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Seifert-Van Kampen theorem

InfernalAtom683 — Thu, 14 May 2026 10:54:34 GMT

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	== Statement ==		== Statement ==
	Let <math>X</math> be a topological space, and <math>U~~, V~~\~~subset X~~</math> be open ~~sets such that~~ <math>X ~~= U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V~~</math> ~~are path-connected~~. ~~Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps:~~		Let <math>X</math> be a topological space and let <math>\mathcal{U}=\{U_i\}_{i\in I}</math> be an open cover of <math>X</math>.

	<math display="block">i\~~colon U~~\~~cap V~~\~~hookrightarrow U,~~\~~quad j~~\~~colon U~~\~~cap V~~\~~hookrightarrow V,~~\~~quad k~~\~~colon U~~\~~hookrightarrow X~~,\~~quad l~~\~~colon V~~\~~hookrightarrow X,~~</math>		Let <math>\mathcal{F}</math> be the set of all finite non-empty intersections of members of <math>\mathcal{U}</math>:
			<math display="block">\mathcal{F}=\left\{\left.\bigcap_{i\in J}U_i\right\|\emptyset\neq J\subseteq I, \|J\|<\infty\right\}</math>.

	~~then the following diagram is a pushout:~~		Regard <math>\mathcal{F}</math> as a category whose objects are the elements of <math>\mathcal{F}</math> and in which there is a unique morphism <math>V\to W</math> whenever <math>V\subseteq W</math>.
	~~<div style="display: flex; justify-content: center; gap: 40px;">~~
	<~~kroki lang="tikz"~~>
	\~~documentclass[tikz]~~{~~standalone~~}
	\~~usepackage~~{~~quiver~~}
	~~\begin{document}~~
	~~\begin{tikzpicture}[baseline=(current bounding box.center)]~~
	~~\node[scale=1.5] {~~
	~~\begin{tikzcd}~~
	~~{\pi_1(U\cap~~ V~~,x_0)} & {~~\~~pi_1(U,x_0)} \\~~
	~~{\pi_1(~~V~~,x_0)} & {~~\~~pi_1(X,x_0)}~~
	~~\arrow["{i_*}", from=1-1, to=1-2]~~
	~~\arrow["{j_*}"', from=1-1, to=2-1]~~
	~~\arrow["{k_*}", from=1-2, to=2-2]~~
	~~\arrow["{l_*}"', from=2-1, to=2-2]~~
	~~\end{tikzcd}~~
	};
	~~\end{tikzpicture}~~

	\~~end~~{~~document~~}		Let <math>\Pi_1\colon \mathcal{F}\to \mathsf{Gpd}</math> be the functor that sends each <math>V\in \mathcal{F}</math> to its fundamental groupoid <math>\Pi_1(V)</math> and each inclusion <math>V\hookrightarrow W</math> to the induced morphism of groupoids.
	</~~kroki~~>
	</~~div~~>		Then the canonical morphism
			<math display="block">\operatorname{colim}_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>
			is an isomorphism of groupoids.

Seifert-Van Kampen theorem

InfernalAtom683 — Thu, 14 May 2026 07:37:20 GMT

Created page with "== Statement == Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps: <math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math> then the following d..."

New page

== Statement ==
Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps:

<math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math>

then the following diagram is a pushout:
<div style="display: flex; justify-content: center; gap: 40px;">
<kroki lang="tikz">
\documentclass[tikz]{standalone}
\usepackage{quiver}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[scale=1.5] {
\begin{tikzcd}
{\pi_1(U\cap V,x_0)} & {\pi_1(U,x_0)} \\
{\pi_1(V,x_0)} & {\pi_1(X,x_0)}
\arrow["{i_*}", from=1-1, to=1-2]
\arrow["{j_*}"', from=1-1, to=2-1]
\arrow["{k_*}", from=1-2, to=2-2]
\arrow["{l_*}"', from=2-1, to=2-2]
\end{tikzcd}
};
\end{tikzpicture}

\end{document}
</kroki>
</div>

Commutator

InfernalAtom683 — Wed, 29 Apr 2026 13:57:44 GMT

Created page with "A '''commutator''' is an algebraic expression that measures the failure of two elements to commute. It occurs throughout abstract algebra, particularly in group theory, ring theory, and linear algebra. If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of i..."

New page

A '''commutator''' is an algebraic expression that measures the failure of two elements to [[commute]]. It occurs throughout [[abstract algebra]], particularly in [[group theory]], [[ring theory]], and [[linear algebra]].

