speculation

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## Speculative philosophy

In the 19th century, _speculation_, from Latin _speculari_ = _observing_, referred to a method in [[philosophy]] by which truths were meant to be "observed" in a kind of [[mysticism|mystic]] insight. Specifically in the school of [[idealism]] this _speculation_ is used with positive connotation and is meant to be closer to the truth.

Evidently, later this attitude widely changed.

## Hegel on speculation

Various paragraphs by [[Hegel]] on _speculative philosophy_:

According to the _[[Lectures on the History of Philosophy]]_ ([here ](Lectures+on+the+History+of+Philosophy#DescartesNonSpeculative)) this kind of speculation originates with [[Fichte]].

> [WdL §3](Science+of+Logic#3) Indem so die Wissenschaft und der gemeine Menschenverstand sich in die Hände arbeiteten, den Untergang der Metaphysik zu bewirken, so schien das sonderbare Schauspiel herbeigeführt zu werden, ein gebildetes Volk ohne Metaphysik zu sehen, – wie einen sonst mannigfaltig ausgeschmückten Tempel ohne Allerheiligstes. – Die Theologie, welche in früheren Zeiten die Bewahrerin der spekulativen Mysterien und der obzwar abhängigen Metaphysik war, hatte diese Wissenschaft gegen Gefühle, gegen das Praktisch-Populäre und gelehrte Historische aufgegeben.

> Science and ordinary common sense thus co-operating to bring about the downfall of metaphysics, there was seen the strange spectacle of a cultured nation without metaphysics – like a temple richly ornamented in other respects but without a holy of holies. Theology, which in former times was the guardian of the speculative mysteries and of metaphysics (although this was subordinate to it) had given up this science in exchange for feelings, for what was popularly matter-of-fact, and for historical erudition.

In _[[Encyclopedia of the Philosophical Sciences]]_ §82 it says (not in the first editions, beware) that speculation in this sense is the same that at other times has been called [[mysticism]]:

> [Enc§82d](Science+of+Logic#Enc82d) Hinsichtlich der Bedeutung des Spekulativen ist hier noch zu erwähnen, daß man darunter dasselbe zu verstehen hat, was früher, zumal in Beziehung auf das religiöse Bewußtsein und dessen Inhalt, als das Mystische bezeichnet zu werden pflegte.  Wenn heutzutage vom Mystischen die Rede ist, so gilt dies in der Regel als gleichbedeutend mit dem Geheimnisvollen und Unbegreiflichen, und dies Geheimnisvolle und Unbegreifliche wird dann, je nach Verschiedenheit der sonstigen Bildung und Sinnesweise, von den einen als das Eigentliche und Wahrhafte, von den anderen aber als das dem Aberglauben und der Täuschung Angehörige betrachtet.

> Hierüber ist zunächst zu bemerken, daß das Mystische allerdings ein Geheimnisvolles ist, jedoch nur für den Verstand, und zwar einfach um deswillen, weil die abstrakte Identität das Prinzip des Verstandes, das Mystische aber (als gleichbedeutend mit dem Spekulativen) die konkrete Einheit derjenigen Bestimmungen ist, welche dem Verstand nur in ihrer Trennung und Entgegensetzung für wahr gelten.
Wenn dann diejenigen, welche das Mystische als das Wahrhafte anerkennen, es gleichfalls dabei bewenden lassen, daß dasselbe ein schlechthin Geheimnisvolles sei, so wird damit ihrerseits nur ausgesprochen, daß das Denken für sie gleichfalls nur die Bedeutung des abstrakten Identischsetzens hat und daß man um deswillen, um zur Wahrheit zu gelangen, auf das Denken verzichten oder, wie auch gesagt zu werden pflegt, daß man die Vernunft gefangennehmen müsse.

> [Enc§82e](Science+of+Logic#Enc82e) Nun aber ist, wie wir gesehen haben, das abstrakt verständige Denken so wenig ein Festes und Letztes, daß dasselbe sich vielmehr als das beständige Aufheben seiner selbst und als das Umschlagen in sein Entgegengesetztes erweist, wohingegen das Vernünftige als solches gerade darin besteht, die Entgegengesetzten als ideelle Momente in sich zu enthalten.
Alles Vernünftige ist somit zugleich als mystisch zu bezeichnen, womit jedoch nur so viel gesagt ist, daß dasselbe über den Verstand hinausgeht, und keineswegs, daß dasselbe überhaupt als dem Denken unzugänglich und unbegreiflich zu betrachten sei."

In [Hegel, Lectures on Religion](#HegelLectures):

> ([Hegel, Lectures on Religion](#HegelLectures)) Spekulative Philosophie ist das Bewusstsein der Idee, so daß alles als Idee aufgefasst wird; die Idee aber ist das Wahre in Gedanken, nicht als bloße Anschauung oder Vorstellung. Das Wahre in Gedanken ist näher dieses, daß es konkret sei, in sich entzweit gesetzt, und zwar so, daß die zwei Seiten des Entzweiten entgegengesetzte Denkbestimmungen sind, als deren Einheit die Idee gefasst werden muss. Spekulativ denken heißt, ein Wirkliches auflösen und dieses in sich so entgegensetzen, daß die Unterschiede nach Denkbestimmungen entgegengesetzt sind und der Gegenstand als Einheit beider aufgefasst wird. .

## References

* German Wikipedia, _[Spekulation (Philosophie)](http://de.wikipedia.org/wiki/Spekulation_%28Philosophie%29)_

* [[Georg Hegel]], _[[Encyclopedia of the Philosophical Sciences]]_

* {#HegelLectures} [[Georg Hegel]], _[[Lectures on the Philosophy of Religion]]_

[[!redirects speculations]]

[[!redirects speculative philosophy]]


speed

## Related concepts

* [[speed of light]]

* [[velocity]], [[rapidity]]

* [[position]]

* [[mass]], [[charge]]

## References

* Wikipedia, _[Speed](https://en.wikipedia.org/wiki/Speed)_

[[!redirects speeds]]


speed of light

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## Idea

The [[speed]] of [[light]]; a [[physical unit]].

## Related concepts

* [[Michelson-Morley experiment]]

* [[special relativity]], [[general relativity]]

[[!include fundamental scales -- contents]]

## References

See also

* Wikipedia, _[Speed of light](https://en.wikipedia.org/wiki/Speed_of_light)_

Visualization in relation to scales in the solar system:

* [www.techexplorist.com/3-animations-provided-nasa-scientist-suggests-the-speed-of-light-frustratingly-slow/20215/](https://www.techexplorist.com/3-animations-provided-nasa-scientist-suggests-the-speed-of-light-frustratingly-slow/20215/)


Spencer Bloch

* [webpage](http://www.math.uchicago.edu/~bloch/)

## Selected writings

On the [[Mordell conjecture]]:

* [[Spencer Bloch]], _The Proof of the Mordell Conjecture_, Math. Int. **6** 2 (1984) 41-47 [[pdf](https://link.springer.com/content/pdf/10.1007/BF03024155.pdf), [doi:10.1007/BF03024155](https://doi.org/10.1007/BF03024155)]

On [[secondary characteristic classes]]:

* {#BlochEsnault97} [[Spencer Bloch]], [[Hélène Esnault]], _Algebraic Chern-Simons theory_, American Journal of Mathematics, Volume 119, Number 4, August 1997, pp. 903-952 ([pdf](http://www.mi.fu-berlin.de/users/esnault/preprints/helene/39-algebraic-chern-simons.pdf))

category: people


Spencer Breiner

Former PhD student of [[Steve Awodey]] at CMU. Currently at National Institute of Standards and Technology.

### Related pages

* [[logical scheme]]

###Publications

* PhD thesis, _Scheme representation for first-order logic_, ([arXiv:1402.2600](http://arxiv.org/abs/1402.2600))

category: people


Spencer cohomology

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## Idea

The [[de Rham cohomology]] and [[Dolbeault cohomology]] can be viewed as [[abelian sheaf cohomology]] theories with [[coefficients]] in the sheaf of [[locally constant functions]], respectively [[holomorphic functions]]. Spencer cohomology is a generalization of these two cohomologies for the case of the solution sheaf of an arbitrary [[linear operator|linear]] [[differential operator]].

