Here is the question I am trying to solve:
Let $G$ be a topological group and $p: X \to G$ be a covering. Let $e \in G$ be the identity element of $G$ and choose $f \in p^{-1}(e).$
$(a)$ Show that $X$ has a unique topological group structure with identity element $f,$ such that $p$ is a (continuous) group homomorphism.
$(b)$ Show that $X$ is abelian if $G$ is.
Some thoughts and hints :
1- I know that I should produce a multiplication map $\mu^{'}$ (Look at the diagram below) and this mutiplication map must be associative and unital (**I do not know why I should show that if later on I also should prove that there exists an inverse, could someone clarify this to me please? **)
I also do not know how to produce this map but I am guessing that we should use the lifting theorem but still I do not know how, could anyone help me in this please?
2- I also know that we want to produce an inverse map $\chi$ on $X$ according to the following diagram
More precisely, since the inverse on a group means the following diagram:
Where we wanna prove that the left arrow of the diagram factors through $e$ i.e., we have $G \to e \to G$ i.e., we have the following diagram:
more precisely, we have the following diagram:
i.e., the equation that should be satisfied is $e = g^{-1} g$ which means that there is a group homomorphism that sends all of $G$ to the trivial group and then includes the trivial group into $G.$
Still I do not know how to show the existence of this inverse.
Note That:
In general, I know I will be using the lifting theorem, in the first case, I wanna show that $X \times X \to G \times G \to G$ lifts over the covering map i.e., we wanna show that the hypotheses of the lifting theorem are satisfied and similarly for the second case.
For (b) I do not know what exactly should I do but I will post it in a separate question
Edit:
1-Is the map $X \times X \to G \times G$ is just $p \times p$?
2- In the link provided by @OsamaGhani (this one Covering space of a topological group is itself a topological group) in the comments, why the author tended to take the identity as the second component in the multiplication defined on the covering space?
3- Also, in the path lifting property, we usually are lifting a path, where is the path we are lifting here? are we using the path lifting property or the homotopy lifting property? I am confused (my gut feeling we will be using homotopy lifting property). No,again, I think there is no given homotopy here to lift
4- I think also there is a typo in the link mentioned above. In this term $\mu(\widetilde{e},x)=x$, I think $\mu$ can not take the first argument an element of $\widetilde{G}.$ Am I correct?
5- I am not accustomed in general to defining a lifting may in two components, how I should choose these two components in this case?
6- How we will keep the endpoint of the first path, the beginning of the second path in our case here?
7- will this link If a covering space of a topological space X has a topological group structure, when we transfer this structure on X? help in defining the multiplication?
EDIT 2
I think if someone can show me why the multiplication map $X \times X \to X$ exists that will be great, I believe we wanna show that $f_* (\pi_1 (Y, y_0)) \subset p_* (\pi_1 (\tilde X, \tilde x_0))$ but I do not know how. Any help will be greatly appreciated!