torsion constraints in supergravity

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## Idea

The [[equations of motion]] of [[supergravity]] typically imply -- or are even equivalent to ([Candiello-Lechner 93](#CandielloLechner93), [Howe 97](#Howe97)), that the [[supertorsion|super-]][[torsion of a Cartan connection|torsion]] of the super-[[vielbein fields]] vanishes. At least in some cases these _supergravity torsion constraints_ may naturally be understood as saying that supergravity solutions are ([[higher Cartan geometry|higher]]) [[super-Cartan geometry]] modeled on [[extended super Minkowski spacetime]] _with_ its canonical [[torsion of a G-structure]], due to the fact that the [[left invariant 1-forms]] on super-Minkowski space are not closed.

### From the canonical torsion of super-Minkowski spacetime
 {#CanonicalTorsionOfSuperMinkowskiSpacetime}

The torsion constraint is naturally understood by regarding [[supergravity]] as [[Cartan geometry]] for the inclusion of the [[orthogonal group]] into a [[super Poincare group]] and by noticing that the corresponding local model space, which is [[super-Minkowski spacetime]] $\mathbb{R}^{d|N}$, canonically has non-vanishing torsion.

Let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d|N}$ underlying the [[super translation group]]. Then the [[left-invariant 1-forms]] are

* $\psi^\alpha = d \theta^\alpha$.

* $e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta$.

Here the extra summand in the equation for $e^a$ (necessary to make it left-invariant) causes it to be non-closed:

$$
  \begin{aligned}
    d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta)
    \\
    & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta
    \\
    & = \frac{i}{2} \overline{\psi}\Gamma^a \psi
  \end{aligned}
  \,.
$$

Taking the [[spin connection]] $(\omega^a{}_b)$ on $\mathbb{R}^{d|N}$ to vanish, as usual, this means that there is non-vanishing [[torsion]]:

$$
  \begin{aligned}
    \tau^a & = \mathbf{d} e^a + \omega^a{}_b \wedge e^b
    \\
    & = \mathbf{d} e^a
    \\
    & = \frac{i}{2} \overline{\psi}\Gamma^a \psi
  \end{aligned}
$$

Depending on perspective one might say that it is the [[supertorsion]] that vanishes (see at _[[super-Minkowski spacetime]]_ and at _[[D'Auria-Fre formulation of supergravity]]_ for this perspective), or, alternatively, that one is dealing with [[Cartan geometry]]/[[G-structure]] whose local model space carries non-vanishing torsion, see [below](#InTermsOfTorsionTwistedGStructure).

Notice that the torison-full but left-invariant forms are of course obtained from the torsion-free but non-left-invartiant forms by a $GL(\mathbb{R}^{d|N})$-valued function:

$$
  \left(
     \array{
        e^a
        \\
        \psi^\alpha
     }
  \right)
  =
  \left(
    \array{
      id & \tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha
      \\
      0 & id
    }
  \right)
  \left(
    \array{
      \mathbf{d}x^a
      \\
      \mathbf{d}\theta^\alpha
    }
  \right)
$$

$$
  \left(
    \array{
      \mathbf{d}x^a
      \\
      \mathbf{d}\theta^\alpha
    }
  \right)
  =
  \left(
    \array{
      id & -\tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha
      \\
      0 & id
    }
  \right)
  \left(
     \array{
        e^a
        \\
        \psi^\alpha
     }
  \right)
$$

This shows that regarding

$$
  (E^A) \coloneqq (E^a, E^\alpha) \coloneqq (e^a, \Psi^\alpha)
$$

as a [[super-vielbein]] is consistent: this is indeed a [[homotopy]] in

$$
  \array{
     \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N})
     \\
     &
     {}_{\mathllap{\tau_{\mathbb{R}^{d|N}}}}\searrow
     &
     \swArrow_{E}
     &
     \swarrow_{\mathrlap{O(\mathbb{R}^{d|N})\mathbf{Struc}}}
     \\
     && \mathbf{B}GL(\mathbb{R}^{d|N})
  }
$$

but not the tautological one given by

$$
  \array{
     \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N})
     \\
     &
     \searrow
     &
     \downarrow
     &
     \swarrow
     \\
     && \mathbf{B}GL(\mathbb{R}^{d|N})
  }
$$

where the left triangle is that which exhibits the canonical trivialization of the [[frame bundle]] of $\mathbb{R}^{d|N}$.

### In terms of torsion-twisted $G$-structure
 {#InTermsOfTorsionTwistedGStructure}

Given a [[subgroup]] $G\hookrightarrow GL(V)$ of the [[general linear group]] of a linear model space $V$ (e.g. [[super-Minkowski spacetime]] $\mathbb{R}^{d|N}$), then a [[G-structure]] is [[integrability of G-structure|first-order integrable]] if on the first-order [[infinitesimal neighbourhoods]] of any point it is equal to the canonical (trivial) $G$-structure on $V$. Ordinarily the standard [[torsion]] on $V$ vanishes, and if so then so does that of any first-order integrable $G$-structure, which is the reason why for these the [[torsion of a G-structure]] vanishes.

But in the situation of $V$ being [[super-Minkowski spacetime]] as [above](#CanonicalTorsionOfSuperMinkowskiSpacetime), the torsion of the local model space does not vanish, and so accordingly neither does that of a first-order integrable $G$-structure in this case.

This perspective on the torsion constraints in supergravity is adopted in ([Lott 01](#Lott01)), see there around (38) of the original article or section 4 of the review on the arXiv.

## Properties

### Relation to supergravity equations of motion

The [[supergravity]] [[equations of motion]] typically imply the torsion constraints. See at [super p-brane -- On curved spacetimes](Green-Schwarz+action+functional#OnCurvedSpacetime) for more.

With enough [[supersymmetry]], the torsion constraints (always together with the [[Bianchi identities]] on the superfields, see at _[[D'Auria-Fre formulation of supergravity]]_) may even become equivalent to the supergravity equations of motion. This is so for [[11-dimensional supergravity]] ([Candiello-Lechner 93](#CandielloLechner93), [Howe 97](#Howe97), see [Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4](#CederwallGranNilssonTsimpis04)) and maybe its maximally supersymmetric [[KK-compactifications]]. See at _[Examples -- 11d SuGra](#Examples11dSuGra)_.

### Relation to CR-geometry

A close [[analogy]] between [[CR geometry]] and [[supergravity]] [[superspacetimes]] (as both being [[torsion of a G-structure|torsion-ful]] [[integrable G-structures]]) is pointed out in ([Lott 01 exposition (4.2)](#Lott01)).

## Examples

In accord with the [above](#CanonicalTorsionOfSuperMinkowskiSpacetime), typically the [[equations of motion]] of a [[supergravity]] theory constrain the spinorial part of the torsion to have components $(\Gamma^a)_{\alpha \beta}$.

### 11d supergravity
 {#Examples11dSuGra}

The torsion constraint for [[11-dimensional supergravity]] is discussed for instance in ([Bergshoeff-Sezgin-Townsend 87, equation (14)](#BergshoeffSezginTownsend87)).

Here something special happens:

The articles ([Candiello-Lechner 93 (5.6)](#CandielloLechner93), [Howe 97](#Howe97), see [Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4](#CederwallGranNilssonTsimpis04)) show that imposing the torsion constraint (on some chart) $\mathbf{d} E^a + \omega^{a}{}_b \wedge E^b - \bar \psi \Gamma^a \psi  = 0$ as well as $(\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi)_{\theta \theta} = 0$ implies the equations of motion of [[11d supergravity]].

Moreover, setting $\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi = 0$ generally (not just the component proportional to the wedge product of two fermionic 1-forms, hence requiring the full supertorsion tensor to be that of super-Minkowski spacetime)) then ([Candiello-Lechner 93, (5.8) with (6.5)](#CandielloLechner93))  this in addition puts the [[field strength]] of the [[supergravity C-field]] to 0. Hence this implies solutions to the ordinary vacuum [[Einstein equations]] in 11d. Such solutions are considered notably in the context of [[M-theory on G2-manifolds]] (e.g. [Acharya 02, p. 9](M-theory+on+G2-manifolds#Acharya02)). See also at _[M-theory on G2-manifolds -- Details -- Vacuum solution and torsion constraints](M-theory+on+G2-manifolds#VacuumSolutionsAndTorsion)_.

### 10d Heterotic supergravity
 {#HeteroticSupergravity}

For [[heterotic supergravity]] in 10d the [[equations of motion]] are equivalent to the condition that

1. the super-torsion of the bosonic part $\{e^a\}$ of the [[super vielbein]] is a bosonic form

   $$
     \mathcal{D}e^a - \overline{\psi} \Gamma^a \psi = T^a{}_{b c} e^b \wedge e^c
   $$

1. the super-torsion of the odd part $\psi^\alpha$ of the [[super vielbein]] is of the form

   $$
    \mathcal{D} \psi^\alpha = T^\alpha{}_{b c} e^b \wedge e^c  + \overline{\psi}^\beta \Gamma_a{}_{\beta \gamma} \phi^{\alpha \gamma}
   $$

   for

   $$
     \phi^{\alpha \beta} \propto tr(\chi^\alpha \chi^\beta) - tr(T^\alpha T^\beta)
   $$

   proportional to the bispinor formed by tracing the square of the [[gaugino]] field $\chi$

1. the [[curvature]] 2-form of the [[gauge field]] has vanishing bispinorial component:

   $$
     F_{\alpha \beta} = 0
   $$

   (this is the [[10d super Yang-Mills theory]] sector)

This is due to ([Witten 86 (5)+(27)](#Witten86)), see also ([Atick-Dhar-Ratra 86 (4.1)](#AtickDharRatra86)).
These authors do not state explicitly that $\phi^{\alpha \beta} \propto tr(\lambda^\alpha \lambda^\beta) - tr (T T)$. (Among authors using a similar but different parameterization this statement is made explicit in [Candiello-Lechner 93 (2.5) with (2.29)](#CandielloLechner93)). But this follows by taking the differential of the bispinorial part of the 3-form field (which is the cocycle term for the heterotic [[Green-Schwarz superstring]])

$$
  d
  \left(
    \overline{\psi}
      \wedge
      \Gamma_a
    \psi
  \wedge e^a
  \right)
  \propto
  \underset{i}{\sum}
  \underset{= F^i_{(1,1)}}{
  \underbrace{
    \left(
      \overline{\psi} \Gamma_a \chi^i
    \right)
    \wedge e^a
  }
  }
  \wedge
  \underset{ = F^i_{(1,1)}}{
  \underbrace{
    \left(
      \overline{\chi^i}
      \Gamma_b
      \psi
    \right)
  \wedge e^b
  }
  }
  - tr(R R)_{(2,2)}
$$

where we used the relation ([Witten 86 (8)](#Witten86)) (recalled for instance in [Bonora-Bregola-Lechner-Pasti-Tonin 87 (2.28)](#BonoraBregolaLechnerPastiTonin87), [Lechner-Tonin 08 (2.13)](#LechnerTonin08)).

According to ([Bonora-Bregola-Lechner-Pasti-Tonin 90](#BonoraBregolaLechnerPastiTonin90)) in fact all these constraints follow from just $T^a_{\alpha \beta}  \propto \Gamma^a_{\alpha \beta}$.

## Related concepts

* [[CR manifold]]

* [[D'Auria-Fre formulation of supergravity]]

## References

### General

The formulation of supergravity equations of motion in terms of constraints on the torsion tensor goes back to

* [[Julius Wess]] [[Bruno Zumino]], _Superspace formulation of supergravity_, Phys. Lett. B66 (1977), 361–364.

