 )]

[[!redirects spins]]


spin bordism

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* tabe of contents
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## Idea

[[B-bordism]] for [[spin structure]]

## References

* [Manifold Atlas](http://www.map.mpim-bonn.mpg.de/Main_Page), _[Spin bordism](http://www.map.mpim-bonn.mpg.de/Spin_bordism)_

[[!redirects Spin bordism]]


spin chain

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Heisenberg model]]

* [[Bethe ansatz]]

* [[integrable system]]

* [[Kitaev spin chain]]

## References

### General

Review:

* [[Ingmar Saberi]], _An introduction to spin systems for mathematicians_ [[arXiv:1801.07270](https://arxiv.org/abs/1801.07270)]

The original articles:

* [[Hans Bethe]], _Zur Theorie der Metalle_, Z. Phys.71, 205 (1931) ([doi:10.1007/BF01341708](https://doi.org/10.1007/BF01341708))

Further development:

* Mari Carmen Banuls, Michal P. Heller, Karl Jansen, Johannes Knaute, Viktor Svensson, _From spin chains to real-time thermal field theory using tensor networks_ ([arXiv:1912.08836](https://arxiv.org/abs/1912.08836))

  (relation to [[tensor networks]])

See also:

* Taro Kimura, Rui-Dong Zhu, *Bethe/Gauge Correspondence for $SO$/$Sp$ Gauge Theories and Open Spin Chains*, JHEP **2021** 227 (2021) [[arXiv:2012.14197](https://arxiv.org/abs/2012.14197), doi:10.1007/JHEP03(2021)227]

* Ziwei Wang, Rui-Dong Zhu, *Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries* [[arXiv:2401.00764](https://arxiv.org/abs/2401.00764)]

### For single trace operators in super Yang-Mills theory

Identifiication of [[spin chain]] dynamics in the [[action]] of the [[dilatation operator]] in [[super Yang-Mills theory]], specifically [[D=4 N=4 super Yang-Mills theory]], on [[single trace operators]]/[[BMN operators]] and correspondence to [[Green-Schwarz superstrings]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS-CFT correspondence]]:

* {#MinahanZarembo02}  J. A. Minahan, [[Konstantin Zarembo]], _The Bethe-Ansatz for $N=4$ Super Yang-Mills_, JHEP 0303 (2003) 013 ([arXiv:hep-th/0212208](https://arxiv.org/abs/hep-th/0212208))

* {#BeisertStaudacher03} [[Niklas Beisert]], [[Matthias Staudacher]], _The $\mathcal{N}=4$ SYM Integrable Super Spin Chain_,
Nucl. Phys. B670:439-463, 2003 ([arXiv:hep-th/0307042](https://arxiv.org/abs/hep-th/0307042))

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

Review includes

* {#BBGK04} A. V. Belitsky, [[Volker Braun]], A. S. Gorsky, G. P. Korchemsky, _Integrability in QCD and beyond_, Int. J. Mod. Phys. A19:4715-4788, 2004 ([arXiv:hep-th/0407232](https://arxiv.org/abs/hep-th/0407232))

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

For [[D3-D7 brane bound states]] and [[D3-D5 brane bound states]]:

* Marius de Leeuw, Tamás Gombor, Charlotte Kristjansen, Georgios Linardopoulos, Balázs Pozsgay, _Spin Chain Overlaps and the Twisted Yangian_ ([arXiv:1912.09338](https://arxiv.org/abs/1912.09338))

On [[supersymmetry|supersymmetric]] [[spin chains]] carrying [[braid group representations]] ("[[anyons]]"):

* Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, *Topological Quantum Computation on Supersymmetric Spin Chains* [[arXiv:2209.03822](https://arxiv.org/abs/2209.03822)]

On [[topological phases]] (i.e. gapped  [[ground states]]) of spin chains:

* [[Houssam Abdul-Rahman]], Marius Lemm, Angelo Lucia, [[Bruno Nachtergaele]], Amanda Young,  *A class of two-dimensional AKLT models with a gap*, in: *Analytic Trends in Mathematical Physics*,  Contemporary Mathematics **741**,  AMS (2020) [[arXiv:1901.09297](https://arxiv.org/abs/1901.09297), [doi:10.1090/conm/741/14917](https://doi.org/10.1090/conm/741/14917)]