## References

### General

* [[eom]], _[Spencer cohomology](http://www.encyclopediaofmath.org/index.php/Spencer_cohomology)_

* Wikipedia, _[Spencer cohomology](http://en.wikipedia.org/wiki/Spencer_cohomology)_

Discussion of Spencer cohomology as the home of the higher-order [[torsion invariants of G-structures]] originates in

* {#Guillemin65} [[Victor Guillemin]], _The integrability problem for $G$-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. ([JSTOR](http://www.jstor.org/stable/1994134))

### In classification of SuGra backgrounds

On [[Spencer cohomology]] applied to the classification of [[BPS state|BPS solutions]] of [[D=11 supergravity]]:

* {#FFS17} [[José Figueroa-O’Farrill]], Andrea Santi, _Spencer cohomology and eleven-dimensional supergravity_, Comm. Math. Phys. 349 (2017), 627-660 ([arXiv:1511.08737](https://arxiv.org/abs/1511.08737))


Spencer Ernst Dickson

## Selected writings

Introducing the notion of [[torsion theory]] for [[abelian categories]]:

* {#Dickson63} [[Spencer Ernst Dickson]], _Torsion theories for abelian categories_, Thesis, New Mexico State University (1963) ([ProQuest](https://www.proquest.com/openview/0c7793e99ae9368bd940699a03d70132/1))

* {#Dickson66} [[Spencer Ernst Dickson]], *A Torsion Theory for Abelian Categories*, Transactions of the American Mathematical Society **121** 1 (1966) 223-235 ([doi:10.2307/1994341](https://doi.org/10.2307/1994341), [jstor:1994341](https://www.jstor.org/stable/1994341))

category: people

[[!redirects Spencer E. Dickson]]


SPHERE

SPHERE is a CNRS laboratory that concentrates on **S**cience, **P**hilosophy and **H**istory (which explains the first three letters of its title). It has organised  several interesting conferences on Historical and Philosophical aspects of Category Theory and its higher dimensional cousins.

[Webpage](http://www.sphere.univ-paris-diderot.fr/spip.php?rubrique122&lang=fr)

category: philosophy
[[!redirects Laboratoire SPHERE]]


sphere

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## Definition

### Finite-dimensional spheres

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

The $n$-dimensional unit **sphere** , or simply **$n$-sphere**, is the [[topological space]] given by the [[subset]] of the $(n+1)$-dimensional [[Cartesian space]] $\mathbb{R}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$

$$
  S^n = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = 1 \}
  \,.
$$

The $n$-dimensional sphere of radius $r$ is
$$ S^n_r = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = r \} .$$
Topologically, this is equivalent ([[homeomorphism|homeomorphic]]) to the unit sphere for $r \gt 0$, or a [[point]] for $r = 0$.

This is naturally a [[smooth manifold]] of [[dimension]] $n$, with the [[smooth structure]] induced by the standard smooth structure on $\mathbb{R}^n$.

=--

### Infinite dimensional spheres
 {#InfiniteDimensionalSphere}

One can also talk about the [[infinite-dimensional sphere]] in an arbitrary (possibly infinite-dimensional) [[normed vector space]] $V$:
$$ S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} .$$

If a [[LCTVS|locally convex topological vector space]] admits a continuous linear injection into a [[normed vector space]], this can be used to define its sphere.  If not, one can still define the sphere as a *quotient* of the space of non-zero vectors under the scalar action of $(0,\infty)$.

Homotopy theorists (e.g. [tom Dieck 2008, example 8.3.7](#tomDieck2008)) define $S^\infty$ as the [[directed colimit]] of the $S^n$:
$$ S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty .$$
Note that this $S^\infty$ is _not_ homeomorphic to the sphere of any metrizable space as defined above, since the metrizable [[CW complex|CW-complexes]] are precisely the  locally finite [[CW complex|CW-complexes]] ([Fritsch–Piccinini 1990: 48, prop. 1.5.17](#FP1990)), which [$S^\infty$ is not](#CWstructures) (every open $n$-cell intersects all closed $m$-cells with $m \ge n$)

In themselves, infinite-dimensional spheres provide nothing new to [[homotopy theory]], as they are at least weakly contractible and usually [[contractible space|contractible]].  However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of [[classifying spaces]].

If the vector space is a [[shift space]], then contractibility is straightforward to prove.

+-- {: .num_theorem #theorem}
###### Theorem
Let $V$ be a [[shift space]] of some order.  Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors).  Then $S V$ is contractible.
=--

+-- {: .proof}
###### Proof
Let $T \colon V \to V$ be a [[shift map]].  The idea is to homotop the sphere onto the image of $T$, and then down to a point.

It is simplest to start with the non-zero vectors, $V \setminus \{0\}$.  As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$.  Define a homotopy $H \colon [0,1] \times (V \setminus
 \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$.  It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$.  To see that it is well-defined, we need to show that $H_t(v)$ is never zero.  The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a [[shift map]] then it has none.

As $T$ is a [[shift map]], it is not surjective and so we can pick some $v_0$ not in its image.  Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$.  As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$.  Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$.

If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function.  If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.
=--

## Properties

### Basic

* The $n$-sphere is the [[boundary]] of the $(n+1)$-[[ball]].

* These spheres, or rather their underlying [[topological spaces]] or [[simplicial set|simplicial sets]], are fundamental in (ungeneralised) [[homotopy theory]].  In a sense, [[Whitehead's theorem]] says that these are all that you need; no further [[generalized (Eilenberg-Steenrod) homotopy theory|generalised homotopy theory]] (in a sense [[Eckmann–Hilton duality|dual]] to [[Eilenberg–Steenrod cohomology theory]]) is needed.

* [[positive dimension spheres are H-cogroup objects]], and this is the origin of the [[group]] structure on [[homotopy groups]]).

\begin{proposition}
The [[topological complexity]] of the sphere is
\[
  \label{TopologicalComplexityOfSpheres}
    TC
    \big(
      S^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       2 & ; n \quad odd
       \\
       3 & ; n \quad even
    \end{array}
    \right.
\]
\end{proposition}

([Farber 01, Theorem 8](#Farber01))

This proposition can be generalized:

\begin{proposition}
The [[topological complexity]] of a product of spheres is
\[
  \label{TopologicalComplexityOfProductOfSpheres}
    TC
    \big(
      (S^m)^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       n+1 & ; m \quad odd
       \\
       2n+1 & ; m \quad even
    \end{array}
    \right.
\]
\end{proposition}

([Farber 01, Theorem 13](#Farber01))

A special case of this proposition is $TC(T^n)=n+1$ for the [[topological complexity]] of the [[torus]].

### CW-structures
 {#CWstructures}

The $n$-sphere is an [[dimension of a cell complex|$n$-dimensional]] [[CW complex]] in several ways:

* The $n$-sphere ($n \ge 0$) admits, for every point $x_0 \in S^n $, a [[CW complex|CW-structure]] with one $0$-cell $x_0$ and one $n$-cell $S^n \setminus \{ x_0 \}$, by [[stereographic projection]]. ([tom Dieck 2008, example 8.3.7](#tomDieck2008))

* The $n$-sphere ($n \ge 0$) can also be constructed from the $(n-1)$-sphere by attaching $n$-cells (the north and south hemispheres) to the equator $(n-1)$-sphere. Iteratively applying this construction starting with $S^{-1}=\varnothing$ yields a [[CW complex]] $S^n$ with two $k$-cells in each dimension $k \le n$, and subcomplex inclusions $S^n \subseteq S^{n+1}$ for all $n \in \mathbf{N}$; the colimit of this sequence is (by definition) $S^\infty$. ([tom Dieck 2008, example 8.3.7](#tomDieck2008))

### Coset space structure
 {#LabelCosetSpaceStructure}

#### As quotients of compact Lie groups
 {#AsQuotientsOfCompactLieGroups}

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

For $n \in \mathbb{N}$ the [[n-spheres]] are [[coset spaces]] of [[orthogonal groups]]

$$
  S^n \;\simeq\; O(n+1)/O(n)
  \,.
$$

Similarly for the corresponding [[special orthogonal groups]]

$$
  S^n \;\simeq\; SO(n+1)/SO(n)
$$

and [[spin groups]]

$$
  S^n \;\simeq\; Spin(n+1)/Spin(n)
$$

and [[pin groups]]

$$
  S^n \;\simeq\; Pin(n+1)/Pin(n)
  \,.
$$

=--

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

Fix a [[unit vector]] in $\mathbb{R}^{n+1}$. Then its [[orbit]] under the defining $O(n+1)$-[[action]] on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector [[stabilizer group|stabilizes]] it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.

