nLab spherical space form

Idea

A spherical space form is a quotient space $S^n/G$ of a round Riemannian n-sphere ($n \geq 2$) by a subgroup $G$ of its isometry group, which acts freely and properly discontinuously.

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

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 and a complete solution was finally compiled by (Wolf 74) (reviewed in Gadhia 07, section 2.2). A re-proof of the classification is claimed in (Allock 15).

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

For $n =7$ all the groups $G$ in Wolf’s classification act as subgroups of SO(8), the latter equipped with its defining action on $\mathbb{R}^8$ 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).

Given any such lift $\widehat{G} \subset Spin(8)$, one may consider its action on the two 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 by the action of $\widehat{G}$. For $\widehat{G}$ non-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 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):

• $N = 8$: 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 finite subgroup of SU(2), hence a binary dihedral group or the binary tetrahedral group or binary octahedral group or binary icosahedral group, 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).)

$N$ Killing spinors on spherical space form $S^7/\widehat{G}$	$\phantom{AA}\widehat{G} =$	spin-lift of subgroup of isometry group of 7-sphere	3d superconformal gauge field theory on back M2-branes with near horizon geometry $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$	$\phantom{AA}\mathbb{Z}_2$	cyclic group of order 2	BLG model
$\phantom{AA}N = 7\phantom{AA}$	—	—	—
$\phantom{AA}N = 6\phantom{AA}$	$\phantom{AA}\mathbb{Z}_{k\gt 2}$	cyclic group	ABJM model
$\phantom{AA}N = 5\phantom{AA}$	$\phantom{AA}2 D_{k+2}$ $2 T$, $2 O$, $2 I$	binary dihedral group, binary tetrahedral group, binary octahedral group, binary icosahedral group	(HLLLP 08a, BHRSS 08)
$\phantom{AA}N = 4\phantom{AA}$	$\phantom{A}2 D_{k+2}$ $2 O$, $2 I$	binary dihedral group, binary octahedral group, binary icosahedral group	(HLLLP 08b, Chen-Wu 10)

• José Figueroa-O'Farrill et al. 2009 (arXiv:0909.0163, pdf slides)

This analysis applies to the classification of the near horizon geometry of smooth (i.e. non-orbifold) $\geq \tfrac{1}{2}$ BPS 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-dimensional anti de Sitter spacetime with a 7-dimensional spherical space form $S^7/G$ with spin structure and $N \geq 4$, as above (MFFGME 09).

Properties

Concordances of higher gerbes over spherical space forms

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

The claim of Theorem below is that this is equivalently the 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 below) the equivariant homotopy groups of the equivariant classifying space for equivariant $PU(\mathcal{H})$-principal bundles (originally due to Uribe et al., 2014) – 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. generalizes this classification of equivariant $PU(\mathcal{H})$-principal bundles to any braided $n$-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. below) of the smooth Oka principle over that spherical space form.

• Preliminaries

• The classification theorem

• Lemmas and proofs

Preliminaries

Concretely, we consider the following situation:

Definition

Assumption.
In the following, let

1. $G$ be a finite group satisfying the following equivalent conditions (equivalent by the Madsen-Thomas-Wall theorem):

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

   2. $G$ has a continuous free action on the topological d-sphere for some $d \in \mathbb{N}$;

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

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

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

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

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

      (2)$$B Shp(\Gamma) \,\in\, Grp\big( Grpd_\infty \big) \,,$$

   2. truncated, hence a homotopy n-type for some $n \in \mathbb{N}$:

   (3)$$\underset{ n \in \mathbb{N} }{\exists} \;\;\;\;\;\; \tau_n Shp(\Gamma) \,\simeq\, Shp(\Gamma) \;\;\; \in \; Grp_\infty \,.$$

Given $n$ as in (3), possibly increased a little if necessary, we have by (1), a smooth free group action on $S^{n+2}$, whose quotient space is a smooth manifold:

(4)$$S^{n+2} \overset{\phantom{-} q \phantom{-}}{\twoheadrightarrow} S^{n+2}/G \;\;\; \in \; SmthMfd \,.$$

Example

The archetypical examples of topological groups which satisfy Assumption 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 n-truncated for $n \geq 1$ and which is not just braided but even abelian (i.e. $E_\infty$); with delooping $K(\mathbb{Z},2)$;

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

Hence for both of these structure groups, the 7-dimensional spherical space forms discussed above satisfy Assumption .

