Covering map, singular homology

Question

Let $X,Y$ be topological spaces and $q:Y\rightarrow X$ a covering map with $|q^{⁻1}({x} )|=n$ for all $x \in X$. I want to show that the induced map $$H_k(q,\mathbb{Q}):H_k(Y,\mathbb{Q})\rightarrow H_k(X,\mathbb{Q})$$ is surjective for all $k$. I am not sure where to start and would be grateful for any idea. Thank you.

Answer 1

A start: Let $\Delta$ be a closed $1$-simplex in $X$, in other words $\Delta:[0,1]\to X$ with $\Delta(0)=\Delta(1)$, and let $\overline{\Delta}_i$, $i=1,\ldots, n$ denote the various lifts of $\Delta$ to $Y$. $\overline{\Delta}_i$ is not necessarily closed, but we know that $q(\overline{\Delta}(0))=q(\overline{\Delta}(1))$. Show that $\sum_i\overline{\Delta}_i$ is closed, and so $$n[\Delta]\in\mathrm{Im}(H_1(q,\mathbb{Z})).$$It follows that $$[\Delta]\in\mathrm{Im}(H_1(q,\mathbb{Q})).$$