A mathematical formulation in terms of [[torsion of a G-structure|torsion-full]] [[first-order integrable G-structure|first-order integrable]] [[G-structures]] on [[supermanifolds]] (for low dimensional supergravity theories) is given in

* {#Lott01} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_ ([arXiv:0108125](http://arxiv.org/abs/math/0108125), published as: _Torsion constraints in supergeometry_, Comm. Math. Phys. 133 (1990), 563–615 ([doi:10.1007/BF02097010](https://doi.org/10.1007/BF02097010))

which is followed up in

* [[Michel Egeileh]], [[Fida El Chami]], _Some remarks on the geometry of superspace supergravity_, J.Geom.Phys. 62 (2012) 53-60 ([spire](http://inspirehep.net/record/1333125))

### For 11d supergravity

Discussion of torsion constrains for [[11-dimensional supergravity]] from the point of view of consistency of the [[membrane]] [[Green-Schwarz action functional]] is in

* {#BergshoeffSezginTownsend87}  [[Eric Bergshoeff]], [[Ergin Sezgin]], [[Paul Townsend]], _Supermembranes and eleven dimensional supergravity_, Phys.Lett. B189 (1987) 75-78, In [[Mike Duff]],  (ed.), _[[The World in Eleven Dimensions]]_ 69-72 ([pdf](http://streaming.ictp.trieste.it/preprints/P/87/010.pdf), [spire](http://inspirehep.net/record/248230?ln=en))

The claim that this torsion constraint in  [[11-dimensional supergravity]] is already equivalent to all of the [[equations of motion]] is due to

* {#CandielloLechner93} A. Candiello, [[Kurt Lechner]], _Duality in Supergravity Theories_, Nucl.Phys. B412 (1994) 479-501 ([arXiv:hep-th/9309143](http://arxiv.org/abs/hep-th/9309143))

* {#Howe97} [[Paul Howe]], _Weyl Superspace_, Physics Letters B, Volume 415, Issue 2, 11 December 1997, Pages 149–155 ([arXiv:hep-th/9707184](http://arxiv.org/abs/hep-th/9707184))

concisely reviewed in

* {#CederwallGranNilssonTsimpis04} [[Martin Cederwall]], [[Ulf Gran]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], _Supersymmetric Corrections to Eleven-Dimensional Supergravity_, JHEP 0505:052, 2005 ([arXiv:hep-th/0409107](https://arxiv.org/abs/hep-th/0409107))

For commentary see also ([Nilsson 00, section 2](#Nilsson00)) and

* {#CGNN00} [[Martin Cederwall]], [[Ulf Gran]], Mikkel Nielsen, [[Bengt Nilsson]], _Manifestly supersymmetric M-theory_, JHEP 0010 (2000) 041 ([arXiv:hep-th/0007035](http://arxiv.org/abs/hep-th/0007035))

* {#HoweSezgin05} [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, Class.Quant.Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](https://arxiv.org/abs/hep-th/0412245))

also

* {#BrinkHowe80} [[Lars Brink]], [[Paul Howe]], _Eleven-dimensional supergravity on the mass shell in superspace_, Phys. Lett. , B91:384–386, 1980

Discussion of possible deformations of the torsion constraint ([[M-theory]] corrections) includes

* [[Martin Cederwall]], [[Ulf Gran]], Mikkel Nielsen, [[Bengt Nilsson]], _Generalised 11-dimensional supergravity_, in A. Semikhatov, M. Vasiliev and V. Zaikin (eds.) Proceedings of "Quantization, Gauge Theory & Strings", Moscow 2000 ([arXiv:hep-th/0010042](http://arxiv.org/abs/hep-th/0010042))

* [[Paul Howe]], [[Dimitrios Tsimpis]], _On higher-order corrections in M theory_, JHEP 0309 (2003) 038 ([arXiv:hep-th/0305129](http://arxiv.org/abs/hep-th/0305129))

### For 10d heterotic supergravity

Discussion of torsion constraints for [[heterotic supergravity]] goes back to ([Nilsson 81](Green-Schwarz+action+functional#Nilsson81)) and includes

* [[Paul Howe]], A. Umerski, _On superspace supergravity in ten dimensions_, Phys. Lett. B 177 (1986) 163.

* {#AtickDharRatra86} [[Joseph Atick]], Avinash Dhar, and Bharat Ratra, _Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory_, Phys. Rev. D 33, 2824, 1986 ([doi.org/10.1103/PhysRevD.33.2824](https://doi.org/10.1103/PhysRevD.33.2824))

* {#Witten86} [[Edward Witten]], _Twistor-like transform in ten dimensions_, Nuclear Physics B Volume 266, Issue 2, 17 March 1986

* {#BonoraBregolaLechnerPastiTonin87} [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions_, Nuclear Physics B
Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)

* {#BonoraBregolaLechnerPastiTonin90} [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D_,  International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (doi:10.1142/S0217751X90000222)

* [[Paul Howe]], _Heterotic supergeometry revisited_ ([arXiv:0805.2893](http://arxiv.org/abs/0805.2893))

* {#Nilsson00} [[Bengt Nilsson]], _A superspace approach to branes and supergravity_ ([arXiv:hep-th/0007017](http://arxiv.org/abs/hep-th/0007017))

* {#LechnerTonin08} [[Kurt Lechner]], [[Mario Tonin]], _Superspace formulations of ten-dimensional supergravity_, JHEP 0806:021,2008 ([arXiv:0802.3869](https://arxiv.org/abs/0802.3869))

### For 4d supergravity
 {#ReferencesFor4d}

For [[d=4 N=1 supergravity]] the torsion is again constrained to be equal to the left-invariant torsion of super-Minkowski spacetime, see for instance

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 2, (III.2.28a), (III.3.66a) of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* Daniel Patrick Butter, section 2.2.5 of _On conformal superspace and the One-Loop Effective Action in Supergravity_, 2010 ([pdf](http://digitalassets.lib.berkeley.edu/etd/ucb/text/Butter_berkeley_0028E_10582.pdf))

### For 2d supergravity / superstring worldsheets / super Riemann surfaces

* [[Suresh Govindarajan]], [[Burt Ovrut]], _A geometric interpretation for the torsion constrains of $(2,0)$-heterotic worldhseet supergravity_, Mod. Phys. Lett. A6(1991), 3341. ([pdf](http://www.physics.iitm.ac.in/~suresh/sgtalk/talk_html/torsion.pdf))

[[!redirects supergravity torsion constraint]]
[[!redirects supergravity torsion constraints]]

[[!redirects torsion constraint of supergravity]]
[[!redirects torsion constraints of supergravity]]


torsion group

# Torsion groups
* table of contents
{: toc}

## Definition

See at _[[torsion subgroup]]_.

Torsion groups and group [[homomorphisms]] form a [[category]] $Tor$; [[abelian group|abelian]] torsion groups and group homomorphisms form a category $Ab Tor$.

## Related concepts

* [[torsion module]]

* [[torsion sheaf]]

* [[Pontryagin duality for torsion abelian groups]]

## References

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

[[!redirects torsion group]]
[[!redirects torsion groups]]
[[!redirects AbTor]]
[[!redirects Ab Tor]]


torsion module

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## Definition

### For modules over rings

Given a [[ring]] $R$, an element $m$ in an $R$-[[module]] $M$ is **torsion element** if there is a nonzero element $r$ in $R$ such that $r m=0$. In [[constructive mathematics]], given a [[ring]] $R$ with a [[tight apartness relation]] $\#$, an element $m$ in an $R$-[[module]] $M$ is a **torsion element** if there is a element $r$ in $R$ such that $r \# 0$ and $r m=0$.

A **torsion module** is a module whose elements are all torsion. A [[torsion-free module]] is a module whose elements are not torsion, other than $0$.

More generally, given an ideal $\mathfrak{a} \subset R$ then an $\mathfrak{a}$-torsion module is one all whose elements are annihilated by some power of elements in $\mathfrak{a}$.

### For $\infty$-modules over $E_\infty$-rings

Let $A$  be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

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

An $A$-[[∞-module]] $N$ is an _$\mathfrak{a}$-torsion module_ if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

=--

([Lurie "Completions", def. 4.1.3](#LurieCompletions)).

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

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}tor}
  \hookrightarrow
  A Mod
$$

is [[reflective sub-(∞,1)-category|co-reflective]] and the co-reflector $ʃ_{\mathfrak{a}}$ -- the _[[torsion approximation]]_ -- is [[smashing localization|smashing]].

=--

([Lurie "Completions", prop. 4.1.12](#LurieCompletions)).

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

For $N \in A Mod_{\leq 0}$ then [[torsion approximation]], prop. \ref{TorsionApproximation}, intuced a [[monomorphism]] on $\pi_0$

$$
  \pi_0 ʃ_{\mathfrak{a}} N \hookrightarrow \pi_0 N
$$

including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

=--

([Lurie "Completions", prop. 4.1.18](#LurieCompletions)).

## Related concdepts

* [[torsion subgroup]]

* [[torsion-free module]]

## References

* {#Quillen96} [[Daniel Quillen]], _Module theory over nonunital rings_, August 1996 ([[QuillenModulesOverRngs.pdf:file]])

* {#LurieCompletions} [[Jacob Lurie]], section 4.1 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

[[!redirects torsion modules]]


torsion of a Cartan connection

> for disambiguation see at [[torsion]]

***

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## Definiton

Given an $(H \stackrel{i}{\to} G)$-_[[Cartan connection]]_ $\nabla$ on a manifold $X$, its _torsion_ is the component of its [[curvature]] in the [[coimage]] of $i_\ast : \mathfrak{h}\to \mathfrak{g}$.

So if $U \to X$ is a [[cover]] over which the underlying $G$-[[principal bundle]] trivializes (for instance the total space of the $G$-[[principal bundle]] itself if one models the Cartan connection by an [[Ehresmann connection]]) and

$$
  \omega \in \Omega^1(U,\mathfrak{g})
$$

is the Lie algebra valued [[curvature]] form, then the torsion form is

$$
  \tau \coloneqq coim(i)_\ast \omega \in \Omega^1(U,\mathfrak{g}/\mathfrak{h})
  \,.
$$

e.g. ([Sharpe 97, section 5.3, below def 3.1](#Sharpe97), [Cap-Slovák 09, section 1.5.7, p. 85](#CapSlovak09))

## Properties

### Integrability of local charts

If the torsion of a Cartan connection vanishes, then it has [[flat connection|flat]] $\mathfrak{g}/\mathfrak{h}$-valued [[parallel transport]]. In partciular then every point in the base manifold has a neighbourhood $U_x$ over which the given $G$-[[principal bundle]] trivializes and then this parallel trasport gives an identification $U_x \simeq U_{\sigma(x)}(G/H)$ with an open subset in the [[Klein geometry]] $G/H$.

### Relation to integrability of $G$-structure

If the [[Cartan connection]] is regarded as providing, in particular, a [[G-structure]], then the condition that its torsion vanishes is the [[integrability of G-structures]].

## Examples

### (Pseudo-)Riemannian geometry

For $(H \to G)$ being the inclusion $O(d)\hookrightarrow Iso(\mathbb{R}^d)$ of the [[orthogonal group]] into the [[Euclidean group]] or the inclusion $O(d-1,1)\hookrightarrow Iso(\mathbb{R}^{d-1,1})$ of the [[Lorentz group]] into the [[Poincare group]], then an $(H\to G)$-[[Cartan connection]] encodes a ([[pseudo-Riemannian geometry|pseudo]]-)[[Riemannian geometry]] with metric connection. Its torsion then is the [[torsion of a metric connection]]. See also at _[[first-order formulation of gravity]]_.

## Related concepts

* [[torsion of a G-structure]]

## References

Textbook accounts include

* {#Sharpe97} R. Sharpe, _Differential Geometry -- Cartan's Generalization of Klein's Erlagen program_ Springer (1997)

* {#CapSlovak09} [[Andreas Cap]], [[Jan Slovák]], chapter 1 of _Parabolic Geometries I -- Background and General Theory_, AMS 2009

Discussion with an eye towards [[torsion constraints in supergravity]] is in

* {#Lott01} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_ Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

[[!redirects torsion of Cartan connections]]


torsion of a curve

## References

* Wikipedia, _[Torsion of a curve](https://en.wikipedia.org/wiki/Torsion_of_a_curve)_


torsion of a G-structure

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## Idea
 {#Idea}

The [[integrability of G-structures]] exists to first order, precisely if a certain [[torsion]] [[obstruction]] vanishes. This is the first in an infinite tower of tensor invariants in [[Spencer cohomology]] associated with a $G$-structure that obstruct its integrability (local flatness) ([Guillemin 65](#Guillemin65)).