On simulation of quantum spin chains on [[quantum computers]]:

* M. Gluza, M. Kliesch, [[Jens Eisert]], [[Leandro Aolita]], *Fidelity witnesses for fermionic quantum simulations*, Phys. Rev. Lett. **120** 190501 (2018) [[arXiv:1703.03152](https://arxiv.org/abs/1703.03152), [doi:10.1103/PhysRevLett.120.190501](https://doi.org/10.1103/PhysRevLett.120.190501)]

In [[holographic CMT]]:

* Naoto Yokoi, Yasuyuki Oikawa, Eiji Saitoh, *Holographic Dual of Quantum Spin Chain as Chern-Simons-Scalar Theory* [[arXiv:2310.01890](https://arxiv.org/abs/2310.01890)&rbrack:

[[!redirects spin chains]]


Spin Chern-Simons theory

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#### Quantum field theory
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#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Spin Chern-Simons theory_ is a [[prequantum field theory]]/[[quantum field theory]] like [[Chern-Simons field theory]] but defined on/restricted to 3-manifolds equipped with [[spin structure]] and making use of that structure to divide the [[action functional]] (in the exponent) by 2.

(Beware that there is also ordinary $G$-Chern-Simons theory for [[gauge group]] $G = Spin(n)$ a [[spin group]], which in traditional parlance one might also pronounce as "Spin Chern-Simons theory", but which is different, in general, from Spin Chern-Simons theory in the sense discussed here.)

The division by 2 makes the [[AdS3-CFT2 and CS-WZW correspondence|holographically dual]] theory in 2d be the correct [[self-dual higher gauge theory|self-dual theory]].  The generalization of the [[Spin structure]] to [[higher dimensional Chern-Simons theory]] is that of [[integral Wu structure]]. In the next relevant case of [[7d Chern-Simons theory]] this is related to the flux quantizaton condition on the [[supergravity C-field]] wholse holographically related [[self-dual higher gauge field]] is the 2-form-field in the [[6d (2,0)-superconformal QFT]] on the [[M5-brane]].

## Properties

### Relation to framed Chern-Simons theory

Notice that if a 3-manifold admits a [[spin structure]] then it also admits a [[framing]]. (...)

## Related concepts

* [[analytically continued Chern-Simons theory]]

* [[holomorphic Chern-Simons theory]]

* [[eta invariant]]

[[!include square roots of line bundles - table]]

## References

For general ([[compact Lie group|compact]]) [[Lie groups]] as [[gauge groups]] spin Chern-Simons theory is discussed in

* [[Jerome Jenquin]], _Classical Chern-Simons on manifolds with spin structure_ ([arXiv:0504524](http://arxiv.org/abs/math/0504524))

* [[Jerome Jenquin]], _Spin Chern-Simons and Spin TQFTs_ ([arXiv:0605239](http://arxiv.org/abs/math/0605239))

For [[abelian group|abelian]] [[gauge groups]] Spin Chern-Simons theory is discussed in

* Dmitriy Belov, [[Gregory Moore]], _Classification of abelian spin Chern-Simons theories_ ([arXiv:hep-th/0505235](http://arxiv.org/abs/hep-th/0505235))

and with emphasis on the holographically dual [[self-dual higher gauge theory]] in

* {#BelovMoore06} Dmitriy Belov, [[Gregory Moore]], _Holographic Action for the Self-Dual Field_ ([arXiv:hep-th/0605038](http://arxiv.org/abs/hep-th/0605038))

This is based on work by Witten and Hopkins-Singer, see the references at _[[self-dual higher gauge theory]]_.

[[!redirects spin Chern-Simons theory]]


spin connection

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## Idea

The term _spin connection_ is traditionally used in [[physics]] -- for instance in [[first-order formulation of gravity]] -- to denote a [[connection on a bundle|connection]] on the [[tangent bundle]] of a [[manifold]] with [[spin structure]] given as a [[special orthogonal Lie algebra]]-valued connection on the underlying [[special orthogonal group]]-[[principal bundle]]. The term is mainly used to distinguish from the the connection expressed in [[Christoffel symbol]]s.  Since for this connection alone the [[spin structure]] is not important, the term _spin connection_ is often used for $\mathfrak{so}$-connecition on any orientable manifold.

