Theorem. Let $X$ be a topological space, let $U,V$ be connected open subsets of $X$ such that $W=U\cap V$ is connected and let $x$ be a point of $U\cap V$. Then, the fundamental group $\pi_1(X,x)$ is the amalgamated product $\pi_1(U,x) *_{\pi_1(W,x)} \pi_1(V,x)$, that is, the quotient of the free product of the groups $\pi_1(U,x)$ and $\pi_1(V,x)$ by the normal subgroup generated by the elements of the form $i_U(c)i_V(c)^{-1}$, where $i_U$ and $i_V$ are the natural injections from the groups $\pi_1(U,x)$ and $\pi_1(V,x)$ respectively in their free product.
The classical proof of this result in topology books relies decomposes a loop at $x$ as a product of loops at $x$ which are either contained in $U$, or in $V$.
(In fact, van Kampen proves a theorem which is quite different at first sight.)
It has been long recognized that there is a completely different approach is possible, from which all loops are totally absent. For this proof we make a supplementary assumption, namely that our spaces are « semi-locally simply connected » : Any point $a$ of $X$ has a neighborhood $A$ such that the morphism $\pi_1(A,a)\to \pi_1(X,a)$ is trivial.
When $X$ is a connected slsc space together with a point $x$, the theory of the fundamental group is related to the theory of coverings,under the form of an equivalence of categories between coverings of $X$ and sets with an action of $\pi_1(X,x)$. The equivalence of categories is explicit; it maps a covering $p\colon Y\to X$ to the fiber $p^{-1}(x)$ on which $\pi_1(X,x)$ acts naturally via the path-lifting property of coverings (given $y\in p^{-1}(x)$, any loop $c$ at $x$ lifts uniquely to a path with origin $y$, the endpoint of which is $c\cdot y$).
Given this equivalence, one can prove the van Kampen Theorem very easily in two steps. First of all, one observes that it is equivalent to have a covering of $X$ as to have a covering of $U$ and a covering of $V$ together with an identification of these coverings above $W$. A covering of $U$ corresponds to a set $A$ with an action of $\pi_1(U,x)$; a covering of $V$ corresponds to a set $B$ with an action of $\pi_1(V,x)$; an identification of these coverings above $W$ corresponds to a bijection from $A$ to $B$ which is compatible with the two actions of $\pi_1(W,x)$ acting on $A$ via the morphism $\pi_1(W,x)\to \pi_1(U,x)$ and on $B$ via the morphism $\pi_1(W,x)\to \pi_1(V,x)$. It is harmless to assume that $A=B$ and that the bijection from $A$ to $B$ is the identity. Now, a covering of $X$ corresponds to a set $A$ together with two actions of the groups $\pi_1(U,x)$ and $\pi_1(V,x)$ such that the two actions of $\pi_1(W,x)$ are equal. This is precisely the same as a set $A$ together with an action of the amalgamated product $\pi_1(U,x)*_{\pi_1(W,x)}\pi_1(V,x)$. CQFD.
The same proof applies and allows to describe the fundamental group of an union of spaces in more general contexts. For example, let us use the same method to understand the fundamental group of the circle $\mathbf S_1$. It is clear that a circle is nothing but an interval $ [0,1]$ of which the two endpoints are glued, and a covering of the circle corresponds to a covering $p\colon X\to [0,1]$ of the interval $[0,1]$ together with an identification of the fibers at $0$ and~$1$. Now, a covering of the interval can be written as a product $A\times [0,1]$ (where $A$ is the fiber at $0$, say). Consequently, identifying the fibers at $0$ and $1$ means giving yourself a bijection of $A$ to $A$. In other words, a covering of the circle « is » a set $A$ together with a permutation of $A$, in other words, a set $A$ with an action of the additive group $\mathbf Z$. Moreover, the obvious loop is the image by the glueing map $[0,1]\to\mathbf S_1$ of the obvious path joining $0$ to $1$ so that this loop is the generator of $\pi_1(\mathbf S_1,p(0))$.