The _torsion of a $G$-structure_ is defined to be the space in which the invariant part of the [[torsion of a Cartan connection]] takes values, for any [[Cartan connection]] compatible with the $G$-structure (see at _[Cartan connection -- Examples -- G-Structure](Cartan%20connection#ExampleGStructures)_) ([Sternberg 64, from p. 317 on](#Sternberg64), [Guillemin 65, section 4](#Guillemin65)), for review see also ([Lott 90, p.10](#Lott90), [Joyce 00, section 2.6](#Joyce00)).

The order $k$-torsion of a $G$-structure (counting may differ by 1) is an element in a certain [[Spencer cohomology]] [[cohomology group|group]] ([Guillemin 65, prop. 4.2](#Guillemin65)) and is the [[obstruction]] to lifting an order-$k$-[[integrable G-structure]] to order $k+1$ ([Guillemin 65, theorem 4.1](#Guillemin65)).

## Related concepts

* [[torsion of a Cartan connection]]

## References

The concept goes back to the work of [[Eli Cartan]] ([[Cartan geometry]]).

Textbook accounts include

* {#Sternberg64} [[Shlomo Sternberg]], section VII of _Lectures on differential geometry_, Prentice Hall 1964; Russian transl. Mir 1970

Discussion including the higher order obstructions in [[Spencer cohomology]] to [[integrability of G-structures]] is in

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

Formalization in [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

Discussion with an eye towards [[torsion constraints in supergravity]] is in

* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

Discussion with an eye towards [[special holonomy]] is in

* {#Joyce00} [[Dominic Joyce]], section 2.6 of _Compact manifolds with special holonomy_, Oxford University Press 2000

Further mentioning of the higher order torsion invariants includes

* {#Bryant05} [[Robert Bryant]], section 4.2 of _Some remarks on $G_2$-structures_, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 [pdf](http://gokovagt.org/proceedings/2005/ggt05-bryant.pdf)

Discussion specifically for kinematical groups:

* [[José Figueroa-O'Farrill]], _On the intrinsic torsion of spacetime structures_ ([arXiv:2009.01948](https://arxiv.org/abs/2009.01948))

See also

* Wikipedia, _[Torsion of a G-structure](https://en.wikipedia.org/wiki/G-structure#Torsion_of_a_G-structure)_

[[!redirects torsion of G-structures]]

[[!redirects torsion invariants of G-structures]]


torsion of a metric connection

+-- {: .rightHandSide}
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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

> For other notions of _[[torsion]]_ see there.

***

#Contents#
* table of contents
{:toc}

## Definition

A ([[pseudo-Riemannian metric|pseudo]]) [[Riemannian metric]] with metric-compatible [[Levi-Civita connection]] on a [[smooth manifold]] $X$ may be encoded by a [[connection on a bundle|connection]] with values in the [[Poincaré Lie algebra]] $\mathfrak{iso}(p,q)$.

This [[Lie algebra]] is the [[semidirect product]]

$$
  \mathfrak{iso}(p,q) \simeq \mathfrak{so}(p,q) \ltimes \mathbb{R}^{p+q}
$$

of the [[special orthogonal Lie algebra]] and the abelian translation Lie algebra. Accordingly, a connection 1-form has two components

* $\Omega \in \Omega^1(U,\mathfrak{so}(p,q))$ (sometimes called the "[[spin connection]]");

* $E \in \Omega^1(U,\mathbb{R}^{p+q})$ (sometimes called the "[[vielbein]]").

The metric itself is

$$
  g = \langle E \otimes E \rangle
  \,.
$$

Accordingly also the [[curvature]] 2-form has two components:

* $R = d \Omega + [\Omega \wedge \Omega] \in \Omega^2(U, \mathfrak{so}(p,q))$ -- the [[Riemann curvature]];

* $\tau = d E + [\Omega \wedge E]$ -- the **torsion**.

This is the special case of the more general concept of [[torsion of a Cartan connection]].

## Generalizations

In [[supergeometry]] a metric structure is given by a connection with values in the [[super Poincaré Lie algebra]]. The corresponding notion of torsion has an extra contribution from [[spinor]] fields: the [[super torsion]].

See also

* [[integrability of G-structures]]

## Related concepts

* [[first-order formulation of gravity]]

[[!include local and global geometry - table]]


torsion points of an elliptic curve

\tableofcontents

\section{Idea}

Recall that an [[elliptic curve]] $E$ is an [[abelian variety]]; in particular, its set of points admits a group structure. A point of $E$ is a torsion point if it is a [[torsion element]] of this group structure.

If $E$ is defined over a [[number field]] $F$, the [[Galois theory]] of $F$ interacts very well with the torsion points of $E$, as we describe below.

\section{Definition}

\begin{defn} Let $E$ be an [[elliptic curve]] over a field $F$, and let $l \geq 2$ be an integer. An _$l$-torsion point of $E$_ is a point $x$ of $E$ such that $l x = 0$, that is, $\underbrace{x + x + \cdots + x}_{l} = 0$ in the (abelian) group structure on (the set of points of) $E$, where $0$ denotes the identity element of the group structure (often taken to be the point at infinity).

A _torsion point_ of $E$ is a point of $E$ which is an $l$-torsion point of $E$ for some integer $l \geq 2$.
\end{defn}

\begin{rmk} For a fixed integer $l \geq 2$, the set of $l$-torsion points of $E$ assembles into an [[abelian group]] with respect to the group structure of $E$. The same is true of the set of all torsion points of $E$. \end{rmk}

\begin{notn} For a fixed integer $l \geq 2$, the set of $l$-torsion points of $E$ is often denoted $E[l]$. \end{notn}

\section{Splitting}

The following observation is used frequently when working with torsion points of an elliptic curve over a [[number field]].

\begin{proposition} Let $E$ be an elliptic curve defined over $\mathbb{Q}$, the [[rational number|rationals]]. Then for any integer $l \geq 2$, there is an [[isomorphism]] of [[abelian group|abelian groups]] $E[l] \cong \mathbb{Z} / l\mathbb{Z} \oplus \mathbb{Z} / l\mathbb{Z}$. \end{proposition}

\begin{rmk} \label{RemarkAutomorphismsOfTorsionPointsAsMatrixGroup} Thus if $l$ is a [[prime number|prime]], so that $\mathbb{Z} / l \mathbb{Z}$ is a [[field]] $\mathbb{F}_{l}$, we can think of the set of automorphisms of the abelian group $E[l]$ as the [[matrix group]], specifically the [[general linear group]], $GL_2\left( \mathbb{F}_{l} \right)$. \end{rmk}

\section{Galois theory}

Given an elliptic curve $E$ defined over a [[number field]] $F$, the following observation ties the Galois theory of $F$ to the torsion points of $E$. We shall denote the algebraic closure of $F$ by $\overline{F}$.

\begin{proposition} \label{PropositionGaloisGroupActsOnEllipticCurveAndItsTorsionPoints} Let $E$ be an elliptic curve defined over a [[number field]] $F$. Let $F'$ be an [[field extension|extension]] of $F$. The [[Galois group]] $Gal\left(F' / F \right)$ acts on all of (the set of points of) $E / F'$, the (abelian group of) torsion points of $E / F'$, and the (abelian group of) $l$-torsion points of $E / F'$ for any fixed integer $l \geq 2$, in the obvious way: given an automorphism $\sigma$ of $F'$ which fixes $F$, we send a point $x$ of $E / F'$ to $\sigma(x)$. \end{proposition}

\begin{rmk} The principal point is that given $\sigma$ and $x$ as in Proposition \ref{PropositionGaloisGroupActsOnEllipticCurveAndItsTorsionPoints}, one can check that $\sigma(x)$ is still a point of $E$, and is still an $l$-torsion point for some $l \geq 2$ if $x$ is. \end{rmk}

Given Remark \ref{RemarkAutomorphismsOfTorsionPointsAsMatrixGroup}, we can, as with any [[group action]], reformulate Proposition \ref{PropositionGaloisGroupActsOnEllipticCurveAndItsTorsionPoints} in the case of $l$-torsion points for a fixed prime $l \geq 2$ as follows.

\begin{corollary} \label{CorollaryHomomorphismFromGaloisGroupToTorsionPoints} Let $l \geq 2$ be a [[prime number|prime]]. Let $E$ be an elliptic curve defined over a [[number field]] $F$. Let $F'$ be an [[field extension|extension]] of $F$. Then the application of automorphisms of $F'$ to points of $E$ determines a [[homomorphism|group homomorphism]] $Gal\left(F' / F\right) \rightarrow GL_2\left( \mathbb{F}_{l} \right)$. \end{corollary}

\begin{rmk} Given the homomorphism of Corollary \ref{CorollaryHomomorphismFromGaloisGroupToTorsionPoints}, we can take its kernel. This subgroup of $Gal\left( F' / F \right)$ determines, by [[Galois theory]], a [[field extension]] of $F$. Let us denote it by $K$. In fact, $K$ is simply the [[tensor product]] of $E[l]$ with $F$; from that point of view, we have demonstrated that this tensor product is a finite [[Galois extension]] of $F$. \end{rmk}


torsion sheaf

#Contents#
* table of contents
{:toc}

## Idea

An [[abelian sheaf]] ([[sheaf of abelian groups]]) is called a _torsion sheaf_ if it is a [[sheafification]] of a [[presheaf]] that takes values in [[torsion groups]].

## References

* [[Günter Tamme]], section II 9.1 of _[[Introduction to Étale Cohomology]]_

[[!redirects torsion sheafs]]


torsion subgroup

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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition
 {#Definition}

Torsion and torsion-free classes of objects in an [[abelian category]] were introduced axiomatically as a [[torsion theory]] (or torsion pair) in ([Dickson 1963](#Dickson63)).

Beware that there are other, completely independent, concepts referred to as _[[torsion]]_. See there for more.

### In groups

The **torsion subgroup** of an [[abelian group]] $G \in $ [[Grp]] is the [[subgroup]] of all those [[elements]] $g \,\in\, G$, which have [[finite number|finite]] [[order]], i.e. those for which  some power is the [[neutral element]]
$$
  g \in G \;\text{is torsion}
  \;\;\;\;\;\;\Leftrightarrow\;\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\exists}
  \;\;\;
  g^n
  \;\coloneqq\;
  \underset{
    n\; factors
  }{
    \underbrace{
      g \cdot g \cdots g
    }
  }
  \,=\, \mathrm{e}
  \,.
$$

A group is

* *pure torsion* if it coincides with its torsion subgroup,

* *torsion-free* if its torsion subgroup is [[trivial subgroup|trivial]].

Notice that for [[abelian groups]] $A \in $ [[AbGrp]] the group operation is often written additively, namely as "$+$" and the [[neutral element]] is written as zero, whence this reads:

$$
  a \in A \;\text{is torsion}
  \;\;\;\;\;\Leftrightarrow\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\exists}
  \;\;\;
  n \cdot a
  \;\coloneqq\;
  \underset{
    n\; summands
  }{
    \underbrace{
      a + a \cdots + a
    }
  }
  \,=\, 0
  \,.
$$

### In monoids

The situation with monoids is very similar to the situation with groups.

The **torsion subgroup** of a [[commutative monoid]] $M \in $ [[Mon]] is the [[submonoid]] of all those [[elements]] $m \,\in\, M$ for which some power is the [[neutral element]]
$$
  m \in M \;\text{is torsion}
  \;\;\;\;\;\;\Leftrightarrow\;\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\exists}
  \;\;\;
  m^n
  \;\coloneqq\;
  \underset{
    n\; factors
  }{
    \underbrace{
      m \cdot m \cdots m
    }
  }
  \,=\, \mathrm{e}
  \,.
$$

Every such submonoid is a group, which is why the set of all such elements is called a torsion subgroup.