## Related entries

See at _[[field (physics)]]_ the section _[Ordinary gravity](field%20%28physics%29#OrdinaryGravity)_.

## References

* _[[geometry of physics]]_

[[!redirects spin connections]]


spin foam

#Contents#
* table of contents
{:toc}

## Idea

*Spin foam* models are particular models aimed at finding a theory of  quantum gravity. Their relation to Einstein-Hilbert gravity in classical limit seems however not to be convincingly argued for in the literature.

## Related concepts

* [[spin network]]

* [[quantum gravity]]

* [[Boulatov model]], [[group field theory]], [[zoranskoda:ncFourier]], [[Etera R. Livine]]

## References

* [[John Baez]], _Spin foam models_, Class. Quantum Grav. __15__ (1998), 1827-1858, [gr-qc/9709052](http://arxiv.org/abs/gr-qc/9709052)
* Winston J. Fairbairn, Etera R. Livine,  _3D spinfoam quantum gravity: matter as a phase of the group field theory-_, Classical Quantum Gravity __24__ (2007), no. 20, 5277–5297 [MR2351342](http://www.ams.org/mathscinet-getitem?mr=2351342),[doi](http://dx.doi.org/10.1088/0264-9381/24/20/021)
* D. V. Boulatov, _A model of three-dimensional lattice gravity_,
Modern Phys. Lett. A __7__ (1992), no. 18, 1629–1646, [MR94c:83039](http://www.ams.org/mathscinet-getitem?mr=1168831), [doi](http://dx.doi.org/10.1142/S0217732392001324)
* A. Perez, _Spin foam models for quantum gravity_, Class. Quant. Grav. 20 (2003) R43, arXiv:gr-qc/0301113
* Sergei Alexandrov, Philippe Roche, _Critical overview of loops and foams_, [arxiv/1009.4475](http://arxiv.org/abs/1009.4475)
* Winston Fairbairn, [[Etera R. Livine]], _3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory_, Class.Quant.Grav. __24__:5277-5297, 2007, [arxiv/gr-qc/0702125](http://arxiv.org/abs/gr-qc/0702125), [doi](http://dx.doi.org/10.1088/0264-9381/24/20/021)

[[!redirects spinfoam]]
[[!redirects spin foam model]]
[[!redirects spinfoam model]]


Spin geometry

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=--
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This page collects links related to the book

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]]

  _Spin geometry_

  Princeton University Press (1989)

  [ISBN 9780691085425](https://press.princeton.edu/books/hardcover/9780691085425/spin-geometry-pms-38-volume-38)

on [[spin geometry]].

#Contents#
* table of contents
{:toc}

## Chapter I Clifford algebras, Spin groups and Their Representations

### 1 Clifford algebras

* [[Clifford algebra]]

### 2 The group $Pin$ and $Spin$

* [[spin group]]

### 3 The algebras $CL_n$ and $Cl_{r,s}$

### 4 Classification

### 5 Representations

* [[spin representation]]

### 6 Lie algebra structures

### 7 Some direct applications to geometry

### 8 Some further applications to the theory of Lie groups

### 9 K-Theory and the Atiyah-Bott-Shapiro construction

* [[K-theory]]

* [[Atiyah-Bott-Shapiro isomorphism]]

### 10 KR-theory and the $(1,1)$-Periodicity theorem

* [[KR-theory]]

## Chapter II Spin Geometry and the Dirac Operator

### 1 Spin structures on vector bundles

* [[spin structure]]

### 2 Spin manifolds and Spin cobordism

* [[spin cobordism]]

### 3 Clifford and spinor bundle

* [[Clifford bundle]]

* [[spinor bundle]]

### 4 Connection on spinor bundle

* [[connection on a bundle]]

* [[spin connection]]

### 5 The Dirac operators

* [[Dirac operator]]

### 6 The fundamental elliptic operators

* [[elliptic operator]]

