Theorem 1

Let X be a compact contractible topological space and let $x$ be a point of X. The Poincaré group $\pi_1(X, x)$ has a finite presentation.

The arcwise connected component of $x$ in X is open, closed and contractible; this makes it possible to suppose that the space X is connected. Since the space X is contractible, the X-space $E =\lambda_x(X)$, endowed with the terminal mapping, is a nonempty arcwise simply connected covering (IV, p. 342, th. 1). The group G = Aut$_X(E)$ is isomorphic to $\pi_1(X, x)$; it therefore suffices to prove that the group G has a finite presentation.

Since X is compact, every point $x$ of X possesses a compact neighbourhood $K_x$ over which the covering E is trivializable (TG, I, p. 65, corollary). Since X is locally connected, every $x\in X$ possesses an open connected neighbourhood $W_x$, contained in $K_x$. Let F be a finite subset of X such that the $W_x$, for $x\in F$, cover X. Let $n$ be the cardinal of F.

We shall prove by induction that, for every integer $k$ such that 1 $\leqslant k\leqslant n$, there exist a subset A of cardinal $k$ contained in F and, for $x\in A$, a section $s_x$ of $p$ over $K_x$, such that the union of the $s_x(W_x)$, for $x\in A$, is a connected subset of E. The assertion is true for $k= 1$. Suppose it true for an integer $k$ such that $1\leqslant k < n$ and let us prove that it is true for $k+ 1$. Let A be a subset of F of cardinal $k$ and, for every $x\in A$, let $s_x$ be a section of E over $K_x$, such that $\bigcup_{x\in A}s_x(W_x)$ is connected. The open sets $\bigcup_{x\in A}W_x$ and $\bigcup_{x\in F-A}W_x$ are nonempty and cover X; their intersection is therefore nonempty, since X is connected. There thus exist $x\in A,y\in F-A$ and $z\in W_x\cap W_y$. Put $A'= A\cup \{y\}$ and choose a section $s_y$ of E over $K_y$ such that $s_y(z) =s_x(z)$. The connected open sets $\bigcup_{p\in A}s_p(W_p)$ and $s_y(W_y)$ are nonempty and have a common point; their union is therefore connected. This proves the assertion for $k+ 1$. By induction, it is therefore true for every integer $k\in \{1, . . . , n\}$.

Applying the preceding to $k=n$, there then exists, for every $x\in F$, a section $s_x$ of E over $K_x$, in such a way that $U =\bigcup_{x\in F}s_x(W_x)$ is a connected open subset of E. We have $p(U) = X$. Since the group G operates transitively in each fibre of the covering E, we have GU = E.

The closure of U is contained in $\bigcup_{x\in F}s_x(K_x)$ and therefore is compact. The operation of G in E is proper (I, p. 96, cor. 1), hence the set of pairs $(g, x)\in G\times E$ such that $x\in \overline{U}$ and $gx\in \overline{U}$ is a compact subset of $G\times E$ (TG, I, p. 77, prop. 6). It follows that the set of $g\in G$ such that $\overline{U}\cap g\overline{U}=\not\emptyset$ is compact. Since it is discrete, it is finite. A fortiori, the set of $g\in G$ such that $U\cap gU=\not\emptyset$ is finite. The theorem thus follows from prop. 10 of I, p. 136.