Seifert-Van Kampen theorem - Revision history

InfernalAtom683 at 10:54, 14 May 2026

2026-05-14T10:54:34Z

← Older revision		Revision as of 18:54, 14 May 2026
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	Then the canonical morphism		Then the canonical morphism
	<math display="block">\~~colim_~~{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>		<math display="block">\operatorname{colim}_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>
	is an isomorphism of groupoids.		is an isomorphism of groupoids.

InfernalAtom683 at 10:52, 14 May 2026

2026-05-14T10:52:25Z

← Older revision		Revision as of 18:52, 14 May 2026
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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">\colim_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math>
			is an isomorphism of groupoids.

InfernalAtom683: Created page with "== Statement == Let $X$ be a topological space, and $U, V\subset X$ be open sets such that $X = U\cup V$ , and $U$ , $V$ and $U\cap V$ are path-connected. Take a basepoint $x_0\in U\cap V$ with inclusion maps: $i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,$ then the following d..."

2026-05-14T07:37:20Z

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>