Covers for Free Groups

due by Monday, Nov 4, 2024

Let $X$ be the wedge of two circles at the point $x_0$. Recall that the fundamental group $\pi_1(X,x_0)$ is the free group on two generators $\alpha$ and $\beta$.

1. Show that $X$ has a connected Galois 3-fold cover $p: Y \to X$.
2. Show that for connected Galois 3-fold covers $p: Y \to X$, the space $Y$ is unique up to homeomorphism but the covering $p: Y \to X$ is not unique up to covering isomorphism (i.e., homeomorphisms commuting with covering maps).
3. Describe the Galois covering of $X$ corresponding to the abelianisation homomorphism $\langle \alpha,\beta\rangle\to \Z^2$.
4. Construct an infinite connected covering of $X$ that is not Galois.