Definition

(topological groupoids and their underlying D-topological stacks)
We will write:

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

for

1. the topological groupoids which are

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

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

2. their associated D-topological stacks, regarded as objects in $SmthGrpd_\infty$.

Notice, as indicated in the last row of (5), that the smooth stack correspinding to the action groupoid of $G$ acting on $S^{n+2}$ is equivalent to the quotient smooth manifold $S^{n+2}/G$ itself, due to the condiiton that the $G$-action is free.

Lemma

(canonical Cech groupoid of $\Gamma$-principal bundles on spherical space form)
Under the truncation condition (3), the groupoid of $\Gamma$-principal bundles over $S^{n+2}/G$ (internal to TopSp) is equivalent to the groupoid of topological functors out of the action groupoid into the delooping groupoid (5), with 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 $k \in \mathbb{N}$:

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

Proof

The $n$-truncation condition (3) 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 theory, that every $\Gamma$-principal bundle on $S^{n+2}$ is isomorphic to the trivial bundle:

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

Therefore, every $\Gamma$-principal bundle on the spherical space form $S^n/G$ trivializes after being pulled back along the coprojection $q$ (4). The corresponding 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 $\Gamma \rightrightarrows \ast$:

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 $(c_1 \rightrightarrows \ast)$.

Definition

($\infty$-groupoids of concordances of principal bundles on spherical space forms)
Consider the following two $\infty$-groupoids:

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

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

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

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

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 snooth shape via the smooth path $\infty$-groupoid (by this Prop.),

2. the canonical Cech groupoid-presentation of groupoids of principal bundles, from Lem. ,

3. the fact that the simplicial homotopy colimit of $\infty$-groupoids presented by simplicial sets is given by the diagonal of the corresponding bisimplicial set (by this Prop.).

The classification theorem

Proposition

Under Assumption , the $\infty$-groupoid of concordances of $\Gamma$-principal bundles on $S^{n+2}/G$ is equivalent to the hom $\infty$-groupoid from $B G$ to $B \Gamma$:

$$\esh \, Maps \big( S^{n+2}/G ,\, \mathbf{B}\Gamma \big) \;\; \simeq \;\; Maps \big( B G ,\, B \Gamma \big) \,.$$

Proof

Since $S^{n+2}/G$ is a smooth manifold, the smooth Oka principle (here) 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 in $Grpd_\infty$:

(9)$$\begin{array}{lll} Maps \big( Shp(S^{n+2}/G) ,\, B \Gamma \big) & \;\simeq\; Maps \big( Shp(S^{n+2} \sslash G) ,\, B \Gamma \big) \\ & \;\simeq\; Maps \big( Shp(S^{n+2}) \sslash G ,\, B \Gamma \big) \\ & \;\simeq\; Maps \big( \underset{\longrightarrow}{\lim} \, Shp(S^{n+2}) \times G^{\times_\bullet} ,\, B \Gamma \big) \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, Maps \big( Shp(S^{n+2}) \times G^{\times_\bullet} ,\, B \Gamma \big) \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, Maps \big( G^{\times_\bullet} ,\, B \Gamma \big) & \text{ by (3) } \\ & \;\simeq\; Maps \big( \underset{\longrightarrow}{\lim} \, G^{\times_\bullet} ,\, B \Gamma \big) \\ & \;\simeq\; Maps \big( B G ,\, B \Gamma \big) \end{array}$$

Moreover:

Definition

There is a canonical comparison morphism between the shapes of mapping stacks in Def. , given by precomposition with the terminal morphism $S^{n+2} \xrightarrow{\; p\;} \ast$:

(10)$$\esh \, Map \Big( \mathbf{B}G ,\, \mathbf{B}\Gamma \Big) \xrightarrow {\;\;\; \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma) \;\;\;} \esh \, Map \Big( S^{n+2}/G ,\, \mathbf{B}\Gamma \Big) \, \,.$$

Theorem

Under Assumption , the comparison morphism (10) is an equivalence of $\infty$-groupoids.