=--

Similarly, the analogous argument for [[unit spheres]] inside (the [[real vector spaces]] underlying) [[complex vector spaces]], we have

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

For $k \in \mathbb{N}$ the [[n-sphere|(2k+1)-sphere]] $S^{2k+1}$ is the [[coset space]] of [[special unitary groups]]:

$$
  S^{2k+1}
  \;\simeq\;
  SU(k+1)/SU(k)
  \,.
$$

=--

And still similarly, the analogous argument for [[unit spheres]] inside (the [[real vector spaces]] underlying) [[quaternionic vector spaces]], we have

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

For $k \in \mathbb{N}$, $k \geq 1$ the [[n-sphere|(4k-1)-sphere]] $S^{4k-1}$ is the [[coset space]] of [[quaternionic unitary groups]]:

$$
  S^{4k-1}
  \;\simeq\;
  Sp(k)/Sp(k-1)
  \,.
$$

=--

Generally:

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

The [[connected topological space|connected]] [[compact Lie groups]] with [[effective action|effective]] [[transitive actions]]  on [[n-spheres]] are precisely (up to [[isomorphism]]) the following:

* [[SO(n)]]

* [[U(n)]]

* [[SU(n)]]

* [[Sp(n)]]

* [[Sp(n).Sp(1)|Sp(n).SO(2)]]

* [[Sp(n).Sp(1)]]

* [[G2]]

* [[Spin(7)]]

* [[Spin(9)]]

with [[coset spaces]]

$$
  \begin{aligned}
    SO(n)/SO(n-1)
    & \simeq S^{n-1}
    \\
    U(n)/U(n-1) & \simeq S^{2n-1}
    \\
    SU(n)/SU(n-1) & \simeq S^{2n-1}
    \\
    Sp(n)/Sp(n-1) & \simeq S^{4n-1}
    \\
    Sp(n)\cdot SO(2)/Sp(n-1)\cdot SO(2) & \simeq S^{4n-1}
    \\
    Sp(n)\cdot Sp(1)/Sp(n-1)\cdot Sp(1) & \simeq S^{4n-1}
    \\
    G_2/SU(3) & \simeq S^6
    \\
    Spin(7)/G_2 & \simeq S^7
    \\
    Spin(9)/Spin(7) & \simeq S^{15}
  \end{aligned}
$$

=--

This goes back to [Montgomery & Samelson (1943)](#MontgomerySamelson43), see [Gray & Green (1970), p. 1-2](#GrayGreen70), also [Borel & Serre (1953), 17.1](#BorelSerre53).

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

The [[isomorphisms]] in Prop. \ref{nSphereAsCosetSpace} and Prop. \ref{OddDimSphereAsSpecialUnitaryCoset} above hold in the [[category]] of [[topological spaces]] ([[homeomorphisms]]), but in fact also in the [[category]] of [[smooth manifolds]] ([[diffeomorphisms]]) and even in the [[category]] of [[Riemannian manifolds]] ([[isometries]]).

The other [[coset space]] realizations of some [[n-spheres]] in Prop. \ref{TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres} are [[homeomorphisms]], but not necessarily [[isometries]] ("[[squashed spheres]]"). There is also a [[double coset space]] realization which is not even a [[diffeomorphisms]] ("[[exotic sphere]]", the [[Gromoll-Meyer sphere]]).

For more see _[7-sphere -- Coset space realization](7-sphere#CosetSpaceRealization)_.

=--

[[!include coset space structure on n-spheres -- table]]

#### As quotients of Lorentz groups
 {#AsQuotientsOfLorentzGroups}

If one drops the assumption of [[compact Lie group|compactness]], then there are further coset space realizations of $n$-spheres, notably as [[quotient group|quotients]] of [[Lorentz groups]] by [[parabolic subgroups]]: *[[celestial spheres]]*, e.g.: [Toller (2003, p. 18)](#Toller03), [Varlamov (2006, p. 6)](#Varlamov06), [Math.SE:a/4092474](https://math.stackexchange.com/a/4092474/58526).

### Spin structure

\begin{example}
The $n$-sphere, for each $n \in \mathbb{N}$, carries a canonical [[spin structure]], induced from its [[coset space]]-realization $S^n \simeq Spin(n+1)/Spin(n)$
([above](#LabelCosetSpaceStructure)), as a special case of the canonical $H$-structure on $G/H$ ([this example](G-structure#CanonicalHStructureOnFModH)).
\end{example}

Other ways to see this:

* {#Nowaczyk15} Nikolai Nowaczyk, Theorem A.6.6 in: _Dirac Eigenvalues of higher Multiplicity_, Regensburg 2015 ([arXiv:1501.04045](https://arxiv.org/abs/1501.04045))

* {#SpinorsInGeometryAndPhysics} S. Gutt, _Killing spinors on spheres and projective spaces_, p. 238-248 in: A. Trautman, G. Furlan (eds.) _Spinors in Geometry and Physics -- Trieste 11-13 September 1986_, World Scientific 1988
([doi:10.1142/9789814541510](https://doi.org/10.1142/9789814541510),  [GBooks, p. 243](https://books.google.ae/books?id=d14GCwAAQBAJ&lpg=PA243&ots=_tH_He8UFg&dq=%22spin%20structure%20on%20spheres%22&pg=PA243#v=onepage&q=%22spin%20structure%20on%20spheres%22&f=false))

\linebreak

### Parallelizability

* Precisely four spheres are [[parallelizable]], and three of these are so via [[Lie group]] structure (hence are the only spheres with Lie group structure) (see at _[[Hopf invariant one theorem]]_):

  * $S^0$ (the [[group of order two]], the group of units of the [[real numbers]]);

  * $S^1$ (the [[circle group]], the group of unit [[complex numbers]]);

  * $S^3$ (the [[special unitary group]] $SU(2)$, the group of unit [[quaternions]]);

  * $S^7$ (the [[Moufang loop]] of unit [[octonions]])

### Branched covers

Every $n$-[[dimension|dimensional]] [[PL manifold]] admits a [[branched covering]] of the [[n-sphere]] ([Alexander 20](branched+cover#Alexander20)).

By the [[Riemann existence theorem]], every [[connected topological space|connected]] [[compact topological space|compact]] [[Riemann surface]] admits the [[structure]] of a branched cover by a [[holomorphic function]] to the [[Riemann sphere]]. See [there](branched+cover+of+the+Riemann+sphere#RiemannSurfaces) at _[[branched cover of the Riemann sphere]]_.

\begin{center}
<img src="https://ncatlab.org/nlab/files/TorusBranchedCoverOverSphere.jpg" width="500">
\end{center}

> graphics grabbed from [Chamseddine-Connes-Mukhanov 14, Figure 1](branched+cover#ChamseddineConnesMukhanov14), [Connes 17, Figure 11](branched+cover#Connes17)

For [[3-manifolds]] branched covering the [[3-sphere]] see ([Montesinos 74](branched+cover#Montesinos74)).

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ branched covers of the  [[4-sphere]] ([Piergallini 95](#Piergallini95), [Iori-Piergallini 02](branched+cover#IoriPiergallini02)).

But the [[n-torus]] for $n \geq 3$ is _not a [[cyclic group|cyclic]]_ branched over of the [[n-sphere]] ([Hirsch-Neumann 75](branched+cover#HirschNeumann75))

### Iterated loop spaces

+-- {: .num_prop #RationalCohomologyOfIteratedLoopSpaceOf2kSphere}
###### Proposition
**([[rational cohomology]] of [[iterated loop space]] of the [[n-sphere|2k-sphere]])**

Let

$$
  1 \leq D \lt n = 2k \in \mathbb{N}

$$

(hence two [[positive number|positive]] [[natural numbers]], one of them required to be [[even number|even]] and the other required to be smaller than the first) and consider the [[iterated loop space|D-fold loop space]] $\Omega^D S^n$ of the [[n-sphere]].

Its [[rational cohomology|rational]] [[cohomology ring]] is the [[free construction|free]] [[graded-commutative algebra]] over $\mathbb{Q}$ on one [[generators and relations|generator]] $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:

$$
  H^\bullet
  \big(
    \Omega^D S^n
    ,
    \mathbb{Q}
  \big)
  \;\simeq\;
  \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big]
  \,.
$$

=--

([Kallel-Sjerve 99, Prop. 4.10](#KallelSjerve99))

## Examples
 {#Examples}

*  The $(-1)$-sphere is the [[empty space]].