A monoid is

* *pure torsion* if it coincides with its torsion subgroup (and is thus the same as a pure torsion group),

* *torsion-free* if its torsion subgroup is [[trivial subgroup|trivial]].

Notice that for [[commutative monoids]] $C \in $ [[CMon]], where the monoid operation is traditionally written as addition $+$ and the [[neutral element]] is written as zero, this reads:

$$
  c \in C \;\text{is torsion}
  \;\;\;\;\;\Leftrightarrow\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\exists}
  \;\;\;
  n \cdot c
  \;\coloneqq\;
  \underset{
    n\; summands
  }{
    \underbrace{
      c + c \cdots + c
    }
  }
  \,=\, 0
  \,.
$$

### In modules

Given a [[ring]] $R$, an element $m$ in an $R$-[[module]] $M$ is a **torsion element** if there is a nonzero element $r$ in $R$ such that $r m=0$. In [[constructive mathematics]], given a [[ring]] $R$ with a [[tight apartness relation]] $\#$, an element $m$ in an $R$-[[module]] $M$ is a **torsion element** if there is a element $r$ in $R$ such that $r \# 0$ and $r m=0$.

A **[[torsion module]]** is a module whose elements are all torsion. A [[torsion-free module]] is a module whose elements are not torsion, other than $0$.

## Properties

### Relation to the $Tor$-functor

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

For $A$ an [[abelian group]], its torsion subgroup is isomorphic to the value of the degree-1 [[Tor|Tor functor]] $Tor^\mathbb{Z}_1(\mathbb{Q}/\mathbb{Z}, A)$.

=--

See at _[Tor - relation to torsion subgroups](Tor#RelationToTorsionGroups)_ for more.

### Relation to flatness

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

An [[abelian group]] is torsion-free precisely if regarded as a $\mathbb{Z}$-[[module]] it is a [[flat module]].

=--

This is a special case of a [more general result](http://ncatlab.org/nlab/show/principal+ideal+domain#flat) for modules over a [[principal ideal domain]]. See also _[flat module - Examples](flat+module#Examples)_ for more. However, it is unclear whether this result holds in [[constructive mathematics]], since the $\mathbb{Z}$ is not a [[principal ideal domain]] unless [[excluded middle]] holds.

## Examples and applications
 {#ExamplesAndApplications}

\begin{example}\label{FiniteGroupsArePureTorsion}
**(finite groups are pure torsion)**
\linebreak
Every [[finite group]] is pure torsion, hence is its own maximal torsion subgroup.
\end{example}

\begin{example}
**(tensoring with the rational numbers removes torsion subgroups)**
\linebreak
Given an [[abelian group]] $A$ which is pure torsion (e.g. a [[finite abelian group]], by Ex. \ref{FiniteGroupsArePureTorsion}), its [[tensor product of abelian groups|tensor product]] with the additive group of [[rational numbers]] is the [[trivial group|trivial]] abelian group:

$$
  A \,\text{torsion}
  \;\;\;
  \Rightarrow
  \;\;\;
  A \otimes_{\mathbb{Z}} \mathbb{Q}
  \;\simeq\;
  0
  \;\;\;
  \in
  \;
  AbGrp
  \,.
$$

Because, for $a \in A$ with
$\underset{ n\;summands }{\underbrace{a + a + \cdots + a}} =  0$ and for
$p,q \in \mathbb{Z} \subset \mathbb{Q}$ with $q \neq 0$ we have, by the definition of [[tensor product of abelian groups]]:

$$
  \begin{aligned}
    a \otimes \frac{p}{q}
    & \;=\;
    a \otimes
    \underset{n\; summands}{
    \underbrace{
      \Big(
       \frac{p}{n \cdot q}
         + \frac{p}{n \cdot q}
         + \cdot  +
          \frac{p}{n \cdot q}
      \Big)
    }}
    \\
    & \;=\;
    \underset{n\;summands}{
    \underbrace{
      \Big(a \otimes \frac{p}{n \cdot q}\Big)
      + \cdots +
      \Big(a \otimes \frac{p}{n \cdot q}\Big)
    }}
    \\
    & \;=\;
     \underset{
       n \; summands
     }{
     \underbrace{
       ( a + \cdots + a )
     }}
     \otimes
     \frac{p}{n \cdot q}
    \\
    & \;=\;
    0 \otimes \frac{p}{n \cdot q}
    \\
    & \;=\; 0
    \,.
  \end{aligned}
$$

In [[rational homotopy theory]] one considers the higher [[homotopy groups]] $\pi_n(X)$ of [[topological spaces]] $X$ tensored over [[rational numbers|$\mathbb{Q}$]]: the resulting groups $\pi_n(X) \otimes_{\mathbb{Z}} \mathbb{Q}$ are then necessarily torsion-free -- in this sense [[rational homotopy theory]] studies spaces "up to torsion".
\end{example}

\begin{example}
**(torsion homotopy groups of spheres)**
\linebreak
  By the [[Serre finiteness theorem]], the [[homotopy groups of spheres]] are [[finite groups]], and hence pure torsion by Ex. \ref{FiniteGroupsArePureTorsion}, except in the degree of the dimension of the sphere and, for even-dimensional spheres, twice its dimension minus one:

  $$
    \pi_k\big( S^n \big)
    \;\;
    \simeq
    \;
    \left\{
    \begin{array}{ll}
      \mathbb{Z} & k = n
      \\
      \mathbb{Z} \oplus \text{torsion} &
      k = 2n-1  \;\text{and}\; n\;\text{even}
      \\
      \text{torsion} & \text{otherwise}
    \end{array}
    \right.
  $$
\end{example}

\begin{example}
**(torsion in elliptic curves)**
\linebreak
The torsion elements of an [[elliptic curve]] as a group are important in [[number theory]] and [[arithmetic geometry]]. See [[torsion points of an elliptic curve]].
\end{example}

## Related concepts

* [[torsion-free module]]

* [[p-torsion]]

* [[torsion approximation]]

* [[Pontryagin duality for torsion abelian groups]]

* [[Bockstein homomorphism]]

## References

On axiomatization of [[torsion theory]] in [[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))

[[!redirects torsion subgroup]]
[[!redirects torsion subgroups]]

[[!redirects torsion group]]
[[!redirects torsion groups]]
[[!redirects torsion element]]
[[!redirects torsion elements]]

[[!redirects torsion-free group]]
[[!redirects torsion-free abelian group]]
[[!redirects torsion-free groups]]
[[!redirects torsion-free abelian groups]]


torsion theory

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[[!include additive and abelian categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __torsion theory__ in an [[abelian category]] $A$ is a [[pair]] $(T,F)$ of [[additive category|additive]] [[subcategories]], called the __torsion class__ $T$ and the __torsion free class__ $F$, such that the following conditions hold:

* $Hom(T,F) = 0$

(in other words, $A(X,Y) = 0$ if $X \in Ob T$ and $Y\in Ob F$).

* $Hom(T,Y) = 0 \Rightarrow Y\in Ob F$

* $Hom(X,F) = 0 \Rightarrow X\in Ob T$

* for all $X\in Ob A$, there exists $Y\subset X$, $Y\in Ob T$ and $X/Y\in Ob F$

Equivalently, a torsion theory in $A$ is a pair $(T,F)$ of [[strictly full subcategories]] of $A$ such that the first and last conditions in the above list hold. Alternatively, we can require the last condition
and the following 3: $T\cap F=\{0\}$, $T$ is closed under quotients
and $F$ under subobjects.  It follows also that $T$ and $F$ are stable under extensions.

### Torsion part of an object

If the [[abelian category]] $A$ satisfies [[Pierre Gabriel|Gabriel]]'s [[property (sup)]] then for every [[object]] $X$ there exists the largest [[subobject]] $t(X)\subset X$ which is in $T$, which is called the _torsion part_ of $X$ (sometimes written as $X_T$). Under the [[axiom of choice]], $t: X\to t(X)$ can be extended to a functor.

### Hereditary torsion theories

A torsion theory is called __hereditary__ if $T$ is closed under [[subobjects]], or equivalently, $t$ is [[left exact functor]].
For some authors (e.g. Golan) torsion theory is assumed to
be hereditary.

## Properties

If $(T,F)$ is a torsion class then $T$ and $F$ both contain the [[zero object]] and are closed under [[biproducts]] ([Borceux II 1.12.3](ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra)). Presentation of an object $X$ in $Ob A$ as an [[extension]] $0\to Y\to X\to X/Y\to 0$,  $Y$ in $Ob T$ by $X/Y$ in $Ob F$ is unique up to an [[isomorphism]] of [[short exact sequences]] ([Borceux II 1.12.4](ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra)).

Given an [[abelian category]] $A$ there is a [[bijection]] between universal [[closure operator|closure operations]] on $A$, hereditary torsion theories in $A$ ([Borceux II 1.12.8](ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra)) and, if $A$ is a [[locally finitely presentable category]] also with [[left exact functor|left exact]] [[localizations]] of $A$ admitting a [[right adjoint]] and with [[localizing subcategories]] of $A$ ([Borceux II 1.13.15](ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra)).

## Examples

The basic example of a torsion class is the class of [[torsion subgroup|torsion]] [[abelian groups]] within the [[category]] $A =$ [[Ab]] of all [[abelian groups]]. The torsion theories are often used as a means to formulate [[localization]] theory in [[abelian categories]].

## Literature

The notion originates with:

* {#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))

Comprehensive accounts:

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, vol. 2

* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

* [[Joachim Lambek]], _Torsion theories, additive semantics, and rings of quotients_, with app. by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics __177__, Springer-Verlag 1971, vi+94 pp. [MR284459](http://www.ams.org/mathscinet-getitem?mr=284459)

For a unified treatment in abelian and in [[triangulated categories]] see

* Apostolos Beligiannis, Idun Reiten, _Homological and homotopical aspects of torsion theories_,  Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. [pdf](http://www.math.uoi.gr/~abeligia/torsion.pdf)

As explained there, in triangulated context, torsion pairs are in 1-1 correspondence with [[t-structures]]. One could
also study a relation between torsion theories on an abelian
category with tilting theory and $t$-structures on the derived category:

* Dieter Happel, Idun Reiten, Sverre O. Smalø, _Tilting in abelian categories and quasitilted algebras_, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88

* Riccardo Colpi, Luisa Fiorot, Francesco Mattiello, _On tilted Giraud subcategories_, [arxiv/1307.1987](http://arxiv.org/abs/1307.1987)

Other references in abelian context include

* Lia Vaš, _Differentiability of torsion theories_, ([pdf](http://www.usciences.edu/~lvas/homepage_sa_umdja/difftt_single.pdf))

For analogues in nonadditive contexts see

* [[Michael Barr]], _Non-abelian torsion theories_, Canad. J. Math. 25 (1973) 1224--1237

* Basil A. Rattray, _Torsion theories in non-additive categories_, Manuscripta Math. __12__ (1974), 285–305 [MR340360](http://www.ams.org/mathscinet-getitem?mr=340360) [doi](http://dx.doi.org/10.1007/BF01155518)

* [[Jiří Rosický ]], [[Walter Tholen]], _Factorization, fibration and torsion_, [arxiv/0801.0063](http://arxiv.org/abs/0801.0063), Journal of homotopy and Related Structures

* M. M. Clementino, D. Dikranjan, [[Walter Tholen]], _Torsion theories and radicals in normal categories_, J. of Algebra __305__ (2006) 92-129

* [[Dominique Bourn]], [[Marino Gran]], _Torsion theories in homological categories_, J. of Algebra __305__ (2006) 18–47 [MR2007k:18018](http://www.ams.org/mathscinet-getitem?mr=2262518) [doi](http://dx.doi.org/10.1016/j.jalgebra.2006.07.011)

Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

[[!redirects torsion theories]]

[[!redirects pretorsion theory]]
[[!redirects pretorsion theories]]

[[!redirects pre-torsion theory]]
[[!redirects pre-torsion theories]]


torsion-free module

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#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[module]] over a [[ring]] whose underlying [[abelian group]] has trivial [[torsion subgroup]] is called _torsion-free_.