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. below, while for $\pi_1$ this is Prop. below, whose proof has an evident generalization to all $n$.

Corollary

Under Assumption , there is an equivalence of $\infty$-groupoids of the form:

(11)$$\esh \, Map \big( \mathbf{B}G ,\, \mathbf{B}\Gamma \big) \;\; \simeq \;\; Map \big( \esh \, \mathbf{B}G ,\, \esh \, \mathbf{B}\Gamma \big) \;=\; Map \big( B G ,\, B \Gamma \big)$$

Proof

This is the composite of Prop. with Thm. .

Remark

Cor. 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 (11) fail in general, unless some extra condition like Assumption is imposed.

For example, in the case that both $G$ and $\Gamma$ are compact Lie groups, the $\infty$-groupoids on the left of (11) are the hom $\infty$-groupoids of the global orbit category for compact Lie equivariance groups.

Example

(equivariant classifying space of $PU(\mathcal{H})$-bundles)
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. 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, see also Uribe & Lück 2014, Thm. 15.17.

Lemmas and proofs

Example

A 1-morphism in (8) 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:

Lemma

The simplicial set presentations for shapes of mapping stacks in Def. have at least all 2-horn fillers.

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:

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

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

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:

and is filled by the evident directly analogous procedure.

$\,$

Lemma

Under the truncation assumption (3), 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.

Proof

The homotopy class of the classifying map of a principal bundle is represented by the topological realization of any of its Cech cocycles. Therefore, Lemma implies that in the following commuting diagram the left and the composite function are bijections:

It follows by 2-out-of-3 that also the function on the right is a bijection:

(12)$$\tau_0 \, \mathrm{TopFunc} \Big( \big( S^{n+2} \times G \rightrightarrows S^{n+2} \big) ,\, \big( \Gamma \rightrightarrows \ast \big) \Big) \xrightarrow{\;\; \sim \;\;} \tau_0 \, \mathrm{Maps} \Big( \mathrm{Shp} \big( S^{n+2} /\!/ G \big) ,\, B \Gamma \Big)$$

Observe that the composite of the equivalences in (9) 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 functoriality of the nerve-operation followed by topological realization, the following diagram commutes:

Here

• the top and bottom horizontal functions are bijections by (12),

• the right vertical functions are bijections by (9).

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.

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

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

Lemma

Every 1-morphism in $\esh \,Maps\big( S^{n+2}/G ,\, \mathbf{B}\Gamma \big)$ (Exp. ) is equivalent (homotopic relative its endpoints) to one of the form

(13)$$\array{ P \times \Delta^1 &\xrightarrow{\phantom{--}}& P' \\ \big\downarrow {}^{\mathrlap{p \times id_{[0,1]}}} && \big\downarrow {}^{\mathrlap{p'}} \\ S^{n+2}/G \times \Delta^1 &=& S^{n+2}/G \times \Delta^1 }$$

Proof

By the classification theory of principal bundles (or, more concretely, by the proof of this Prop.), 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. we write $\ell$ for such an isomorphism onto its $P_0$-component:

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

(14)$$\ell\vert_0 \;\colon\; P_0 \xrightarrow{\; id \;} P_0 \mathrlap{\,.}$$

From this we may construct the following 2-morphism in (8):

(15)

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

Therefore the 2-face of the above 2-morphism (15) is degenerate (where the last step uses (14)):

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

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 (13):

$$\begin{aligned} \delta_1^\ast \big( s_0^\ast(\phi) \circ s_0^\ast(\ell) \big) & \;=\; \delta_1^\ast \big( s_0^\ast( \phi \circ \ell ) \big) \\ & \;=\; \phi \circ \ell \;\colon\; P_0\vert_0 \xrightarrow{\;} P_1 \mathrlap{\,.} \end{aligned}$$

Hence the 2-morphism (15) exhibits the claimed homotopy relative endpoints.