*  The [[0-sphere]] is the [[disjoint union]] of two [[points]] (the classical [[boolean domain]]).

*  The [[1-sphere]] is the [[circle]].

*  The [[2-sphere]] is usual sphere from ordinary geometry. This canonically carries the structure of a [[complex manifold]] which makes it the [[Riemann sphere]].

* The [[3-sphere]] and [[4-sphere]], [[5-sphere]] and [[6-sphere]] and [[7-sphere]] are interesting, too.

## Related concepts

* [[round sphere]]

* [[fuzzy sphere]], [[Podleś sphere]]

* [[hemisphere]], [[equator]]

* [[unit sphere]]

* [[stereographic projection]]

* [[sphere fiber bundle]]

* [[sphere type]]

* [[polar coordinates]]

* [[Reeb sphere theorem]]

* [[homotopy groups of spheres]]

* [[rational n-sphere]]

* [[sphere spectrum]]

* [[spherical fibration]]

* [[geometric quantization of the 2-sphere]]

* [[representation sphere]]

* [[motivic sphere]]

* [[group actions on spheres]]

* [[homology sphere]]

* [[homotopy sphere]]

* [[sphere packing]]

* [[Tate sphere]]

* The [[non-abelian cohomology|non-abelian]] [[generalized cohomology theory]] [[representable functor|represented]] by [[n-spheres]] is [[Cohomotopy cohomology theory]].

## References

* {#FP1990} [[Rudolf Fritsch]], [[Renzo A. Piccinini]], _Cellular structures in topology_, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948))

* {#tomDieck2008} [[Tammo tom Dieck]],  _Algebraic topology_. European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://doi.org/10.4171/048))

### Formalization

Axiomatization of the [[homotopy type]] of the 1-sphere (the [[circle]]) and the 2-sphere, as [[higher inductive types]], is in

* [[Univalent Foundations Project]], section 6.4 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

Visualization of the idea of the construction for the 2-sphere is in

* [[Andrej Bauer]], _HoTT $S^2$_ ([video](https://vimeo.com/119606901))

### Group actions on spheres

Discussion of [[free actions|free]] [[group actions on spheres]] by [[finite groups]] includes

* {#Wall78} [[C. T. C. Wall]], _Free actions of finite groups on spheres_, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/wall7.pdf))

* [[Alejandro Adem]], _Constructing and deconstructing group actions_ ([arXiv:0212280](http://arxiv.org/abs/math/0212280))

The subgroups of [[special orthogonal group|SO(8)]] which act freely on $S^7$ have been classified in

* J. A. Wolf, _Spaces of constant curvature_, Publish or Perish, Boston, Third ed., 1974

and lifted to actions of [[Spin group|Spin(8)]] in

* {#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/))

Discussion of [[transitive actions]] on $n$-spheres by [[compact Lie groups]]:

* {#MontgomerySamelson43} [[Deane Montgomery]], [[Hans Samelson]], _Transformation Groups of Spheres_, Annals of Mathematics Second Series **44** 3 (1943) 454-470 [[jstor:1968975](https://www.jstor.org/stable/1968975)]

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

Further discussion of these actions is in

* {#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))

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

where they are related to the [[black brane|black]] [[M2-brane]] [[BPS state|BPS]]-solutions of [[11-dimensional supergravity]] at [[ADE-singularities]].

See also the [[ADE classification]] of such actions on the [[7-sphere]] (as discussed there)

Discussion of actions of [[Lorentz groups]] on [[celestial spheres]]:

* {#Toller03} [[Marco Toller]], *Homogeneous Spaces of the Lorentz Group* [[arXiv:math-ph/0301014](https://arxiv.org/abs/math-ph/0301014)]

* {#Varlamov06} V. V. Varlamov, *Relativistic Spherical Functions on the Lorentz Group*, J. Phys. A: Math. Gen. **39** (2006) 805-822 [[doi:10.1088/0305-4470/39/4/006](https://doi.org/10.1088/0305-4470/39/4/006)]

### Geometric structures on spheres

Coset space structures on spheres:

* {#BorelSerre53} [[Armand Borel]], [[Jean-Pierre Serre]], _Groupes de Lie et Puissances Reduites de Steenrod_, American Journal of Mathematics, Vol. 75, No. 3 (Jul., 1953), pp. 409-448 ([jstor:2372495](https://www.jstor.org/stable/2372495))

The following to be handled with care:

* [[Michael Atiyah]], _The non-existent complex 6-sphere_, [arxiv/1610.09366](https://arxiv.org/abs/1610.09366)

### Embeddings of spheres

The ([[isotopy]] [[equivalence class|class]] of an) [[embedding of differentiable manifolds|embedding]] of a [[circle]] (1-sphere) into the [[3-sphere]] is a _[[knot]]_. Discussion of embeddings of spheres of more general dimensions into each other:

* {#Haefliger66} [[André Haefliger]], _Differentiable Embeddings of $S^n$ in $S^{n+q}$ for $q \gt 2$_, Annals of Mathematics Second Series, Vol. 83, No. 3 (May, 1966), pp. 402-436 ([jstor:1970475](https://www.jstor.org/stable/1970475))

### Iterated loop spaces

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structure of Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

### Topological complexity

On [[topological complexity]] of spheres and [[product topological space|products]] of spheres (including [[tori]] as special case):

* {#Farber01} [[Michael Farber]], _Topological complexity of motion planning_ (2001), [arXiv:math/0111197](https://arxiv.org/abs/math/0111197);

[[!redirects n-sphere]]
[[!redirects n-spheres]]

[[!redirects d-sphere]]
[[!redirects d-spheres]]

[[!redirects spheres]]

[[!redirects infinite sphere]]

[[!redirects 1-sphere]]
[[!redirects 1-spheres]]

[[!redirects 2-sphere]]
[[!redirects 2-spheres]]


sphere eversion

## References

* Wikipedia, _[Smale's paradox](http://en.wikipedia.org/wiki/Smale's_paradox)_

[[!redirects Smale's paradox]]


sphere fiber bundle

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

#Contents#
* table of contents
{:toc}

## Idea

A _sphere fiber bundle_ is a [[fiber bundle]] whose [[fibers]] are [[spheres]] $S^n$ of some [[dimension]] $n$.

Often, but not always, this is considered in [[homotopy theory]] or even in [[stable homotopy theory]], hence for [[fibers]] which have the ([[stable homotopy type|stable]]) [[homotopy type]] of a sphere, in which case one speaks of _[[spherical fibrations]]_. See there for more.

## Properties
 {#Properties}

### Vertical tangent bundles of sphere bundles
 {#VerticalTangentBundlesOfSphereBundles}

The following appears in [[schreiber:Twisted Cohomotopy implies M5-brane anomaly cancellation|FSS 20, Sec. 3]] (somewhat implicit in v1, explicitly in v2):

\begin{prop}
  Let
  \begin{tikzcd}[column sep=tiny]
    S^n \!/\!\!/ O(n+1)
    \ar[d]
    \ar[
      r,
      -,
      shift left=1pt
    ]
    \ar[
      r,
      -,
      shift right=1pt
    ]
    &
    B O(n)
    \ar[
      d,
      "B i"
    ]
    \\
    B O(n+1)
    \ar[
      r,
      -,
      shift left=1pt
    ]
    \ar[
      r,
      -,
      shift right=1pt
    ]
    &
    B O(n+1)
  \end{tikzcd}
denote the universal $n$-[[spherical fibration]] over the [[classifying space]] of the [[orthogonal group]], where

  \begin{tikzcd}[row sep=0pt]
    O(n)
    \ar[
      r,
      hook
    ]
    &
    O(n+1)
    \\
    A
    \ar[
      r,
      phantom,
      "\mapsto"{
        description
      }
    ]
    &
    \mathrm{diag}(1,A)
  \end{tikzcd}

is the canonical inclusion.