### 7 $Cl_k$-linear Dirac operators

* [[fiber integration in K-theory]]

* [[index]]

### 8 Vanishing theorems and some applications

## Chapter III Index Theorems

* [[index theory]]

* [[index of a Dirac operator]]

## Chapter IV Applications in Geometry and Topology

* [[spin geometry]]

category: reference


spin geometry

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{:toc}

## Idea

The [[differential geometry]] of [[manifolds]] with [[spin structure]] is called **spin geometry**. It studies [[spin group]]-[[principal bundles]], [[spin representations]], the [[associated bundle|associated]] [[spinor bundles]], and the [[Dirac operators]] acting on [[spaces of sections]] of these bundles, hence also their [[index theory]] and generally [[K-theory]].

## In physics

The relevance of spin geometry in [[physics]] rests on the fact that in [[quantum mechanics]] and [[quantum field theory]] in general and in the [[standard model of particle physics]] in particular, [[fermions]] such as the [[electron]] are mathematically modeled as [[sections]] of spin-bundles. The very term _spin_ originates in the fact that the quanta of these fields behave to some extent as if they had an intrinsic [[angular momentum]], as if they were _spinning_ about an axis as a classical top.

Spin geometry also plays a central role in supersymmetric quantum field theory such as [[supergravity]].

## Related concepts

* [[spin group]]. [[spin representation]], [[spinor]], [[spin]]

* [[Fierz identity]]

* [[geometric algebra]]

## References

The classical monograph on spin geometry is

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

Discussion with an eye towards [[mathematical physics]]:

* [[Gerd Rudolph]], [[Matthias Schmidt]], Chapter 5 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

Lecture notes:

* {#FigueroaFarrill10} [[José Figueroa-O'Farrill]], _[PG course on Spin Geometry](https://empg.maths.ed.ac.uk/Activities/Spin/)_, 2010

Fundamentals of the relevant [[supergeometry]] are in

* [[Pierre Deligne]], [[Daniel Freed]], _Supersolutions_ ([arXiv:hep-th/9901094](http://arxiv.org/abs/hep-th/9901094))

  in [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

Discussion in the context of [[conformal geometry]] is in

* [[Pierre Anglès]], section 2.2.1.2 of _Conformal Groups in Geometry and Spin Structures_, Progress in Mathematical Physics 2008


spin group

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#Contents#
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## Idea

The __spin group__ $Spin(n)$ is the [[universal covering space]] of the [[special orthogonal group]] $SO(n)$. By the usual arguments it inherits a group structure for which the operations are smooth and so is a [[Lie group]] like $SO(n)$.

For special cases in low dimensions see at:  [[Spin(2)]], [[Spin(3)]], [[Spin(4)]], [[Spin(5)]], [[Spin(6)]], [[Spin(7)]], [[Spin(8)]]

## Definition

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

A _quadratic vector space_ $(V, \langle -,-\rangle)$ is a [[vector space]] $V$ over finite [[dimension]] over a [[field]] $k$ of [[characteristic]] 0, and equipped with a symmetric [[bilinear form]] $\langle -,-\rangle \colon V \otimes V \to k$.

=--

Conventions as in ([Varadarajan 04, section 5.3](#Varadarajan04)).

We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding [[quadratic form]].

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

The _[[Clifford algebra]]_ $CL(V,q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[associative algebra]] over $k$ which is the [[quotient]]

$$
  Cl(V,q)
    \coloneqq
  T(V)/I(V,q)
$$

of the [[tensor algebra]] of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.

=--

Since the [[tensor algebra]] $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.

Let $(\mathbb{R}^n, q = {\vert -\vert})$ be the $n$-dimensional [[Cartesian space]] with its canonical [[scalar product]]. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the [[complexification]] of its [[Clifford algebra]].

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

There exists a unique [[complex number|complex]] [[representation]]

$$
  Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)
$$

of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest [[dimension]]

$$
  dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]}
  \,.
$$

=--

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

The [[Pin group]] $Pin(V;q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[subgroup]] of the [[group of units]] in the [[Clifford algebra]] $Cl(V,q)$

$$
  Pin(V,q) \hookrightarrow GL_1(Cl(V,q))
$$

on those elements which are multiples $v_1 \cdots v_{n}$ of elements $v_i \in V$ with $q(v_i) = 1$.