Lemma

The comparison morphism (Def. ) is injective on connected components.

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 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.

Lemma

The comparison morphism (Def. ) is surjective on connected components.

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 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.

Proposition

The comparison morphism (Def. ) is bijective on connected components:

$$\pi_0 \big( \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma) \big) \;\; \colon \;\; \pi_0 \bigg( \esh \, Map \Big( \mathbf{B}G ,\, \mathbf{B}\Gamma \Big) \bigg) \xrightarrow {\;\;\; \sim \;\;\;} \pi_0 \bigg( \esh \, Map \Big( S^{n+2}/G ,\, \mathbf{B}\Gamma \Big) \bigg)$$

Proof

By Lemma it is injective, and by Lemma it is surjective.

Lemma

The comparison morphism (Def. ) is injective on fundamental groups.

Proof

Since both simplicial sets have all 2-horn fillers, by Lem. , 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. 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.

Lemma

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

$$c(-) \xrightarrow{ \widehat{\gamma}(1)(-) } c(-)^{ \widehat{\gamma}(1)(-) } \,.$$

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 $c(1)^{\gamma} \,=\, c(1)$):

Lemma

The comparison morphism (Def. ) is surjective on fundamental groups.

Proof

Since both simplicial sets have all 2-horn fillers, by Lem. , 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. it is sufficient to assume that the cocycle $c(0)$ is already constant along $S^{n+2}$.

Hence considering this case, Prop. 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 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 (3).

Remark

With (2) in Assumption , Prop. implies that the space of concordances has $\infty$-group structure:

$$\esh \, Maps \big( S^{n+2}/G ,\, \mathbf{B}\Gamma \big) \;\;\; \in \; Grp(Grpd_\infty) \,.$$

Proposition

The comparison morphism (Def. ) is an isomorphism on fundamental groups (for any basepoint):

$$\pi_1 \big( \esh \, Map(p/\!\!/G,\,\mathbf{B}\Gamma) \big) \;\; \colon \;\; \pi_1 \bigg( \esh \, Map \Big( \mathbf{B}G ,\, \mathbf{B}\Gamma \Big) \bigg) \xrightarrow {\;\;\; \sim \;\;\;} \pi_1 \bigg( \esh \, Map \Big( S^{n+2}/G ,\, \mathbf{B}\Gamma \Big) \bigg)$$

Proof

By Lemma this function is injective on fundamental groups.

In order to see surjectivity, we may use that the $\infty$-groupoid of concordances has group structure (Rem. ), 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 gives the surjectivity.

(…)

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:

• Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen, Math. Ann. 39 (1891), 257–278 (doi:10.1007/BF01206655)

Solution of the classification problem:

• Joseph Wolf, Spaces of constant curvature, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 (doi:10.1090/chel/372)

Streamlined re-proof:

• Daniel Allcock, Spherical space forms revisited, Trans. Amer. Math. Soc. 370 (2018), 5561-5582 (arXiv:1509.00906, doi:10.1090/tran/7167)

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

• 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, pdf slides)

• Sunil Gadhia, Supersymmetric quotients of M-theory and supergravity backgrounds, PhD thesis, School of Mathematics, University of Edinburgh, 2007 (spire: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:

• 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 (doi:10.1016/0040-9383(76)90031-8)

• 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)

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, ISBN:9781571463012)

Smooth spherical space forms

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

• Madsen, Thomas and Wall 1983

• Ib Madsen, Smooth spherical space forms, Geometric Applications of Homotopy Theory I. Springer 1978. 303-352 (doi:10.1007/BFb0069224, pdf)