Then its [[homotopy fiber]] inclusion is the classifying map $\vdash Fr(S^n)$ of the orthonormal [[frame bundle]] of the [[n-sphere]]:

\begin{tikzcd}
  S^n
  \ar[
    rr,
    "
     \vdash \mathrm{Fr}(S^n)
    "
  ]
  \ar[d]
  \ar[
    drr,
    phantom,
    "\mbox{\tiny\textrm(pb)}"{
      description
    }
  ]
  &&
  B O(n)
  \ar[d]
  \\
  \ast
  \ar[
    rr
  ]
  &&
  B O(n+1)
\end{tikzcd}

\end{prop}
\begin{proof}
  By the [[pasting law]] we find that the [[homotopy fiber]] of the homotopy fiber inclusion, and hence (by the discssion at [[principal infinity-bundle]]) the total space of the bundle it classifies, is $\Omega B O(n+1) \simeq O(n+1)$:

\begin{tikzcd}
  O(n)
  \ar[
    r,
    hook,
    "i"
  ]
  \ar[d]
  \ar[
    dr,
    phantom,
    "\mbox{\tiny\textrm(pb)}"{
      description
    }
  ]
  &
  O(n+1)
  \ar[
    r
  ]
  \ar[
    d
  ]
  \ar[
    dr,
    phantom,
    "\mbox{\tiny\textrm(pb)}"{
      description
    }
  ]
  &
  \ast
  \ar[
    d
  ]
  \\
  \ast
  \ar[r]
  &
  S^n
  \ar[
    r,
    "
     \vdash \mathrm{Fr}(S^n)
    "{
      description
    }
  ]
  \ar[d]
  \ar[
    dr,
    phantom,
    "\mbox{\tiny\textrm(pb)}"{
      description
    }
  ]
  &
  B O(n)
  \ar[
    d,
    "B i"
  ]
  \\
  &
  \ast
  \ar[
    r
  ]
  &
  B O(n+1)
\end{tikzcd}

Moreover, we have an evident [[isomorphism]]

\begin{tikzcd}
  O(n+1)
  \ar[
    out=180-66,
    in=66,
    looseness=3.5,
    "
    \scalebox{.77}{$
      \phantom{\cdot}
      \mathclap{
        O(n)
      }
      \phantom{\cdot}
    $}
    "{
      description
    },
    shift right=1
  ]
  \ar[
    rr,
    "\sim"
  ]
  &&
  \mathrm{Fr}(S^n)
  \ar[
    out=180-66,
    in=66,
    looseness=3.5,
    "
    \scalebox{.77}{$
      \phantom{\cdot}
      \mathclap{
        O(n)
      }
      \phantom{\cdot}
    $}
    "{
      description
    },
    shift right=1
  ]
\end{tikzcd}

given by acting with $O(n+1)$ on the canonical [[orthonormal basis]] $(v_0, v_1, \cdots, v_n)$ of $\mathbb{R}^{n+1}$, regarded as a point $v_0$ on $S^n = S(\mathbb{R}^{n+1})$ equipped with a frame $(v_1, \cdots, v_n)$ of its [[tangent space]] $T_{v_0} S(\mathbb{R}^{n+1})$.

This isomorphism is manifestly $O(n)$-equivariant, and its quotient on both sides is manifestly $S^n$, so that this is actually an isomorphism of $O(n)$-principal bundles.
\end{proof}

In parametrized generalization of this situation, it follows that:

\begin{corollary}\label{OnceStabilizedVerticalTangentBundlesOfSphereBundlesArePulledBackFromBase}
**(once-[[stable tangent bundle|stabilized]] [[vertical tangent bundles]] to [[sphere-fiber bundles]] are [[pullback bundle|pulled back]] from base)**
  Let
  $
    S(p) \colon S(\mathcal{V}) \to X
  $
  be an $n$-[[spherical fibration]], associated to an [[orientation|oriented]] [[real vector bundle]] $p \colon \mathcal{V} \to X$,
hence fitting into a [[homotopy pullback]]-diagram as shown here:
  \begin{tikzcd}[column sep=small]
    S(\mathcal{V})
    \ar[
      rrr,
      bend left=15pt,
      "\vdash T_{S(p)} S(\mathcal{V})"
    ]
    \ar[
      rr
    ]
    \ar[
      d,
      "
        S(p)
      "{
        left
      }
    ]
    \ar[
      drr,
      phantom,
      "\mbox{\tiny\textrm(pb)}"
    ]
    &
    {\phantom{AA}}
    &
    S^n /\!\!/ O(n+1)
    \ar[
      r,
      -,
      shift left=1pt
    ]
    \ar[
      r,
      -,
      shift right=1pt
    ]
    \ar[d]
    &
    B O(n)
    \ar[
      dl,
      "
        \;\;
        B i
      "{
        right
      }
    ]
    \\
    X
    \ar[
      rr,
      "
        \vdash \mathcal{V}
      "{
        above
      }
    ]
    &&
    B O(n+1)
  \end{tikzcd}
Then:

1. the top map shown classifies the [[vertical tangent bundle]] $T_{S(p)} S(\mathcal{V})$;

1. hence the homotopy-commutativity of the diagram says that the  once-[[stable tangent bundle|stabilized]] vertical tangent bundle is the [[pullback bundle|pullback]] of the original bundle on the base:

   $$
     T_{S(p)}
     S(\mathcal{V})
     \times \mathbb{R}
     \;\simeq_{{}_{X}}\;
     S(p)^\ast
     \big(
       \mathcal{V}
     \big)
     \,.
   $$

\end{corollary}

### Tangent bundles of sphere bundles

The following generalizes Cor. \ref{OnceStabilizedVerticalTangentBundlesOfSphereBundlesArePulledBackFromBase} to the full [[tangent bundle]] of sphere-fiber bundles, now assuming that the base is a [[smooth manifold]] and giving a more traditional [[differential geometry|differential-geometric]] [[proof]] (the following statement appears, without proof, as [Crowley-Escher 03, Fact. 3.1](#CrowleyEscher03), apparently reading between the lines in [Milnor 56, p. 403](#Milnor1956)):

\begin{prop}\label{StableTangentBundleOfUnitSphereBundle}
**([[stable tangent bundle]] of [[unit sphere bundle]])** \linebreak
  The once-[[stable tangent bundle|stabilized]] [[tangent bundle]] of a [[unit sphere bundle]] $S(\mathcal{V})$ in a [[real vector bundle]] $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a [[smooth manifold]] $M$ is [[isomorphism|isomorphic]] to the [[pullback bundle|pullback]]  of the [[direct sum of vector bundles|direct sum]] of the [[stable tangent bundle]] of the base manifold with that vector bundle:

$$
  T S(\mathcal{V}) \times \mathbb{R}
  \;
  \simeq
  \;
  S(p)^\ast
  \big(
    T M
    \oplus_M
    \mathcal{V}
  \big)
  \,.
$$
\end{prop}

\begin{proof}\label{ProofOfStableTangentBundleOfUnitSphereBundle}
  Consider first the actual [[tangent bundle]] but to the [[open ball]]/[[disk]]-[[fiber bundle]] $D(\mathcal{V})$ that fills the given sphere-fiber bundle: By the standard splitting ([this Prop.](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)) this is the [[direct sum of vector bundles|direct sum]]

$$
  T
  \big(
   D(\mathcal{V})
  \big)
  \;\simeq\;
  \big(
     D(p)^\ast T M
  \big)
  \oplus_M
  T_p D(\mathcal{V})
  \,,
$$

where $T_p D(\mathcal{V})$ is the [[vertical tangent bundle]] of the disk bundle. But, by definition of disk bundles, this is the restriction of the vertical tangent bundle of the vector bundle $\mathcal{V}$ itself, and that is just the pullback of that vector bundle along itself (by [this Example](vertical+vector+field#SplittingOfTotalSpaceTangentBundle)):

$$
  \begin{aligned}
  \cdots
  &
  \simeq\;
  \big(
    D(p)^\ast T M
  \big)
  \oplus_M
  \big(
    D(p)^\ast \mathcal{V}
  \big)
  \\
  & \simeq\;
  D(p)^\ast
  \big(
    T M \oplus_M \mathcal{V}
  \big)
  \,.
  \end{aligned}
$$

To conclude, it just remains to observe that the [[normal bundle]] of the [[n-sphere]]-boundary inside the $(n+1)$-ball is manifestly [[trivial bundle|trivial]], so that the restriction of the tangent bundle of $D(\mathcal{V})$ to $S(\mathcal{V})$ is the [[stable tangent bundle]] of $S(\mathcal{V})$.
\end{proof}

\begin{remark}
  Prop. \ref{StableTangentBundleOfUnitSphereBundle} implies that every *stable* [[characteristic class]] of the [[tangent bundle]] of an orthogonal [[sphere-fiber bundle]] -- i.e all [[polynomials]] in its [[Pontryagin classes]] -- are *[[basic differential form|basic]]*, i.e. [[pullback in cohomology|pulled back]] from the base space.
\end{remark}

## Examples

### Unit sphere bundles

\begin{example}\label{UnitSphereBundles}
A key example of sphere fiber bundles are the _unit sphere bundles_ inside of [[real vector bundles]] that are equipped with [[orthogonal structure]]: the bundles whose [[fibers]] are the [[unit spheres]] in the corresponding fiber of the given [[real vector bundle]].
\end{example}

These appear in the discussion of [[Thom spaces]] and hence of [[Thom spectra]], as well as in the discussion of [[wave front sets]].