The [[Spin group]] $Spin(V,q)$ is the further [[subgroup]] of $Pin(V;q)$ on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(v_i) = 1$.

Specifically, "the" Spin group is

$$
  Spin(n) \coloneqq Spin(\mathbb{R}^n)
  \,.
$$

=--

A _[[spin representation]]_ is a [[linear representation]] of the spin group, def. \ref{SpinGroup}.

## Properties

### General

By definition the spin group sits in a [[short exact sequence]] of groups

$$
  \mathbb{Z}_2 \to Spin \to SO
  \,.
$$

### Relation to Whitehead tower of orthogonal group

The spin group is one element in the [[Whitehead tower]] of $O(n)$, which starts out like

$$
  \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n)
  \,.
$$

The [[homotopy group]]s of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$

$$
  \array{
     \pi_{8k+0}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+1}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+2}(O) & = 0
     \\
     \pi_{8k+3}(O) & = \mathbb{Z}
     \\
     \pi_{8k+4}(O) & = 0
     \\
     \pi_{8k+5}(O) & = 0
     \\
     \pi_{8k+6}(O) & = 0
     \\
     \pi_{8k+7}(O) & = \mathbb{Z}
  }
  \,.
$$

By [[Whitehead tower|co-killing]] these groups step by step one gets

$$
  \array{
     cokill\, this &&&& to\,get
     \\
     \\
     \pi_{0}(O) & = \mathbb{Z}_2 &&& SO
     \\
     \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin
     \\
     \pi_{2}(O) & = 0
     \\
     \pi_{3}(O) & = \mathbb{Z} &&& String
     \\
     \pi_{4}(O) & = 0
     \\
     \pi_{5}(O) & = 0
     \\
     \pi_{6}(O) & = 0
     \\
     \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane
  }
  \,.
$$

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

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

### Exceptional isomorphisms
 {#ExceptionalIsomorphisms}

In low [[dimensions]] the [[spin groups]] happens to be [[isomorphism|isomorphic]]  to various other [[classical Lie groups]]. One speaks of _[[exceptional isomorphisms]]_ or _[[sporadic isomorphisms]]_.

See for instance ([Garrett 13](#Garrett13)). See also _[[division algebra and supersymmetry]]_.

In the following $Sp(n)$ denotes the [[quaternionic unitary group]] in [[quaternion|quaternionic]] [[dimension]] $n$.

We have

* in [[Euclidean geometry|Euclidean]] signature

  * $Spin(1) \simeq O(1)$

  * [[Spin(2)]] $\simeq U(1) \simeq SO(2) \simeq S^1$ ([[SO(2)]], the [[circle group]], see also at _[[real Hopf fibration]]_)

    the projection $Spin(2)\to SO(2)$ corresponds to $S^1\stackrel{\cdot 2}{\longrightarrow} S^1$, see also at _[[Theta characteristic]]_

  * [[Spin(3)]] $\simeq Sp(1) \simeq SU(2) \simeq S^3$ (the [[special unitary group]] [[SU(2)]]

    the inclusion $Spin(2) \hookrightarrow Spin(3)$ corresponds to the canonical $S^1 \hookrightarrow S^3$ (see e.g. [Gorbunov-Ray 92](#GorbunovRay92))

  * [[Spin(4)]] $\simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3$

    this is given by identifying $\mathbb{R}^4 \simeq \mathbb{H}$ with the [[quaternions]] and $SU(2) \simeq S^3$ with the group of unit quternions. Then left and right quaternion multiplication gives a [[homomorphism]]

    $$
      SU(2) \times SU(2) \longrightarrow SO(4)
    $$

    $$
      (g,h) \mapsto ( x \mapsto \; g^{-1}  x h )
    $$

    which is a [[double cover]] and hence exhibits the isomorphism.

    In particular therefore the inclusion $Spin(3) \hookrightarrow Spin(4)$ corresponds to the [[diagonal]] $S^3 \hookrightarrow S^3 \times S^3$.

