Here is my problem :
I have convex subsets $X_1, \dots, X_n$ of $\mathbb R^m$ such that $X_i \cap X_j \cap X_k$ is never empty. I have to show that $\bigcup\limits_{i=1}^nX_i$ is simply connected.
This is the chapter about Van Kampen's theorem so I thought we have to use Van Kampen's theorem. Almost all the hypothesis are true.
But I just read that the intersection $\bigcap\limits_{i=1}^nX_i$ is not always non-empty, and a inside this intersection is needed for applying Van Kampen's.
Did I make some mistakes? Maybe there is a smarter way to apply Van Kampen's Theorem here?
Thanks.
Best Answer
Do it by induction on $n$. The case $n = 2$ is exactly Van Kampen's theorem: $X_1$ and $X_2$ are path-connected, and so is their intersection (it's the intersection of two convex sets, hence convex, and it's nonempty, hence path-connected). Thus $$\pi_1(X_1 \cup X_2) \cong \pi_1(X_1) *_{\pi_1(X_1 \cap X_2)} \pi_1(X_2) = 0.$$
Now for the induction step, suppose that you know the result is true for a given $n \ge 2$ and let $X_1, \dots, X_{n+1}$ be convex sets such that $X_i \cap X_j \cap X_k$ is connected for all $i,j,k$. By the induction hypothesis, $Y = X_1 \cup \dots \cup X_n$ is simply connected. It remains to show that $Y \cap X_{n+1}$ is path-connected, and you can apply Van Kampen's theorem again to conclude. (1)
So let's show $Y \cap X_{n+1}$ is path-connected. Clearly $$Y \cap X_{n+1} = (X_1 \cap X_{n+1}) \cup \dots \cup (X_n \cap X_{n+1}).$$
Now suppose $x,y$ belong to $Y$, we are looking for a path from $x$ to $y$. Let $$x \in X_i \cap X_{n+1}, \qquad y \in X_j \cap X_{n+1}.$$ By the hypothesis on the $X_\cdot$, the intersection $X_i \cap X_j \cap X_{n+1}$ is non-empty; choose $z$ inside it. Since $X_i \cap X_{n+1}$ is path-connected (it's the nonempty intersection of two convex sets), there is a path $\gamma$ from $x$ to $z$. Similarly, there is a path $\gamma'$ from $z$ to $y$. Concatenating these two path gives a path $\alpha = \gamma \cdot \gamma'$ from $x$ to $y$. Thus $Y \cap X_{n+1}$ is path-connected and we can conclude (cf. (1)).
Remark. You forgot the assumption (which is written in Hatcher's book) that the sets $X_i$ have to be open. This is crucial to apply van Kampen's theorem, in general it's false.