## Related concepts

* [[Sullivan model of a spherical fibration]]

* [[twisted Cohomotopy]]

## References

Early discussion of sphere bundles (and introducing the general notion of [[fiber bundles]]):

* [[Hassler Whitney]], *On the theory of sphere-bundles*, Proceedings of the National Academy of Sciences **26** 2 (1940) 148-153 [[doi:10.1073/pnas.26.2.148](https://doi.org/10.1073/pnas.26.2.148)]

With focus on [[3-sphere]]-fiber bundles over the [[4-sphere]] and the construction of [[exotic 7-spheres]]:

* {#Milnor1956} [[John Milnor]], _On manifolds homeomorphic to the 7-sphere_, Annals of Mathematics 64 (2): 399–405 (1956) ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/exotic.pdf), [doi:10.1142/9789812836878_0001](https://doi.org/10.1142/9789812836878_0001))

* {#CrowleyEscher03} [[Diarmuid Crowley]], [[Christine Escher]], _A classification of $S^3$-bundles over $S^4$_, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

On sphere fiber bundles as [[Kaluza-Klein compactifications]] in [[supergravity]] and [[string theory]]:

* [[Federico Bonetti]], [[Ruben Minasian]], Valentí Vall Camell, Peter Weck, *Consistent truncations from the geometry of sphere bundles* [[arXiv:2212.08068](https://arxiv.org/abs/2212.08068)]

On sphere bundles with [[formal dg-algebra|formal]] [[Sullivan models]]:

* Jiawei Zhou, *Formality of Sphere Bundles* [[arXiv:2304.09594](https://arxiv.org/abs/2304.09594)]

[[!redirects sphere fiber bundles]]

[[!redirects sphere bundle]]
[[!redirects sphere bundles]]

[[!redirects unit sphere fiber bundle]]
[[!redirects unit sphere fiber bundles]]

[[!redirects sphere-fiber bundle]]
[[!redirects sphere-fiber bundles]]

[[!redirects unit sphere bundle]]
[[!redirects unit sphere bundles]]


sphere packing

+-- {: .rightHandSide}
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#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _sphere packing_ in a [[Euclidean space]] of [[dimension]] $n+1$ is an [[embedding of differentiable manifolds|embedding]] of congruent [[n-spheres]] of some fixed [[radius]], that do not intersect but may touch.

## Sphere packing problem

The _sphere packing problem_ is the problem of finding the densest possible sphere packings in any dimension.

### Low dimensional cases

For $n+1 = 1$ the problem is trivial.

For $n+1 = 2$ it is relatively easy to show that the densest packing is given by spheres centered on the [[vertices]] of the [[honeycomb lattice]].

For $n+1 = 3$ it used to [[Kepler's conjecture]] that the densest packing is that given by spheres centered on the cubical lattice. This was proven only with ample computer assistance by the [[Flyspeck project]].

### Exceptional cases

For $n+1 = 8$ the denest packing is that of spheres centered on the [[E8-lattice]] ([Viazovska 16](#Viazovska16)).

For $n+1 = 24$ the densists packing is that of spheres centered on the [[Leech lattice]] ([CKMRV 17](#CKMRV17)).

An interpretation of the ingredients entering these proofs in terms of bounds on scaling dimensions of fields in [[conformal field theories]], and interpretation of the result, via [[AdS/CFT]] as related to the [[weak gravity conjecture]] in AdS-[[quantum gravity]], is due to ([HMR 19](#HMR19)).

### Irregular cases

For other values of $n$ exact solutions remain unknown, but numerlical simulation shows that the densest sphere packings in the remaining dimensions will not be controlled by [[lattice in a vector space|lattices]], but be highly irregular.

## References

### General

Textbook accounts:

* {#Engel86} [[Peter Engel]], Chapter 10 of: *Geometric Crystallography -- An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$[doi:10.1007/978-94-009-4760-3](https://doi.org/10.1007/978-94-009-4760-3)$]$

  > (which is otherwise concerned with [[crystallography]] via [[crystallographic groups]])

See also:

* Wikipedia, _[Sphere packing](https://en.wikipedia.org/wiki/Sphere_packing)_

Solution to the sphere packing problem in dimensions  8 and 24:

* {#Viazovska16} [[Maryna Viazovska]], _The sphere packing problem in dimension 8_, Annals of Mathematics, **185** 3 (2017) 991-1015 [[arXiv:1603.04246](https://arxiv.org/abs/1603.04246), [doi:10.4007/annals.2017.185.3.7](https://doi.org/10.4007/annals.2017.185.3.7)]

* {#CKMRV17} Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, [[Maryna Viazovska]], _The sphere packing problem in dimension 24_, Annals of Mathematics **185** (2017) 1017-1033 [[arXiv:1603.06518](https://arxiv.org/abs/1603.06518), [doi:10.4007/annals.2017.185.3.8](https://doi.org/10.4007/annals.2017.185.3.8)]

Exposition:

* Ursula Whitcher, *Eight-dimensional spheres and the exceptional $E_8$*, AMS MathVoices Feature Column (Sep. 2022) [[web](https://mathvoices.ams.org/featurecolumn/2022/09/01/eight-dimensional-spheres-and-the-exceptional-e_8/)]

### In string theory
  {#ReferencesInStringTheory}

An interpretation of the ingredients entering these proofs in terms of bounds on scaling dimensions of fields in [[conformal field theories]], and interpretation of the result, via [[AdS/CFT]] as related to the [[weak gravity conjecture]] in AdS-[[quantum gravity]], is due to

* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))

[[!redirects sphere packings]]

[[!redirects sphere packing problem]]
[[!redirects sphere packing problems]]

[[!redirects packing]]
[[!redirects packings]]


sphere spectrum

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

#Contents#
* table of contents
{:toc}

## Definition

The **sphere spectrum** is the [[suspension spectrum]] of the point.

As a [[symmetric spectrum]], see [Schwede 12, example I.2.1](#Schwede12)

## Properties

### Homotopy type

The [[homotopy groups]] of the sphere spectrum are the [[stable homotopy groups of spheres]].

### Truncations

[[homotopy colimits]] of [[simplicial diagrams]] of $k$-[[truncated object in an (infinity,1)-category|truncated]] [[connective spectra]] for $k \in \mathbb{N}$  are [[module spectrum|modules]] over the $k$-truncation $\tau_{\leq k}\mathbb{S}$ of the sphere spectrum.

### As an $E_\infty$-ring

The sphere spectrum is naturally an [[E-∞ ring]] and in fact is the [[initial object]] in the [[(∞,1)-category]] of [[ring spectra]]. It is the higher version of the [[ring]] $\mathbb{Z}$ of [[integer]]s.