If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of invariants such as the [[derived subgroup]], the [[lower central series]], and the [[Lie bracket]].

== Conventions ==

In [[group theory]], two conventions are commonly used:

* '''Left convention''':
<math display="block">
[a,b]=a^{-1}b^{-1}ab
</math>

* '''Right convention''':
<math display="block">
[a,b]=aba^{-1}b^{-1}
</math>

These differ by inversion:
<math display="block">
aba^{-1}b^{-1}=[b,a]^{-1}.
</math>

Unless otherwise stated, this article uses the left convention.

In [[ring theory]] and [[linear algebra]], the standard convention is
<math display="block">
[A,B]=AB-BA.
</math>

== Definition ==

=== Groups ===

Let <math>G</math> be a [[group]], and let <math>a,b\in G</math>. Their commutator is
<math display="block">
[a,b]=a^{-1}b^{-1}ab.
</math>

One has
<math display="block">
[a,b]=e \iff ab=ba,
</math>
where <math>e</math> is the [[identity element]].

=== Rings ===

Let <math>R</math> be a [[ring]], and let <math>A,B\in R</math>. Their commutator is
<math display="block">
[A,B]=AB-BA.
</math>

This vanishes precisely when <math>A</math> and <math>B</math> commute.

=== Linear transformations ===

For [[linear transformation]]s <math>A,B:V\to V</math> on a [[vector space]] <math>V</math>, equivalently for [[square matrix|square matrices]], the commutator is
<math display="block">
[A,B]=AB-BA.
</math>

This is the ring commutator in the [[endomorphism ring]] <math>\operatorname{End}(V)</math>.

== Properties ==

=== Group identities ===

For all <math>a,b,c\in G</math>,
<math display="block">
[a,b]^{-1}=[b,a],
</math>

<math display="block">
[a,a]=e,
</math>

<math display="block">
[a,b]^c=[a^c,b^c],
</math>
where
<math display="block">
x^y=y^{-1}xy.
</math>

Also,
<math display="block">
[ab,c]=[a,c]^b[b,c],
</math>

<math display="block">
[a,bc]=[a,c][a,b]^c.
</math>

=== Ring identities ===

For all <math>A,B,C</math>,
<math display="block">
[A,B]=-[B,A],
</math>

<math display="block">
[A+B,C]=[A,C]+[B,C],
</math>

<math display="block">
[A,BC]=[A,B]C+B[A,C].
</math>

The commutator also satisfies the [[Jacobi identity]]:
<math display="block">
[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.
</math>

For matrices,
<math display="block">
\operatorname{tr}([A,B])=0.
</math>

== Derived subgroup ==

The subgroup generated by all group commutators is the '''[[derived subgroup]]''' or '''commutator subgroup''' of <math>G</math>:
<math display="block">
G'=[G,G]=\langle [a,b]\mid a,b\in G\rangle.
</math>

It satisfies:

* <math>G'</math> is a [[normal subgroup]]
* <math>G/G'</math> is [[abelian group|abelian]]
* <math>G</math> is abelian if and only if <math>G'=\{e\}</math>

The quotient
<math display="block">
G^{\mathrm{ab}}=G/G'
</math>
is called the [[abelianization]] of <math>G</math>.

== Lie algebras ==

In a [[Lie algebra]], the bracket operation often arises from the commutator in an [[associative algebra]]:
<math display="block">
[x,y]=xy-yx.
</math>

Thus every associative algebra determines a Lie algebra by using the commutator as its bracket.

For the matrix algebra <math>M_n(F)</math>, this gives the Lie algebra
<math display="block">
\mathfrak{gl}(n,F).
</math>

== Examples ==

=== Symmetric group ===

In the [[symmetric group]] <math>S_3</math>, let
<math display="block">
a=(1\;2),\qquad b=(2\;3).
</math>

Then
<math display="block">
[a,b]=(1\;3\;2),
</math>
so <math>a</math> and <math>b</math> do not commute.

=== Matrices ===

Let
<math display="block">
A=
\begin{pmatrix}
0&1\\
0&0
\end{pmatrix},
\qquad
B=
\begin{pmatrix}
0&0\\
1&0
\end{pmatrix}.
</math>

Then
<math display="block">
[A,B]=
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}.
</math>

Hence <math>A</math> and <math>B</math> do not commute.

== See also ==

* [[Derived subgroup]]
* [[Lower central series]]
* [[Lie algebra]]
* [[Jacobi identity]]
* [[Abelianization]]
* [[Noncommutative ring]]