    At the level of [[Lie algebras]] $\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4$ and the $\pm 1$-[[eigenspaces]] of the [[Hodge star operator]] $\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4$ gives the [[direct sum]] decomposition $\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)$

  * [[Spin(5)]] $\simeq Sp(2)$ (an indirect consequence of [[triality]], see e.g. [Čadek-Vanžura 97](#CadekVanzura97))

  * [[Spin(6)]] $\simeq SU(4)$ (the [[special unitary group]] [SU(4)](special+unitary+group#SU4))

* in [[Lorentz group|Lorentzian]] signature

  * $Spin(1,1) \simeq GL(1,\mathbb{R})$

  * $Spin(2,1) \simeq SL(2, \mathbb{R})$ -- 2d [[special linear group]] of [[real numbers]]

  * $Spin(3,1) \simeq SL(2,\mathbb{C})$ -- 2d [[special linear group]] of [[complex numbers]]

  * $Spin(4,1) \simeq Sp(1,1)$

  * $Spin(5,1) \simeq SL(2,\mathbb{H})$ -- 2d [[special linear group]] of [[quaternions]]

  * $Spin(9,1) \simeq_{in\;some\;sense} SL(2, \mathbb{O})$ -- 2d [[special linear group]] of [[octonions]] (see [[SL(2,O)]] for more on this would-be isomorphism)

* in [[anti de Sitter group|anti de Sitter]] signature

  * $Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$

  * $Spin(3,2) \simeq Sp(4,\mathbb{R})$

  * $Spin(4,2) \simeq SU(2,2)$

* in mixed signature

  * $Spin(3,3) \simeq SL(4,\mathbb{R})$ ([Garrett 13 (2.12)](#Garrett13))

Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

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

## Examples
 {#ExampleSection}

[[!include low dimensional rotation groups -- table]]

* [[Spin(11,3)]]

## Applications

### Spin geometry

See [[spin geometry]]

### In physics

The name arises due to the requirement that the structure group of the [[tangent bundle]] of [[spacetime]] lifts to $Spin(n)$ so as to 'define particles with spin'... (Someone more awake and focused please put this into proper words!)

See [[spin structure]].

## Related concepts

* [[spinor]], [[spin representation]], [[spinor bundle]]

The [[Whitehead tower]] of the [[orthogonal group]] looks like

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

* [[spin^c group]], [[spin^h group]]

* [[metaplectic group]]

* [[semi-spin group]]

[[!include table of orthogonal groups and related]]

## References

Textbook accounts:

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter I, section 2 of _[[Spin geometry]]_, Princeton University Press (1989)

* [[Howard Georgi]], §21 & 22 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to [[spinors]] in (the [[standard model of particle physics|standard model]] of) [[particle physics]]

See also

* {#Varadarajan04} [[Veeravalli Varadarajan]], section 7 of _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

* wikipedia _[Spin group](http://en.wikipedia.org/wiki/Spin_group)_

Examples of sporadic (exceptional) spin group isomorphisms incarnated as [[isogenies]] onto [[orthogonal groups]] are discussed in

* {#Garrett13} [[Paul Garrett]], _Sporadic isogenies to orthogonal groups_ (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), [[Garrett-SporadicIsogenies.pdf:file]] ]

* {#GorbunovRay92} [[Vassily Gorbunov]], [[Nigel Ray]], _Orientations of $Spin$ Bundles and Symplectic Cobordism_, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]], [doi: 10.2977/prims/1195168855](https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4))

The [[exceptional isomorphism]] [[Spin(5)]] $\simeq$ [[Sp(2)]] is discussed via [[triality]] in

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures  in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))

Discussion of the [[cohomology]] of the [[classifying space]] $B Spin$ includes

* E. Thomas, _On the cohomology groups of the classifying space for the stable spinor groups_, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.