## Related concepts

* [[sphere]]

* [[spherical fibration]]

* [[equivariant sphere spectrum]]

* [[2-periodic sphere spectrum]]

* [[Cohomotopy]]

  * [[stable Cohomotopy]]

  * [[twisted Cohomotopy]]

  * [[equivariant Cohomotopy]]

  * [[differential Cohomotopy]]

* [[superalgebra]]

* [[The Music of the Spheres]]

## References

Lecture notes include

* [[John Rognes]], _The sphere spectrum_ (4 pages) (2004) ([[RognesSphereSpectrum.pdf:file]])

* {#Schwede12} [[Stefan Schwede]], Example I.2.1 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

The [[Postnikov tower]] of (localizations of) the sphere spectrum is discussed in

* [[Karol Szumilo]],  _Postnikov tower of the sphere spectrum_, Master thesis 2009 ([web](https://apd.uw.edu.pl/en/diplomas/bsc-msc/62580/))

* Katja Hutschenreuter, _On rigidity of the ring spectra $P_m \mathbb{S}_{(p)}$ and $k o$_, (2012) ([pdf](http://www.math.uni-bonn.de/people/grk1150/DISS/dissertation-hutschenreuter.pdf))

Specifically the 1-truncation of the sphere spectrum (the [[free construction|free]] [[abelian 2-group]] on a single element) is discussed in

* Niles Johnson, Angélica M. Osorno, _Modeling Stable One-Types_, Theory and Applications of Categories, Vol. 26, 2012, No. 20, pp  ([TAC](http://www.tac.mta.ca/tac/volumes/26/20/26-20abs.html), [arXiv:1201.2686](http://arxiv.org/abs/1201.2686))

The 2-truncation appears for instance in section 3 of

* Norio Iwase. _L-S categories of simply-connected compact simple Lie groups of low rank_ ([arXiv:math/0202122](http://arxiv.org/abs/math/0202122))

[[!redirects S]]


sphere type

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

#Contents#
* table of contents
{:toc}

## Idea

_Sphere types_ are the [[axiom|axiomatization]] of the [[homotopy type]] of the ([[shape modality|shape]] of) the [[spheres]] in the context of [[homotopy type theory]].

## Definition
 {#Definition}

For $n = 0$, the *[[0-sphere]]*-type is defined to the [[boolean domain]] $\mathbf{2}$ (or "[[Bit]]", see [there](boolean+domain#InHomotopyTypeTheory)):

\[
  \label{ZeroSphereType}
  S^0
   \,\coloneqq\,
  \mathbf{2}
  \,.
\]

By [[induction]] on the [[natural number]] $n$: For $n \geq 1$, the *$n$-sphere type* is be defined as

* the (un-reduced) [[suspension type]] $\mathrm{S}(-)$ of the $(n-1)$-sphere type $S^{n-1}$;

or equivalently

* the $n$-fold iterated [[suspension type]] $\mathrm{S}^n(-)$ of $S^0$ (eq:ZeroSphereType):

$$
  S^n
   \;\coloneqq\;
  \mathrm{S} S^{n-1}
    \;=\;
  \mathrm{S}^n S^0
  \,.
$$

## See also

* [[circle type]]

* [[synthetic homotopy theory]]

* [[homotopy groups of spheres]]

## References

* [[Univalent Foundations Project]], §6.4 & §6.5 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects sphere types]]


spheres -- contents

**[[n-sphere]]**

* [[unit sphere]], [[polar coordinates]]

* [[round sphere]], [[squashed sphere]]

* [[hemisphere]], [[equator]]

* [[stereographic projection]]

* [[spherical fibration]], [[twisted Cohomotopy]]

* [[homotopy groups of spheres]]

* [[homotopy sphere]], [[Cohomotopy]]

* [[homology sphere]]

* [[rational n-sphere]]

* [[motivic sphere]]

* [[sphere spectrum]], [[stable Cohomotopy theory]]

* [[group actions on spheres]]

* [[representation sphere]], [[equivariant Cohomotopy]]

* [[Reeb sphere theorem]]

* [[sphere packing]]

* [[Music of the Spheres]]

**[[low dimensional topology|low dimensional]] [[n-spheres]]**

* [[circle]]

  * [[real projective space]] $\,\mathbb{R}P^1$

* [[2-sphere]]

  * [[complex projective line]] $\,\mathbb{C}P^1$: [[Riemann sphere]]

  * [[geometric quantization of the 2-sphere]]

  * [[fuzzy 2-sphere]]

* [[3-sphere]]

  * [[fuzzy 3-sphere]]

* [[4-sphere]]

  * [[quaternionic projective line]] $\,\mathbb{H}P^1$

  * [[fuzzy 4-sphere]]

* [[5-sphere]]

* [[6-sphere]]

* [[7-sphere]]

  * [[exotic 7-sphere]]

* [[8-sphere]]

  * [[octonionic projective line]] $\,\mathbb{O}P^1$

* [[13-sphere]]

* [[15-sphere]]

* [[infinite-dimensional sphere]]


spherical category

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

# Spherical categories
* table of contents
{:toc}

## Idea

A _spherical category_ is a  [[monoidal category]] with [[dualizable object|duals]] that behaves as if its [[morphisms]] can be drawn and moved around on a [[2-sphere]].

## Definition

A **spherical category** is a [[pivotal category]] $\mathcal{C}$ where the left and right [[trace]] operations coincide on all objects.

If $\mathcal{C}$ is in addition a [[tensor category]] so that this trace may be interpreted as an element of the [[ground field]], then the trace $tr\big( id_X\big)$ is called the *[[quantum dimension]]* of the object $X$ (e.g. [FRS02 (2.17)](#FRS02))

## Related concepts

* [[Turaev-Viro model]]

* [[fusion category]]

* [[pivotal category]]

* **spherical category**

## References

The definition is originally due to:

* [[John Barrett]], [[Bruce Westbury]], _Spherical Categories_ ([arXiv:hep-th/9310164](http://arxiv.org/abs/hep-th/9310164))

Review:

* [[Michael Müger]], section 2.3 of: _From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories_ ([arXiv:0111204](http://arxiv.org/abs/math/0111204))

* {#FRS02} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], Section 2.1 of: *TFT construction of RCFT correlators I: Partition functions*, Nucl. Phys. B **646** (2002) 353-497 $[$[arXiv:hep-th/0204148](https://arxiv.org/abs/hep-th/0204148)$]$

More is in:

* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _On braided fusion categories I_ ([arXiv:0906.0620](http://arxiv.org/abs/0906.0620))

[[!redirects spherical category]]
[[!redirects spherical categories]]

[[!redirects quantum dimension]]
[[!redirects quantum dimensions]]


spherical fibration

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

#Contents#
* table of contents
{:toc}

## Idea

A _spherical fibration_ is a [[fiber bundle]] of [[spheres]] of some [[dimension]] (a [[sphere fiber bundle]]). Typically this is considered in [[homotopy theory]] where one considers [[fibrations]] whose [[fibers]] have the [[homotopy type]] of [[spheres]]; and this in turn is often considered in [[stable homotopy theory]] after [[stabilization]] (hence up to tensoring with trivial spherical fibrations)  which makes spherical fibrations models for [[(∞,1)-module bundles]] for the [[sphere spectrum]] regarded as an [[E-∞ ring]].

Every [[real vector bundle]] becomes a spherical fibration in the sense of [[homotopy theory]] upon removing its [[zero section]] and this construction induces a map from vector bundles and in fact from [[topological K-theory]] to spherical fibrations, called the _[[J-homomorphism]]_.

This is closely related to the [[Thom space]]/[[Thom spectrum]] construction for vector bundles.

## Definition

### In components

For $X$ (the [[homotopy type]] of) a [[topological space]], a _spherical fibration over it_ is a [[fibration]] $E \to X$ such that each [[fiber]] has the [[homotopy type]] of a [[sphere]].

Given two spherical fibrations $E_1, E_2 \to X$, there is their fiberwise [[smash product]] $E_1 \wedge_X E_2 \to X$.

For $n \in \mathbb{N}$, write $\epsilon^n \colon X \times S^n \to X$ for the trivial sphere bundle of fiber dimension $n$. Two spherical fibrations $E_1, E_2 \to X$ are _stably fiberwise equivalent_ if there exists $n_1, n_2 \in \mathbb{N}$ such that there is a map

$$
  E_1 \wedge_X \epsilon^{n_1}
  \longrightarrow
  E_2 \wedge_X \epsilon^{n_2}
$$

over $X$ which is fiberwise a [[weak homotopy equivalence]].

One consider the [[abelian group]]

$$
  Sph(X) \in Ab
$$

to be the [[Grothendieck group]] of stable fiberwise equivalence classes of spherical fibrations, under fiberwise smash product.

### Classifying space

There is an associative [[H-space]], $G_n$, of homotopy equivalences of the $(n-1)$-sphere with composition. Then $B G_n$ acts as the [[classifying space]] for spherical fibrations with spherical fibre $S^{n-1}$ ([Stasheff 63](#Stasheff63)).