## Definition

### Torsion-free $\mathbb{Z}$-modules

In classical mathematics, a **torsion-free $\mathbb{Z}$-module** or **torsion free [[abelian group]]** $M$ could be defined using a variant of the zero-divisor property characteristic of [[integral domains]]: for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r m = 0$, then $r = 0$ or $m = 0$, or the [[contrapositive]], if $r \neq 0$ and $m \neq 0$, then $r m \neq 0$.

There is also an equivalent definition: a **torsion-free $\mathbb{Z}$-module** $M$ or **torsion free abelian group** is such that right multiplication by $m$ is injective if $m \neq 0$ and left multiplication by $r$ is injective if $r \neq 0$, where "multiplication" refers to the $\mathbb{Z}$-action.

In [[constructive mathematics]], there are multiple inequivalent ways of defining a torsion-free $\mathbb{Z}$-module. One could define a torsion-free module as a module such that for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r m = 0$, then $r = 0$ and $m = 0$. The first definition is valid in all modules with [[decidable equality]], and could be defined using [[coherent logic]], but is not valid for $\mathbb{R}$-modules.

If the module has a [[tight apartness relation]], then one could define a torsion-free $\mathbb{Z}$-module as a module such that for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r \neq 0$ and $m \# 0$, then $r m \# 0$. This is valid in $\mathbb{R}$, but is no longer capable of being defined in coherent logic. Similarly, one could define a torsion-free $\mathbb{Z}$-module $M$ is such that right multiplication by $m$ is injective if $m \# 0$ and left multiplication by $r$ is injective if $r \neq 0$.

A torsion-free ring is a [[monoid object]] in torsion-free $\mathbb{Z}$-modules.

### Torsion-free $R$-modules

In classical mathematics, given a [[commutative ring]] $R$, a **torsion-free $R$-module** is a module $M$ such that for all $r$ in $Can(R)$, where $Can(R)$ is the [[multiplicative submonoid of cancellative elements]] in $R$ and $m$ in $M$, if $r m = 0$, then $r = 0$ or $m = 0$. Equivalently, the [[contrapositive]], if $m \neq 0$, then $r m \neq 0$. Some authors require $R$ to be an [[integral domain]], where $Can(R)$ is the monoid of nonzero elements in $R$.

In [[constructive mathematics]], given a [[ring]] $R$, there are multiple inequivalent ways of defining a torsion-free $R$-module. One could define a torsion-free module as a module such that for all $r$ in $Can(R)$ and $m$ in $M$, if $r m = 0$, then $m = 0$. The first definition is valid in all modules with [[decidable equality]], and could be defined using [[coherent logic]], but is not valid for $\mathbb{R}$-modules.

If $M$ has a [[tight apartness relations]], then one could define a torsion-free module as a module such that for all $r$ in $Can(R)$ and $m$ in $M$, if $m \# 0$, then $r m \# 0$. This is valid in $\mathbb{R}$-modules, but is no longer capable of being defined in coherent logic.

A torsion-free $R$-[[associative unital algebra|algebra]] is a [[monoid object]] in torsion-free $R$-modules.

## Properties

\begin{proposition}
Every [[divisible group|divisible]] torsion-free $\mathbb{Z}$-module is a [[rational vector space]].
\end{proposition}

\begin{proposition}
Every [[integral domain]] $R$ is a torsion-free $R$-module.
\end{proposition}

## Locality

Being torsion-free is **not** a local property in general.

\begin{proposition}
Let $R$ be a commutative and assume its [[total ring of fractions]] $\mathrm{Q}(R)$ be [[absolutely flat]].
Then for an $R$-module $M$ the following are equivalent :

1. $M$ is torsion-free over $R$ ;
1. $M_\mathfrak{p}$ is torsion-free over $R_\mathfrak{p}$ for every prime ideal $\mathfrak{p} \subset R$ ;
1. $M_\mathfrak{m}$ is torsion-free over $R_\mathfrak{m}$ for every maximal ideal $\mathfrak{m} \subset R$.

\end{proposition}

\begin{proof}

$1 \Rightarrow 2$. Suppose that $M$ is a torsion-free $R$-module and that $\overline{\alpha}\overline{x} = 0$ for a regular $\overline{\alpha} \in R_\mathfrak{p}$ and $\overline{x} \in M_\mathfrak{p}$.
Set $\mathfrak{a}_\mathfrak{p} = \{a ; a s = 0 \quad\text{ for some }\quad s \in R - \mathfrak{p}, a \in R\}$ and $M' = \{m ; s m = 0 \quad\text{ for some }\quad s \in R - \mathfrak{p}, m \in M\}$. Then we may assume that $\overline{\alpha} \in R/\mathfrak{a}_\mathfrak{p} \hookrightarrow R_\mathfrak{p}$ and $\overline{x} \in M/M' \hookrightarrow M_\mathfrak{p}$ by multiplying suitably elements of $R/\mathfrak{a}_\mathfrak{p} - \mathfrak{p}/\mathfrak{a}_\mathfrak{p}$ to $\overline{\alpha}, \overline{x}$.

Denote by $\alpha \in R$ a representative of $\overline{\alpha}$ and
$x \in M$ a representative of $\overline{x}$. Since $\mathrm{Q}(R)$ is absolutely flat, there exists a regular $r \in R$ and $\beta \in R$ such that
$\alpha ( \alpha \beta - r) = 0$.
Because $\overline{\alpha}$ is regular, $\overline{r} = \overline{\alpha} \overline{\beta}$. Then one has $\overline{r} \overline{x} = 0$ which means that there exists $s \in R- \mathfrak{p}$ such that $s r x = 0$ but since $M$ is torsion-free and $r$ is regular, one has $s x = 0$, so $\overline{x} = 0$.

$2 \Rightarrow 3$. Obvious.

$3 \Rightarrow 1$. Let $r \in R$ be a regular element and $x \in M$ such that $r x = 0$. For every maximal ideal $\mathfrak{m} \subset R$, the image $\overline{r} \in R_\mathfrak{m}$ is regular so that $\overline{r} \overline{x} = 0$ in $M_\mathfrak{m}$ means that $\overline{x} = 0$, for every $\mathfrak{m} \subset R$. As a consequence $x = 0$ in $M$.

\end{proof}

## Related concepts

* [[free module]] $\Rightarrow$ [[projective module]] $\Rightarrow$ [[flat module]] $\Rightarrow$ **torsion-free module**

* [[torsion module]]

* [[integral domain]]

* [[multiplicative submonoid of cancellative elements]]

## References

* Shizuo Endo, On semi-hereditary rings, J. Math. Soc. Japan Vol. 13, No. 2, 1961

See also

+ Wikipedia, _[Torsion-free module](https://en.wikipedia.org/wiki/Torsion-free_module)_

[[!redirects torsion-free modules]]
[[!redirects torsion free module]]
[[!redirects torsion free modules]]

[[!redirects torsionfree module]]
[[!redirects torsionfree modules]]

[[!redirects torsion free]]
[[!redirects torsion-free]]

[[!redirects torsion-free abelian group]]
[[!redirects torsion-free abelian groups]]
[[!redirects torsion free abelian group]]
[[!redirects torsion free abelian groups]]

[[!redirects torsion-free Z-module]]
[[!redirects torsion-free Z-modules]]
[[!redirects torsion free Z-module]]
[[!redirects torsion free Z-modules]]

[[!redirects torsion-free ring]]
[[!redirects torsion-free rings]]
[[!redirects torsion-free algebra]]
[[!redirects torsion-free algebras]]


torsor

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## Idea

A _torsor_ (in the [[category of sets]]) is, roughly speaking, a [[group]] that has forgotten its identity element; given any (non-empty) torsor with respect to a group $G$, we recover a group isomorphic to $G$ by making what is known as a _trivialisation_ of the torsor, which roughly corresponds to choosing an identity element. That we wish to keep track of the choice is precisely the reason for working with torsors.

Something analogous is present in the theory of [[Grothendieck fibration|fibrations]], where it can be important to make a choice of lifts ('cloven fibrations').

The notion of a torsor can be [[internalisation|internalised]] to any category with products, and more generally to any category in which the notion of an internal group can be made good sense of. We discuss this general notion below, after first discussing the notion in the category of sets.

## In the category of sets

### Two-sorted definition

\begin{defn} Let $G$ be a [[group]]. A _$G$-torsor_ is a [[set]] $T$ together with an [[action]] $a: G \times T \rightarrow T$ of $G$ on $T$ such that the [[shear map]] $a \times p_2: G \times T \rightarrow T \times T$ is an [[isomorphism]], where $p_2$ is the canonical [[projection]] map $G \times T \rightarrow T$. \end{defn}

If here $T$ is not required to be [[inhabited set|inhabited]] (possibly [[empty set|empty]]) one also speaks of a *[[pseudo-torsor]]*.

\begin{rmk} \label{RemarkTorsorIsomorphicToStructureGroup} If $T$ is non-empty, we shall prove [below](#TrivialisationInSets) that it follows from the definition that $T$ is isomorphic to the underlying set of $G$.  There are many such isomorphisms, and where torsors are used it is often important to choose/fix one. Such a choice is known as a _trivialisation_ of $T$. See [below](#TrivialisationInSets) for more details. \end{rmk}

\begin{rmk} As a consequence of Remark \ref{RemarkTorsorIsomorphicToStructureGroup}, a torsor with respect to some group can be thought of as a [[heap]]. \end{rmk}

\begin{rmk} Asking that the [[shear map]] $a \times p_2$ be an [[isomorphism]] is the same as to say that the action is both [[free action|free]] and [[transitive action|transitive]], hence [[regular action|regular]] if [[inhabited set|inhabited]]: free-ness corresponds to [[injectivity]] of $a \times p_2$, and transitivity corresponds to [[surjectivity]] of the [[shear map]]. \end{rmk}

\begin{example} An [[affine space]] of dimension $n$ over a [[field]] $k$ is a torsor for the additive group $k^n$: this acts by _translation_. \end{example}

\begin{example} A [[unit of measurement]] is (typically) an element in an $\mathbb{R}^\times$-torsor, for $\mathbb{R}^\times$ the multiplicative group of non-zero [[real number]]s: for $u$ any unit and $r \in \mathbb{R}$ any non-vanishing real number, also $r u$ is a unit. And for $u_1$ and $u_2$ two units, one is expressed in terms of the other by a unique $r \neq 0$ as $u_1 = r u_2$. For instance for units of [[mass]] we have the unit of [[kilogram]] and that of gram and there is a unique number, $r = 1000$ with
  $$
    kg = 1000 g.
  $$
\end{example}

\begin{example} \label{ExampleTorsorFromAGroup} Let $G$ be a [[group]]. The action of $G$ on itself equips the underlying set of $G$ with the structure of a $G$-torsor. \end{example}

\begin{rmk} We shall see in Remark \ref{RemarkGroupStructureFromTrivialisation} that all torsors actually arise as in Example \ref{ExampleTorsorFromAGroup}.  \end{rmk}

\begin{example} Given two isomorphic objects $X$ and $Y$ in any category $C$, all isomorphisms between $X$ and $Y$ form a torsor (both for $Aut(X)$ and for $Aut(Y)$, which are mutually isomorphic but not canonically). This is an insight used in (M. Kontsevich, _Operads and motives in deformation quantization_, Lett.Math.Phys.48:35-72 (1999) arXiv:[math/9904055](https://arxiv.org/abs/math/9904055) [doi](https://doi.org/10.1023/A:1007555725247)) explaining period matrices from the point of view of a coordinate ring of an affine torsor. \end{example}

### Single-sorted definition

It is possible to define torsors using a single-sorted [[algebraic theory]].
This is entirely analogous to how [[affine spaces]] can be defined
either as sets with a free and transitive action of a [[vector space]],
or, equivalently, as sets equipped with operations that take
arbitrary affine combinations with coefficients in a given ring.

