Connected Covering Space of a Bouquet of 3 Circles

Question

Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. What is $\pi_{1}(\hat{X})$ (It is a free group of what rank?).

Any hints?

Answer 1

If $p\colon \hat{X}\to X$ is a degree $10$ covering map, then every point in $X$ has a set of preimages in $\hat{X}$ which has cardinality $10$. Given that a covering space of a graph is always a graph, and the preimage of a vertex is a vertex, and an edge is an edge, you should be able to count the edges and vertices of the graph $\hat{X}$ and so find its euler characteristic. The Euler characteristic then gives you the rank of the homology of $\hat{X}$ (remember $\hat{X}$ is connected) and so, because the fundamental group of a graph is free, you also get the rank of $\pi_1(\hat{X})$.