There is an inclusion of the [[orthogonal group]] $O(n)$ into $G_n$.

Suspension gives a map $G_n \to G_{n+1}$ whose limit is denoted $G$. Then $B G$ classifies stable spherical fibrations.

### As $(\infty,1)$-module bundles

(...)

## Properties

### Adams conjecture

The [[Adams conjecture]] (a theorem) characterizes certain spherical fibrations in the image of the [[J-homomorphism]] as trivial.

### Gysin sequence

The [[long exact sequence in cohomology]] induced by a spherical fibration is called a _[[Gysin sequence]]_.

### Rational homotopy type

See _[[Sullivan model of a spherical fibration]]_.

## Related concepts

* [[sphere spectrum]]

* [[Thom spectrum]]

* [[twisted cohomotopy]]

## References

### General

An original reference is

* [[Albrecht Dold]], [[Richard Lashof]], _Principal quasifibrations and fibre homotopy equivalence of bundles_, 1958 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/doldlashof.pdf))
 {#DoldLashof58}

Treatment of the classifying space for spherical fibrations is in

* {#Stasheff63} [[James Stasheff]], _A classification theorem for fibre spaces_, Topology Volume 2, Issue 3, October 1963, Pages 239-246.

Reviews include

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

* Howard Marcum, Duane Randall, _The homotopy Thom class of a spherical fibration_, Proceedings of the AMS, volume 80, number 2 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/marcum1.pdf))

* Per Holm, Jon Reed, section 7 of _Structure theory of manifolds_, Seminar notes 1971[pdf](http://www.mn.uio.no/math/tjenester/kunnskap/kompendier/seminar-holm.pdf)

* Oliver Straser, Nena Röttgens, _Spivak normal fibrations_ ([pdf](http://www.map.mpim-bonn.mpg.de/images/b/be/Regensburg2012Talk5.pdf))

* S. Husseini, _Spherical fibrations_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/husseini2.pdf))

### In rational homotopy theory

Discussion in [[rational homotopy theory]] (for more see at _[[Sullivan model of a spherical fibration]]_):

* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], p. 202 of: _Rational Homotopy Theory_, Graduate Texts in Mathematics **205** Springer (2000)

* {#MollerRaussen85} [[Jesper Møller]], [[Martin Raussen]], _Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces_, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 ([jstor:2000242 ](https://www.jstor.org/stable/2000242))

* {#CohenVoronov05} [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_ ([arXiv:math/0503625](https://arxiv.org/abs/math/0503625))

* {#FelixOpreaTanre16} [[Yves Félix]], John Oprea, [[Daniel Tanré]], Prop. 2.3 in _Lie-model for Thom spaces of tangent bundles_, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 ([pdf](http://www.ams.org/journals/proc/2016-144-04/S0002-9939-2015-12829-8/S0002-9939-2015-12829-8.pdf), [doi:10.1090/proc/12829](https://doi.org/10.1090/proc/12829))

[[!redirects spherical fibrations]]

[[!redirects sphere bundle]]
[[!redirects sphere bundles]]


spherical object

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

# Contents
* table of contents
{: toc}

##Idea

Spherical objects in a general [[pointed category|pointed]]  [[model category]] play the role of the spheres in $Top$.

## Spherical objects

Let $\mathcal{C}$ be a [[pointed category|pointed]] [[model category]].

+--{: .un_defn}
###### Definition
A **spherical object** for $\mathcal{C}$ is a [[model category|cofibrant]] homotopy [[cogroup]] in $\mathcal{C}$.
=--

## Examples

1. The spheres form the obvious examples of spherical objects in the category $Top$, but the rational spheres give other examples.

2. In the category of path connected pointed spaces with action of a discrete group, $Gr.Top^*_0$ and space of form $S^n_G= \bigvee_G S^n$ is a spherical object.(see Baues, 1991, ref. below, p.273).

3. Any rational sphere is a sphere object (in a suitable category for [[rational homotopy theory]]).

4. Let $T$ be a contractible locally finite 1-dimensional simplicial complex, with $T^0$ its 0-skeleton. Let $\epsilon : E'T^0$ be a finite-to-one function. By $S^n_\epsilon$ we mean the space obtained by attaching an $n$-sphere to the vertices of $T$ with at vertex $v$, the spheres attached to $v$ being indexed by $\epsilon^{-1}(v)$. This space $S^n_\epsilon$ is a spherical object in the proper category, $Proper_\infinity^T$, of $T$-based spaces. (In this context $T$ is acting as the analogue of the base point.  It gives a base tree within the spaces.  This is explored a bit more in [[proper homotopy theory]].)

For instance, take $T = \mathbb{R}_{\geq 0}$, made up of an infinite number of closed unit intervals (end-to-end), then $S^n_\epsilon$ will be the infinite string of spheres considered in the entry on the [[Brown-Grossmann homotopy groups]] if we take $\epsilon$ to be the identity function on $T^0$.

+-- {: .un_defn}
###### Definition
By a  **family of spherical objects** for $\mathcal{C}$ is meant a collection of spherical objects in $\mathcal{C}$ closed under suspension.
=--

## The theory $\Pi_\mathcal{A}$

Let $\mathcal{A}$ be such a family of spherical objects. Let $\Pi_\mathcal{A}$ denote the full subcategory of $Ho(\mathcal{C})$, whose objects are the finite coproducts of objects from $\mathcal{A}$.

### Example

For $\mathcal{A} = \{S^n\}^\infty_{n=1}$ in $Top$, $\Pi_\mathcal{A} = \Pi$, the [[algebraic theory|theory]] of [[Pi-algebras]].

Of course, $\Pi_\mathcal{A}$ is a _finite product theory_ in the sense of [[algebraic theories]], and the corresponding models/algebras/modules are called:

## $\Pi_\mathcal{A}$-algebras

We thus have that these are the product preserving functors $\Lambda : \Pi_\mathcal{A}^{op}\to Set_*$.  Morphisms of $\Pi_\mathcal{A}$-algebras are simply the natural transformations.  This gives a category $\Pi_\mathcal{A}-Alg$.

### Properties

*  Such a $\Pi_\mathcal{A}$-algebra, $\Lambda$, is determined by its values $\Lambda(A)\in Set_*$ for $A$ in $\mathcal{A}$, together with, for every $\xi\colon A \to \bigsqcup_{i\in I}A_i$ in $\Pi_\mathcal{A}$, a map

$$\xi^*\colon \prod \Lambda(A_i)\to \Lambda(A).$$

* The object $A$ being a (homotopy) cogroup, $\Lambda(A)$ is a group (but beware the $\xi^*$ need not be group homomorphisms).

### Example

If $X$ is in $\mathcal{C}$, define $\pi_\mathcal{A}(X):= [-,X]_{Ho(\mathcal{C})} : \Pi_{\mathcal{A}}^{op}\to Set_*$.  This is the **homotopy $\Pi_{\mathcal{A}}$-algebra** of $X$.  As with $\Pi$-[[Pi-algebra|algebras]], there is a _realisablity problem_, i.e., given $\Lambda$, find a $X$ and an isomorphism, $\pi_\mathcal{A}(X)\cong \Lambda$.  The realisability problem is discussed in Baues-Blanc (2010) (see below).

## References

Spherical objects are considered in

*  [[Hans-Joachim Baues]] and [[David Blanc]], Comparing cohomology obstructions, (2010), [Arxiv](http://arxiv.org/PS_cache/arxiv/pdf/1008/1008.1712v1.pdf) (to appear JPAA).

Examples are given in earlier work by Baues and by Blanc.

The group action case is in

*  [[Hans-Joachim Baues]],  _Combinatorial Homotopy and 4-Dimensional Complexes_, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

The example from [[proper homotopy theory]] is discussed in

* H.-J. Baues and  Antonio Quintero, _Infinite Homotopy Theory_,  K-monographs in mathematics, Volume 6, Kluwer, 2001.

[[!redirects spherical object]]
[[!redirects spherical objects]]
[[!redirects spherical object and Pi(A)-algebra]]
[[!redirects spherical object and Pi(A)-algebras]]
[[!redirects spherical objects and Pi(A)-algebra]]
[[!redirects spherical objects and Pi(A)-algebras]]


spherical object and Pi(A)-algebra > history

< [[spherical object and Pi(A)-algebra]]
[[!redirects spherical object and Pi(A)-algebra>history]]