More precisely, a __torsor__ (also
known as a [[heap]] when stated in a single-sorted form)
is a set $T$ equipped with a ternary operation
$$t\colon T^3 \to T$$
such that
$$t(a,a,b)=t(b,a,a)=b,\qquad t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).$$
A __homomorphism of torsors__ is a map of sets that preserves this operation.

The equivalence with the two-sorted definition is demonstrated as follows.

Given a $G$-torsor $T$, we send it to the set $T$ equipped with
the ternary operation $t(a,b,c)=g(a,b)c$, where
$g(a,b)$ is the unique element of $G$ such that $g(a,b)b=a$.

Given a torsor $(T,t)$, we send it to the pair $(LTrans(T),T)$,
where $LTrans(T)$ is a subgroup of the group of bijections on the set $T$
comprising precisely the bijections of the form $c\mapsto t(a,b,c)$
for some elements $a,b\in T$.
The group $G$ acts on $T$ by evaluation: $gt=g(t)$.

Mapping $(T,t)\mapsto (LTrans(T),T)\mapsto (T,t)$ gives back the same torsor
$(T,t)$ that we started from.

Mapping $(G,T)\mapsto (T,t)\mapsto (LTrans(T),t)$ produces
a torsor $(LTrans(T),t)$ that is naturally isomorphic to $(G,T)$
via the isomorphism
$$(G,T)\to(LTrans(T),T),\qquad g\mapsto (t\mapsto gt),\qquad t\mapsto t.$$

### Trivialisation {#TrivialisationInSets}

\begin{prpn} \label{PropositionTorsorIsomorphicAsSetToStructureGroup} Let $G$ be a [[group]], and let $\underline{T} = \left( T, a: G \times T \rightarrow T \right)$ be a $G$-torsor. If $T$ is non-empty, it is isomorphic to the underlying set of $G$. \end{prpn}

\begin{proof} Let $t$ be an [[element]] of $T$. The following diagram is [[cartesian square|cartesian]].

\begin{centre}
  \begin{tikzcd}
    G \ar[r, "id \times t"] \ar[d, swap, "a \circ (id \times t)"] & G \times T \ar[d, "a \times p_2"] \\
    T \ar[r, swap, "id \times t"] & T \times T
  \end{tikzcd}
\end{centre}

Since $\underline{T}$ is a $G$-torsor, we have that $a \times p_2$ is an isomorphism. The proposition thus follows immediately from the fact that pullbacks of isomorphisms are isomorphisms (as proven at [[pullback]]).

\end{proof}

\begin{defn} Let $G$ be a [[group]], and let $\underline{T} = \left( T, a: G \times T \rightarrow T \right)$ be a $G$-torsor. A _trivialisation_ of $\underline{T}$ is an [[isomorphism]] between $T$ and the underlying set of $G$. \end{defn}

\begin{rmk} By Proposition \ref{PropositionTorsorIsomorphicAsSetToStructureGroup}, if $T$ is non-empty, $T$ is always isomorphic to the underlying set of $G$. Thus the notion of a trivialisation of $\underline{T}$ is a question of a _choice_ of isomorphism between $T$ and the underlying set of $G$.

The proof of Proposition \ref{PropositionTorsorIsomorphicAsSetToStructureGroup} shows that any choice of element of $T$ gives rise to a trivialisation. Some of these may of course coincide. \end{rmk}

\begin{rmk} \label{RemarkGroupStructureFromTrivialisation} Let $\rho : G \rightarrow T$ be a trivialisation of a torsor $\underline{T}$. The map

$$p_{1} \circ \left( \rho^{-1} \times id \right): T \times T \rightarrow G \times T \rightarrow G$$

where $p_1$ is the canonical map, can be interpreted as a notion of _division_ $d : T \times T \to G$, 'dividing' one element of $T$ by another to obtain an element of $G$. If we further compose $d$ with $\rho$

$$\rho \circ d : T \times T \rightarrow G \rightarrow T,$$

we can think of the resulting map $D$ as a 'division structure' on $T$ for which $\rho$ behaves as an identity, namely such that $D(g, g) = \rho$ for all $g \in G$. In this way, $T$ acquires a group structure isomorphic to that of $G$.

Thus a trivialisation of a torsor equips with it a _choice_ of group structure amongst all of those isomorphic to $G$.

\end{rmk}

### Functoriality (change of structure group)

Torsors can be transported, or in other words pushed forward, along group homomorphisms, as we shall now show. (See also at *[G-space -- change of structure group](topological+G-space#ChangeOfGroupsAndFixedLoci)*).

\begin{prpn} Let $h : G_1 \rightarrow G_2$ be a [[homomorphism|group homomorphism]], and let $\underline{T} = \left( T, a \right)$ be a $G_1$-torsor. Observe that $\left(g_1, g_2, t \right) \mapsto \left( g_2 \cdot h(g_1)^{-1}, a(g_1, t) \right)$ defines an action $G_1 \times G_2 \times T \rightarrow G_2 \times T$ of $G_1$ on $G_2 \times T$. Then $\left(G_2 \times T \right) / G_1$, the quotient of $G_2 \times X$ with respect to this $G_1$-action, defines a $G_2$-torsor with respect to the action of $G_2$ induced by left multiplication, namely that given by $\left( g, \left[ (g', t') \right] \right) \mapsto \left[ \left(g g', t' \right) \right]$, where $\left[ p \right]$, for some $p \in G_2 \times T$, denotes the [[orbit]] of $p$ with respect to the action of $G_1$ on $G_2 \times T$. \end{prpn}

\begin{proof} We shall demonstrate that the map

$$i : \left( G_2 \times T \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right)$$

given by

$$\left( (g, t), \left[ (g', t') \right] \right) \mapsto \left( g\left(g'\right)^{-1}, \left[ (g', t') \right] \right)$$

induces a map

$$\overline{i} : \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right).$$

That is, it respects, in its left factor, the passage to the quotient by the action of $G_1$.

Suppose indeed that we have $(g, t) \in G_2 \times T$ and $\left( g_1, g'', t'' \right) \in G_1 \times G_2 \times T$ such that $(g, t) = \left( g'' \cdot h(g_1)^{-1}, a(g_1, t'') \right)$. We make the following observations.

1. $g''  h(g_1)^{-1} \left(g'\right) ^{-1} = g'' \left( g' h(g_1) \right)^{-1}$
1. $\left[ (g', t') \right] = \left[ \left( g' h\left(g_1^{-1}\right)^{-1}, a(g_1^{-1}, t') \right) \right] = \left[ \left( g' h(g_1), a\left(g_1^{-1}, t'\right) \right) \right]$

Putting these together, we obtain that

$$
\begin{aligned}
i\left( \left( g'', t'' \right), \left[ \left( g', t' \right) \right] \right) &= i\left( \left( g'', t'' \right), \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\
  &= \left( g'' h(g_1)^{-1} \left(g' \right)^{-1}, \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\
  &= \left( g \left(g' \right)^{-1}, \left[ \left( g', t' \right) \right] \right) \\
  &= i\left( g, \left[ \left( g', t' \right) \right] \right),
\end{aligned}
$$

as required.

We now observe that $\overline{i}$ defines an inverse to $a' \times p_{2}$, where $a'$ is the action of $G_2$ on $\left( G_2 \times T \right) / G_1$ induced by left multiplication, and $p_2$ is the projection map $G_2 \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow \left( \left( G_2 \times T \right) / G_1 \right)$. In one direction, for any $g, g' \in G_2$ and $t \in T$, we have that

$$\overline{i}\left( \left[ \left(g' g, t \right) \right], \left[ \left( g, t \right) \right] \right) = \left( g' g g^{-1}, \left[ \left( g, t \right) \right] \right) = \left( g', \left[ \left( g, t \right) \right] \right),$$

as required. In the other direction, suppose that we have $\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) \in \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right)$. Since the map $a \times p_2 : G_1 \times T \rightarrow T \times T$ is an isomorphism, there is a $g_1 \in G_1$ such that $t' = a(g_1, t)$, and we have that $\left[ \left(g, t \right) \right] = \left[ \left( g h(g_1)^{-1}, t' \right) \right]$. We then have that $a' \times p_2$ applied to

$$\overline{i}\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) = \overline{i}\left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) =  \left( g h(g_1)^{-1} \left(g'\right)^{-1}, \left[ \left( g', t' \right) \right] \right)$$

is

$$\left( \left[ \left(g h(g_1)^{-1} \left( g' \right)^{-1} g', t' \right) \right],  \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right),$$

as required.

\end{proof}

\begin{defn} Let $h : G_1 \rightarrow G_2$ be a [[homomorphism|group homomorphism]], and let $\underline{T} = \left( T, a \right)$ be a $G_1$-torsor. We refer to the $G_2$-torsor constructed from $\underline{T}$ using $h$ as in Proposition \ref{PropositionTorsorChangeOfStructureGroup} as the torsor obtained from $\underline{T}$ by _change of structure group_, and denote it $h_{*}\left(\underline{T}\right)$. \end{defn}

## In general

More generally, one may consider torsors over some base space $B$ (in other words, working in the [[topos]] of sheaves over $B$ instead of $Set$). In this case the term __$G$-torsor__ is often used more or less a synonym for the term $G$-[[principal bundle]], but torsors are generally understood in contexts much wider than the term "principal bundle" is usually taken to apply. And a principal bundle is strictly speaking a torsor that is required to be _locally trivial_ . Thus, while the terminology 'principal bundle' is usually used in the setting of [[topological spaces]] or [[smooth manifold]]s, the term _torsor_ is traditionally used in the more general contex of [[Grothendieck topology|Grothendieck topologies]] (faithfully flat and étale topology in particular), [[topos|topoi]] and for generalizations in various category-theoretic setups. While in the phrase '$G$-principal bundle' $G$ is usually a (topological) [[group]] or [[groupoid]], when we say '$G$-torsor', $G$ is usually a [[presheaf]] or [[sheaf]] of group(oid)s, or $G$ is a plain [[category]] (not necessarily even a groupoid).

A __$G$-torsor__, without any base space given, can also simply be an inhabited transitive free $G$-[[action|set]], which is the same as a principal $G$-bundle over the [[point]]. The notion may also be defined in any category with products: a torsor over a [[group object]] $G$ is a [[well-supported object]] $E$ together with a $G$-action $\alpha: G \times E \to E$ such that the arrow

$$\langle \pi_1, \alpha \rangle: G \times E \to E \times E$$

is an [[isomorphism]].

### Definition

Let $G$ be a [[group]] object in some [[category]] $C$, that in the following is assumed, for simplicity, to be a [[cartesian monoidal category]].
The [[object]]s of $C$ we sometimes call [[space]]s. Examples to keep in mind are $C = $ [[Set]] (in which case $G$ is an ordinary [[group]]) or [[Top]] (in which case it is a [[topological group]]) or [[Diff]] (in which case it is a [[Lie group]]).

\begin{defn} A left **$G$-torsor** is an [[inhabited object]] $P$ equipped with a $G$-[[action]], $\rho: G \times P \to P$ (subject to the usual laws for actions) such that the map

$$\langle \rho, \pi_2 \rangle: G \times P \to P \times P$$

is an [[isomorphism]]. \end{defn}

More generally, suppose $C$ is [[finitely complete category|finitely complete]], and let $B$ be an object. Then the [[slice category|slice]] $C/B$ is finitely complete, and the pullback functor $- \times B: C \to C/B$ preserves finite limits. Thus $\pi_2: G \times B  \to B$ acquires a group structure in $C/B$.

\begin{defn} A left **$G$-torsor over $B$** is a $G$-torsor in $C/B$. \end{defn}

Thus, if $B = 1$ is a point, a torsor over a point is the same as an ordinary torsor in $C$, but sometimes the additional "over a point" is convenient for the sake of emphasis.

