Let $X$ be a path-connected, locally path-connected, and semi-locally simply connected space whose fundamental group is isomorphic to $\langle a, b \mid a^5, b^3, aba^{-1}b^{-1} \rangle$. Determine the number of isomorphism classes of coverings of X with a path-connected total space.
I'm confused for the following reason. I'm pretty sure this exercise is about applying a theorem of the lecture which basically states that there is a bijection between pointwise covering spaces $p: \bar X \rightarrow X$ with $\bar X$ path connected and locally path connected and the subgroups of $\pi_1(X,x)$. Please note the bold part: path connected and locally path connected. In the exercise we are only given path connected.
Is this maybe a typo / mistake?
However, assuming it is a typo, this is my solution.
From the lecture we know that $\langle a, b \mid a^5, b^3, aba^{-1}b^{-1} \rangle \cong \mathbb Z _5 \times \mathbb Z_3$. Clearly the subgroups are of this form: "subgroup x subgroup". So all subgroups are: $\mathbb Z_5 \times \mathbb Z_3,\,\, \mathbb Z_5 \times 0 \cong \mathbb Z_5, \,\, 0 \times \mathbb Z_3 \cong \mathbb Z_3, \,\, 0 \times 0 \cong 0$. So the final is answer is: There are 4 isomorphism classes.
So can someone clarify this? Is this most likely a typo or are we supposed to do something different than what I did?
I think there are two parts to this problem. What you did so far is compute the number of .. with a path-connected and locally path-connected total space.
The second part would be showing that if $X$ is locally path-connected and $\tilde{X}$ is a covering of $X$ then $\tilde{X}$ is also locally path-connected.