Observe that the latter example is not an instance of the van Kampen Theorem. One could get it via a groupoid-version of van Kampen.
All of this is more or less explained in the following texts:
The classical proof of this result in topology books relies decomposes a loop at $x$ as a product of loops at $x$ which are either contained in $U$, or in $V$.
(In fact, van Kampen proves a theorem which is quite different at first sight.)
It has been long recognized that there is a completely different approach is possible, from which all loops are totally absent. For this proof we make a supplementary assumption, namely that our spaces are « semi-locally simply connected » : Any point $a$ of $X$ has a neighborhood $A$ such that the morphism $\pi_1(A,a)\to \pi_1(X,a)$ is trivial.
When $X$ is a connected slsc space together with a point $x$, the theory of the fundamental group is related to the theory of coverings,under the form of an equivalence of categories between coverings of $X$ and sets with an action of $\pi_1(X,x)$. The equivalence of categories is explicit; it maps a covering $p\colon Y\to X$ to the fiber $p^{-1}(x)$ on which $\pi_1(X,x)$ acts naturally via the path-lifting property of coverings (given $y\in p^{-1}(x)$, any loop $c$ at $x$ lifts uniquely to a path with origin $y$, the endpoint of which is $c\cdot y$).
Given this equivalence, one can prove the van Kampen Theorem very easily in two steps. First of all, one observes that it is equivalent to have a covering of $X$ as to have a covering of $U$ and a covering of $V$ together with an identification of these coverings above $W$. A covering of $U$ corresponds to a set $A$ with an action of $\pi_1(U,x)$; a covering of $V$ corresponds to a set $B$ with an action of $\pi_1(V,x)$; an identification of these coverings above $W$ corresponds to a bijection from $A$ to $B$ which is compatible with the two actions of $\pi_1(W,x)$ acting on $A$ via the morphism $\pi_1(W,x)\to \pi_1(U,x)$ and on $B$ via the morphism $\pi_1(W,x)\to \pi_1(V,x)$. It is harmless to assume that $A=B$ and that the bijection from $A$ to $B$ is the identity. Now, a covering of $X$ corresponds to a set $A$ together with two actions of the groups $\pi_1(U,x)$ and $\pi_1(V,x)$ such that the two actions of $\pi_1(W,x)$ are equal. This is precisely the same as a set $A$ together with an action of the amalgamated product $\pi_1(U,x)*_{\pi_1(W,x)}\pi_1(V,x)$. CQFD.
The same proof applies and allows to describe the fundamental group of an union of spaces in more general contexts. For example, let us use the same method to understand the fundamental group of the circle $\mathbf S_1$. It is clear that a circle is nothing but an interval $ [0,1]$ of which the two endpoints are glued, and a covering of the circle corresponds to a covering $p\colon X\to [0,1]$ of the interval $[0,1]$ together with an identification of the fibers at $0$ and~$1$. Now, a covering of the interval can be written as a product $A\times [0,1]$ (where $A$ is the fiber at $0$, say). Consequently, identifying the fibers at $0$ and $1$ means giving yourself a bijection of $A$ to $A$. In other words, a covering of the circle « is » a set $A$ together with a permutation of $A$, in other words, a set $A$ with an action of the additive group $\mathbf Z$. Moreover, the obvious loop is the image by the glueing map $[0,1]\to\mathbf S_1$ of the obvious path joining $0$ to $1$ so that this loop is the generator of $\pi_1(\mathbf S_1,p(0))$.
Observe that the latter example is not an instance of the van Kampen Theorem. One could get it via a groupoid-version of van Kampen.
All of this is more or less explained in the following texts:
- Adrien and Régine Douady, Algèbre et théories galoisiennes, Cassini 2005.
- Ronald Brown, Topology and Groupoids, Booksurge Publishing, 2006.
- I remember having read an old Bourbaki Tribu from the 50sby Cartan, Eilenberg and/or Weil explaining this, but I cannot find it anymore on the archive. :-(
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