We restate this definition equivalently in more nuts-and-bolts terms. The ambient category is $C$, as before.

\begin{defn} A left $G$-torsor over $B \in C$ is a [[bundle]] $P\stackrel{\pi}{\to} B$ over $B$ together with a left group [[action]]

$$
  \rho : G\times_B P \to P
$$

which in terms of [[generalized element]]s we write

$$
  (g,p)\to g.p
$$

such that the induced morphism of [[product]]s

$$
  \phi := (\rho, p_2) : G\times_B P \to P\times_B P
$$

which on elements acts as

$$
  (g,p)\to (g.p,p)
$$

is an [[isomorphism]].

\end{defn}

\begin{rmk} As we explain [below](#LocalTrivialization), a torsor is in some tautological sense **locally trivial**, but some care must be taken in interpreting this. One sense is that there is a cover $U$ of $1$ (so that $U \to 1$ is epi, i.e., $U$ is inhabited) such that the torsor, when pulled back to $U$, becomes trivial (i.e., isomorphic to $G$ as $G$-torsor). But this is a very general notion of "cover". A more restrictive sense frequently encountered in the literature is that "cover" means a coproduct of subterminal objects $U_i \hookrightarrow 1$ such that $U = \sum_i U_i$ is inhabited (e.g., an open cover of a space $B$ seen as the terminal object of the sheaf topos $Sh(B)$), and "torsor" would then refer to the local triviality condition for some such $U$. This is the more usual sense when referring to principal bundles as torsors. Or, "cover" could refer to a covering sieve in a [[Grothendieck topology]].

(The condition on the action can be translated to give transitivity etc. in the case of $B$ is a point (left as a standard exercise).)

\end{rmk}

### Examples

#### In topological spaces

Let $C = $ [[Top]], so that all objects are [[topological space]]s and groups $G$ are [[topological group]]s.

A topological $G$-[[principal bundle]] $\pi: P \to B$ is an example of a torsor over $B$ in $Top$. This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of $B$ (see the remarks [below](#LocalTrivialization)).

#### In sheaves

Let $C = Sh(S)$ be a [[category of sheaves]] over a [[site]] $S$.

The canonical example for a torsor in $C$ is the [[trivial torsor]] over a [[sheaf]] of groups, $G$.

#### Group extensions {#GroupExtensions}

Every [[group extension]] $A \to \hat G \to G$ canonically equips $\hat G$ with the structure of an $A$-torsor over $G$. See Group extensions as torsors for details

### Local trivialization {#LocalTrivialization}

In other categories $C$ besides $Set$, we cannot just "pick a point" of $P$ even if $P \to 1$ is an [[epimorphism]], so this argument cannot be carried out, and indeed trivializations may not exist. However, it is possible to construct a local trivialization of a torsor, following a general philosophy from [[topos theory]] that a statement is "locally true" in a category $C$ if it becomes true when reinterpreted in a slice after pulling back $C \to C/U$, where $U$ is inhabited. (This in some sense is the basis of [[Kripke-Joyal semantics]].)

In the present case, we may take $U = P$. Although we cannot "pick a point" of $P$ (= global section of $P \to 1$), we can pick a point of $P$ if we reinterpret it by pulling back to $C/P$. In other words, $\pi_2: P \times P \to 1 \times P \cong P$ does have a global section regarded as an arrow in $C/P$. In fact, there is a "generic point": the diagonal $\Delta: P \to P \times P$. Then, we may mimic the argument above, and consider the pullback diagram

$$\array{
G \times P & \to & G \times P \times P \\
\downarrow & & \downarrow \mathrlap{\langle \rho, \pi_2 \rangle \times id} \\
P \times P & \underset{id \times \Delta}{\to} & P \times P \times P
}$$

living in $C/P$. As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism $\langle \rho, \pi_2 \rangle: G \times P \to P \times P$ we started with!

Thus, a $G$-torsor in a category with products can be tautologically interpreted in terms of $G$-actions on objects $P$ which become trivialized upon pulling back to the slice $C/P$.

## Generalizations

* Instead of a torsor over a group, one can consider a torsor over a [[category]]. See [[torsor with structure category]].

* In [[noncommutative algebraic geometry]], faithfully flat [[Hopf-Galois extension]]s are considered a generalization of (affine) torsors in algebraic geometry.

## Related concepts

* [[homogeneous space]]
* [[principal bundle]] / [[associated bundle]]
* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]
* [[principal 3-bundle]] / [[bundle 2-gerbe]]
* [[principal ∞-bundle]] / [[associated ∞-bundle]]
* [[descent along a torsor]], [[Schneider's descent theorem]]
* [[Hopf-Galois extension]], [[quantum homogeneous space]],
* [[noncommutative principal bundle]], [[quantum heap]]

* [[physical unit]]

## References

Elementary exposition:

* {#Baez} [[John Baez]], _Torsors made easy_, ([web](http://math.ucr.edu/home/baez/torsors.html))

  > (discussion for [[discrete groups]])

Textbook accounts:

* {#Milne80} [[James Milne]], Prop. III.4.1 in: _[[Étale Cohomology]]_, Princeton Mathematical Series **33** (1980) [[jstor:j.ctt1bpmbk1](https://www.jstor.org/stable/j.ctt1bpmbk1), [ISBN:9780691082387](https://press.princeton.edu/books/hardcover/9780691082387/etale-cohomology-pms-33-volume-33)]

  > (discussion for [[algebraic groups]] in [[algebraic geometry]])

For more see the references at _[[principal bundle]]_ (which are torsors in the generality [[internalization|internal]] to [[slice categories]]).

A general [[topos theory|topos theoretic]] account is in

* [[Peter Johnstone]], Section B3.2 of: _[[Sketches of an Elephant]]_ .

See also the references at _[[Diaconescu's theorem]]_.

Discussion in [[homotopy type theory]]/[[univalent mathematics]]:

* {#SymmetryBook} [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]], Def. 4.8.1 in: _[[Symmetry]]_ [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]

* [[David Wärn]], *Eilenberg-MacLane spaces and stabilisation in homotopy type theory* [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685)]

Some further [[category theory|category theoretic]] articles discussing torsors:

* [[Tomasz Brzeziński]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arXiv:0912.0213](http://arxiv.org/abs/0912.0213)

* D. H. Van Osdol, _Principal homogeneous objects as representable functors_,
Cahiers Topologie Géom. Différentielle 18 (1977), no. 3, 271--289, [numdam](http://www.numdam.org/item?id=CTGDC_1977__18_3_271_0)

*  K. T. S. Mohapeloa, _A $2$-colimit characterization of internal categories of torsors_, J. Pure Appl. Algebra 71 (1991), no. 1, 75--91, [doi](http://dx.doi.org/10.1016/0022-4049%2891%2990041-Y)

* Thomas Booker, Ross Street, _Torsors, herds and flocks_ ([arXiv:0912.4551](http://arxiv.org/abs/0912.4551))

* J. Duskin, _Simplicial methods and the interpretation of 'triple' cohomology_, Memoirs AMS __3__, issue 2, n° 163, 1975.  MR393196

* A. Vistoli, _Grothendieck topologies, fibered categories and descent theory_, in: [[FGA explained]], 1--104, Math. Surveys Monogr., 123, AMS 2005, [math.AG/0412512](http://front.math.ucdavis.edu/0412.5512)

* [[Ieke Moerdijk]], _Introduction to the language of stacks and gerbes_, [math.AT/0212266](http://arxiv.org/abs/math/0212266).

Much further material is also in Giraud's book on nonabelian cohomology.

In a [[model theory|model theoretic]] context of [[definable set]]s, principal homogeneous spaces are studied in

* [[Anand Pillay]], _Remarks on Galois cohomology and definability_, The Journal of Symbolic Logic __62__:2 (1997) 487-492 [doi](https://doi.org/10.2307/2275542)

See also

* MathOverflow, _[torsors-for-monoids](http://mathoverflow.net/questions/25863/torsors-for-monoids/25886)_, [torsors-for-finite-group-schemes](https://mathoverflow.net/questions/46678/torsors-for-finite-group-schemes)

[[!redirects torsors]]

[[!redirects principal homogeneous space]]
[[!redirects principal homogeneous spaces]]


torsor with structure category

## Idea

If $C$ is a [[small category]] (or even a [[topological category]]), one can define a $C$-torsor (or _torsor with structure category $C$_) which generalizes the [[torsor]] (principal bundle) with structure group(oid). We present two variants in slightly different context.

## Moerdijk's definition

If $F$ is a [[sheaf]] over $X$, denote by $F_x$ its stalk over $x$ (cf. [[etale space]]).

A __$C$-torsor__ $E$ over a topological space $X$ is given by a functor $E : C\to Shv(X)$ such that

1. (**surjectivity**) every 'total stalk' $\cup_{c\in C_0} E(c)_x$, where $x\in X$, is nonempty;

1. (**transitivity**) for any two germs 'in the same total stalk', $\alpha\in E(c)_x$, $\alpha'\in E(c')_x$, there is a span $c\stackrel{u}\leftarrow b\stackrel{u'}\to c'$ and $\xi\in E(b)_x$ such that $E(u)(\xi)=\alpha$ and $E(u')(\xi)=\alpha'$;

1. (**freeness**) for a parallel pair $u_1,u_2: c\to c'$ of morphisms in $C$, $E(u_1)(\alpha)=E(u_2)(\alpha)$ for some $\alpha\in E(c)_x$ implies there is a morphism $w:b\to c$ and $\zeta\in E(b)_x$ such that $u_1\circ w = u_2\circ w$ and $E(w)(\zeta)=\alpha$.

This definition is from the monograph

* [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_, Springer Lec. Notes Math. __1616__ (1995)

where it is shown that _the [[classifying space]] of a category $C$ classifies $C$-torsors_.

_[[David Roberts]]: This definition should be able to be restated in terms of [[flat functors]]_

## Street's definition

Suppose now $C$ is a finitely complete category with a [[calculus of left fractions]] whose morphisms are called *covers*.

Let $A$ be an [[internal category]] in $C$. An __$A$-torsor trivialized by a cover__ $e : V\to U$ is a [[discrete fibration]] $A\stackrel{p}\leftarrow E\stackrel{q}\to U$ for which there exist a morphism $a : V\to A$ and a commutative diagram
\begin{xymatrix}
A\downarrow{a}\ar[r]^q \ar@/_2pc/[dd]_{p} \ar@<-.5ex>[d]
& V\ar[d]^e \\
E\ar[r]_q\ar[d]_p& U\\
A&
\end{xymatrix}
in which the square is a pullback. Street says __$A$-torsor at $U$__ for an $A$-torsor trivialized by some cover $e : V\to U$.

* [[Ross Street]], _Combinatorial aspects of descent theory_, [pdf](http://www.math.mq.edu.au/~street/DescFlds.pdf) (page 25 in the file)

[[!redirects torsors with structure category]]


Torsten Asselmeyer > history

< [[Torsten Asselmeyer]]


Torsten Asselmeyer-Maluga

* [website](https://www.researchgate.net/profile/Torsten_Asselmeyer-Maluga)

## related $n$Lab entries

* [[exotic smooth structure]]

category: people

[[!redirects Torsten Asselmeyer-Maluga]]
[[!redirects Torsten Asselmeyer]]
[[!redirects Torsten Maluga]]


Torsten Ekedahl

* [Wikipedia entry](http://sv.wikipedia.org/wiki/Torsten_Ekedahl)

category: people


Torsten Wedhorn

* [institute page](https://www.mathematik.tu-darmstadt.de/fb/personal/details/torsten_wedhorn.en.jsp)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Torsten_Wedhorn)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=28577)

## Selected writings

On [[manifolds]], [[sheaves]] and ([[nonabelian cohomology|non-abelian]]) [[Čech cohomology]]:

* [[Torsten Wedhorn]], *Manifolds, Sheaves, and Cohomology*, Springer (2016) [[doi:10.1007/978-3-658-10633-1](https://doi.org/10.1007/978-3-658-10633-1)]

category: people


torus

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#Contents#
* table of contents
{:toc}

## Definition

### In topology and differential geometry

The **torus** is the [[manifold]] (a [[smooth manifold]], hence in particular also a [[topological manifold]]) obtained as the [[quotient]]

$$
  T \coloneqq \mathbb{R}^2 / \mathbb{Z}^2
$$

of the [[Cartesian space|Cartesian plane]], regarded as an [[abelian group]], by the subgroup of pairs of [[integers]].