* {#Pittie91} [[Harsh Pittie]], _The integral homology and cohomology rings of SO(n) and Spin(n)_, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

[[!redirects spin groups]]

[[!redirects Spin group]]
[[!redirects Spin groups]]

[[!redirects Spin]]
[[!redirects Spin(n)]]

[[!redirects spin-group]]

[[!redirects spin groups]]
[[!redirects spin-groups]]


spin Hall effect

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

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] the _spin Hall effect_ refers to phenomena roughly analogous to the standard _[[Hall effect]]_, but with the role of [[electric charge]] played by [[spin]]-polarization of particles/excitations:

A material shows a _spin Hall effect_ if a [[current]] of [[spinning particles]]/excitation through the material induces a separation of particles of different spin polarization. This may happen either due to spin-dependent [[Mott scattering]] or due to intrinsic [[spin-orbit coupling]] ([Dyakonov-Perel 71](#DyakonovPerel71)) playing the
role of the external [[magnetic field]] in the standard [[Hall effect]].

In particular, no external [[magnetic field]] is needed to induce a spin Hall effect; on the contrary, an a external magnetic field may suppress a spin Hall effect, due to the induced spin [[precession]] ([Dyakonov-Perel 71](#DyakonovPerel71)).

## Related concepts

* [[Hall effect]], [[quantum Hall effect]]

* [[quantum spin Hall effect]]

## References

The original description of the effect via [[spin-orbit coupling]]:

* {#DyakonovPerel71} M. I. Dyakonov, V. I. Perel, _Current-induced spin orientation of electrons in semiconductors_, Physics Letters A Volume 35, Issue 6, 12 July 1971, Pages 459-460 (doi:10.1016/0375-9601(71)90196-4)

See also:

* Wikipedia, _[Spin Hall effect](https://en.wikipedia.org/wiki/Spin_Hall_effect)_

[[!redirects spin Hall effects]]


spin network

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[quantum gravity]]

* [[recoupling theory]]

* [[string diagram]]

* [[spin foam]]

* [[representation theory]]

* [[SU(2)]]

* [[10j symbol]]

* [[3-manifolds]]

## References

* Roger Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, 1971, pp. 151–180 *[Penrose 71](https://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf)*

* R. Penrose, Applications of negative dimensional tensors,in Combinatorial Mathematics and its Applications, ed. D. Welsh, Academic Press, New York, 1971, pp. 221-244.

* R. Penrose, On the nature of quantum geometry, in Magic Without Magic,ed. J. Klauder, Freeman,San Francisco, 1972, pp. 333-354. [Penrose72](https://math.ucr.edu/home/baez/penrose/Penrose-OnTheNatureOfQuantumGeometry.pdf)

* R. Penrose, Combinatorial quantum theory and quantized directions, in Advances in Twistor Theory,eds. L. Hughston and R. Ward, Pitman Advanced Publishing Program, San Francisco, 1979, pp.301-317

* C. Rovelli and L. Smolin, Spin networks in quantum gravity, Phys. Rev.D52(1995), 5743-5759.

* J. Barrett and L. Crane, Relativistic spin networks and quantum gravity, J. Math. Phys.39(1998), 3296–3302. *[gr-qc/9709028](https://arxiv.org/abs/gr-qc/9709028)*

* J.C.Baez,``An Introduction to Spin Foam Models of $BF$ Theory and Quantum Gravity,''Lect. Notes Phys. 543, 25-93 (2000), doi:10.1007/3-540-46552-9, [arXiv:gr-qc/9905087](https://arxiv.org/pdf/gr-qc/9905087.pdf).

* J.C.Baez,``Spin network states in gauge theory,'' Adv. Math. 117, 253-272 (1996), doi:10.1006/aima.1996.0012, [arXiv:gr-qc/9411007 [gr-qc]].

* J.C Baez, ``Spin Networks in Nonperturbative Quantum Gravity'', The Interface of Knots and Physics, ed. Louis Kauffman, A.M.S., Providence, 1996, pp. 167-203 [gr-qc/9504036](https://arxiv.org/abs/gr-qc/9504036)

* E.Bianchi, P.Dona and S.Speziale,``Polyhedra in loop quantum gravity,''Phys. Rev. D 83, 044035 (2011),doi:10.1103/PhysRevD.83.044035,[arXiv:1009.3402 [gr-qc]] (https://arxiv.org/pdf/1009.3402v2.pdf).

See also

* Wikipedia, *[Spin network](https://en.wikipedia.org/wiki/Spin_network)*

[[!redirects spin networks]]