As a [[topological space]] this is the [[quotient topological space]] obtained from the [[square]] by identifying opposite sides:

> graphics grabbed from [Lawson 03](#Lawson03)

More generally, for $n \in \mathbb{N}$ any [[natural number]], the _$n$-torus_ is

$$
  T \coloneqq \mathbb{R}^n / \mathbb{Z}^n
  \,.
$$

For $n = 1$ this is the [[circle]].

In this fashion each torus canonically carries the structure of an [[abelian group]], in fact of an abelian [[Lie group]]. Notice that regarded as a [[group]] the torus carries a [[pointed object|base point]] (the neutral element).

### In algebraic geometry
 {#InAlgebraicGeometry}

According to [[SGA|SGA3]], for $X$ a base [[scheme]] then a 1-dimensional torus (in the sense of tori-as-groups) over it is a [[group scheme]] over $X$ which becomes [[isomorphism|isomorphic]] to the [[multiplicative group]] over $X$ after a [[faithfully flat morphism|faithfully flat]] [[group extension]].

In ([Lawson-Naumann 12, def. A.1](#LawsonNaumann12))  this is called "a form of" the multiplicative group over $X$.

By ([Lawson-Naumann 12, prop. A.4](#LawsonNaumann12)) the [[moduli stack of 1-dimensional tori]] $\mathcal{M}_{1dtori}$  in this sense is equivalent to the [[delooping]] of the [[group of order two]]:

$$
  \mathcal{M}_{1dtor}
  \simeq
  \mathbf{B}\mathbb{Z}/2\mathbb{Z}
  \,.
$$

The single nontrival [[automorphism]] of any 1-dimensional toris here is that induced by the canonical automorphism of the multiplicative group

$$
  Aut(\mathbb{G}_m) \simeq \mathbb{Z}/2\mathbb{Z}
$$

which is the [[inversion involution]] (given by sending any element to its [[inverse]] element).

### As a homotopy type
 {#AsAHomotopyType}

As a [[homotopy type]] the torus is the product of two copies of the [circle](https://ncatlab.org/nlab/show/circle#as_a_homotopy_type).

In [[homotopy type theory]] the torus can be formalized as the [[higher inductive type]] generated by a point `base`, two paths, $p$ and $q$, from `base` to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown ([Sojakova15](#Sojakova15)) that this type is equivalent to the product of two circles. For a treatment in [[cubical type theory]], see ([Licata-Brunierie](#Licata-Brunierie)).

$\,$

{#StableHomotopyType} The *[[stable homotopy type]]* of the torus is the [[wedge sum]] of 2 circles and one [[2-sphere]] (eg. [Freed & Moore 13, Thm. 11.8](#FreedMoore13)):

(graphics from [[schreiber:Topological Quantum Computation in TED-K|SS 22]])

## Properties

* The [[character group]] of a torus $T^n$ is isomorphic to its [[fundamental group]], which is $\mathbb{Z}^n$. See also at _[[Pontryagin duality]]_ and at _[[moduli space of flat connections]]_.

\begin{proposition}
One has
$$
TC(T^n)
=n+1
$$
for the [[topological complexity]].
\end{proposition}

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

## Related concepts

* [[solid torus]]

* [[complex torus]]

* [[elliptic curve]]

* [[toric variety]]

* [[noncommutative torus]]

## References

* {#Lawson03} Terry Lawson, _Topology: A Geometric Approach_, Oxford University Press (2003) ([pdf](http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf))

On the [[ordinary cohomology]] of topological tori:

* [[Qiaochu Yuan]], *[The cohomology of the $n$-torus](https://qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus/)* (2013)

The [[moduli stack]] of 1-dimensional tori in [[algebraic geometry]] is discussed (as the cusp point inside the [[moduli stack of elliptic curves]]) in

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], Appendix A of _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_ ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

Discussion of tori the [[homotopy type]] of the torus in [[homotopy type theory]] is in

* {#Sojakova15} Kristina Sojakova, _The equivalence of the torus and the product of two circles in homotopy type theory_, ([arXiv:1510.03918](https://arxiv.org/abs/1510.03918))

* {#Licata-Brunierie} [[Dan Licata]], [[Guillaume Brunerie]], _A cubical approach to synthetic homotopy theory_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/lb15cubicalsynth/lb15cubicalsynth.pdf))

On [[2-group]]-[[higher central extensions|extensions]] of torus groups:

* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))

On the [[stable homotopy type]] of the torus:

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], Thm. 11.8  of: _Twisted equivariant matter_, Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

On [[topological complexity]] of [[tori]] (or more general [[product topological space|products]] of [[spheres]])

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

[[!redirects tori]]

[[!redirects 2-torus]]
[[!redirects 2-tori]]

[[!redirects 3-torus]]
[[!redirects 3-tori]]

[[!redirects 4-torus]]
[[!redirects 4-tori]]

[[!redirects n-torus]]
[[!redirects n-tori]]


torus knot

[[!redirects torus knots]]

+-- {: .rightHandSide}
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###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea##

A torus knot is a [[knot]] that can be drawn on the surface of a [[torus]].

## Definition

 A [[knot]] $K$ is said to be a _torus knot_ if it can be [[embedding|embedded]] in the surface of a [[torus]], that is, we have the map $K : S^1\to \mathbb{R}^3$ factors through the embedding of some torus $T_1\cong S^1\times S^1$ into $\mathbb{R}^3$;

$$S^1\stackrel{K}{\to}T_1\stackrel{embed}{\to} \mathbb{R}^3$$

## Examples

Examples include

* the [[trefoil knot]]:

[[!include trefoil knot - SVG]]

* the cinquefoil knot:

[[!include cinquefoil knot - SVG]]

category: knot theory
[[!redirects torus knots]]


toset > history

< [[toset]]
[[!redirects toset -- history]]


Toshihide Maskawa

* [Wikipedia entry](https://en.wikipedia.org/wiki/Toshihide_Maskawa)

## Selected writings

Early discussion of [[discretized light-cone quantization]]:

* {#MaskawaYamakawi76} [[Toshihide Maskawa]], [[Koichi Yamakawi]], *The Problem of $P^+ = 0$ Mode in the Null-Plane Field Theory and Dirac's Method of Quantization*, Progress of Theoretical Physics **56** 1 (1976) 270–283 [[doi:10.1143/PTP.56.270](https://doi.org/10.1143/PTP.56.270)]

category: people


Toshitaka Kajino

* [Wikipedia entry](https://en.wikipedia.org/wiki/Toshitaka_Kajino)

* [research group page](https://sci.nao.ac.jp/MEMBER/kajino/e/)

## Selected writings

On [[primordial nucleosynthesis]]:

* [[Toshitaka Kajino]], *Cosmological Nucleosynthesis in the Big-Bang and Supernovae*,  Journal of Nuclear Science and Technology **39** sup2 ([doi:10.1080/00223131.2002.10875156](https://doi.org/10.1080/00223131.2002.10875156))

* Grant J. Mathews, Motohiko Kusakabe, [[Toshitaka Kajino]], *Introduction to Big Bang Nucleosynthesis and Modern Cosmology*,  International Journal of Modern Physics EVol. 26, No. 08, 1741001 (2017)  ([arXiv:1706.03138](https://arxiv.org/abs/1706.03138), [doi:10.1142/S0218301317410014](https://doi.org/10.1142/S0218301317410014))

category: people


Toshitake Kohno

* [webpage](https://www.ms.u-tokyo.ac.jp/~kohno/)

## Selected writings

On the [[Drinfeld-Kohno theorem]]:

* [[Toshitake Kohno]], _Monodromy representations of braid groups and Yang-Baxter equations_, Annales de l'institut Fourier __37__, no. 4 (1987), p. 139-160 ([numdam:AIF_1987__37_4_139_0](http://www.numdam.org/item?id=AIF_1987__37_4_139_0))

On [[topology|topological]] aspects of [[2d conformal field theory]] (e.g. [[Knizhnik-Zamolodchikov equation]], and [[braid representations]] of [[conformal blocks]]):

* [[Toshitake Kohno]], _Conformal field theory and topology_, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$[AMS:mmono-210](https://bookstore.ams.org/mmono-210)$]$

On [[braid groups]], [[Yang-Baxter equations]] and [[infinitesimal braid relations]]:

* {#Kohno87} [[Toshitake Kohno]], _Monodromy representations of braid groups and Yang-Baxter equations_, Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 139-160 ([doi:10.5802/aif.1114](https://doi.org/10.5802/aif.1114))

* {#Kohno88} [[Toshitake Kohno]], _Linear representations of braid groups and classical  Yang-Baxter equations_, in: [[Joan S. Birman]], Anatoly Libgober (eds.) _Braids_ Cont. Math. 78 (1988), 339-363 ([doi:10.1090/conm/078](http://dx.doi.org/10.1090/conm/078))

On [[weight systems]]/[[Vassiliev invariants]] as the [[real cohomology]] of [[based loop spaces]] of [[configuration spaces of points]]:

* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_,
In: Yoshiaki Maeda, Hideki Omori and [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

On [[arithmetic topology]]:

* {#KohnoMorishita06} [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), _Primes and Knots_, Contemporary Mathematics, AMS 2006 ([conm:416](http://www.ams.org/bookstore-getitem/item=CONM-416))

On the [[Knizhnik-Zamolodchikov connection]] on [[configuration spaces of points]] and [[conformal blocks]]:

* {#Kohno14} [[Toshitake Kohno]], _Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks_,  Acta Math Vietnam 39, 575–598 (2014)  ([doI;10.1007/s40306-014-0088-6](doi:10.1007/s40306-014-0088-6), [pdf](https://www.ms.u-tokyo.ac.jp/~kohno/papers/kohno_config.pdf))

On [[braid group representations]]:

* [[Toshitake Kohno]], *Introduction to representation theory of braid groups*, Peking 2018 ([pdf](https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf), [[KohnoBraidRepresentations.pdf:file]])

On [[2-representations]] of [[braid cobordism]] via [[higher holonomy]]:

* {#Kohno16} [[Toshitake Kohno]], *Higher holonomy of formal homology connections and braid cobordisms*,  Journal of Knot Theory and Its Ramifications **25** 12  (2016) 1642007 [[doi:10.1142/S0218216516420074](https://doi.org/10.1142/S0218216516420074), [pdf](https://www.ms.u-tokyo.ac.jp/~kohno/papers/hhc.pdf)]

category: people


Toshiya Kawai

* [webpage](http://www.kurims.kyoto-u.ac.jp/en/list/KAWAI,%20Toshiya.html)

## Selected writings

On [[elliptic genera]] as [[partition functions]] of [[2d SCFTs]]:

* [[Toshiya Kawai]], [[Yasuhiko Yamada]], Sung-Kil Yang, _Elliptic Genera and $N=2$ Superconformal Field Theory_, Nucl. Phys. B414:191-212, 1994
([arXiv:hep-th/9306096](http://arxiv.org/abs/hep-th/9306096), doi:10.1016/0550-3213(94)90428-6)

category: people


Tosio Kato

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tosio_Kato)

## Selected writing

Early generalization of the [[adiabatic theorem]]:

* [[Tosio Kato]], *On the Adiabatic Theorem of Quantum Mechanics*, J. Phys. Soc. Jpn. **5** (1950) 435-439 ([doi:10.1143/JPSJ.5.435](https://journals.jps.jp/doi/10.1143/JPSJ.5.435))

category: people