I have been reading Hatcher´s Algebraic Topology, and he wants to prove that if we have a covering space $(E,p)$, with $p(e)=x_0$ then $p_*(\pi_1(E,e)$) consists of the homotopy classes of loops in $X$ starting at $x_0$ such that their lifts are loops in $E$ starting at $e$.To do this in the proof he says that a loop representing an element of the image $p_*$ is homotopic to a loop having such a lift , and intuitively it seems right but i cant seem to see why this is true theoretically , so any help is apreciated, Thanks.
Induced image of the fundamental group of a covering space
algebraic-topologyproof-explanation
Related Solutions
For the first question : $[h_0]=p_*[\gamma]$ doesn't tell you that there is $\delta$ such that $h_0 = p\circ \delta$, it tells you that $h_0$ and $p\circ \gamma$ are homotopic. A priori there is no reason to believe that $h_0$ should itself be the projection of a loop.
The sentence that follows explains why a posteriori it is actually the case that $h_0$, too, is the projection of a loop.
For the second question, $\tilde{U}$ is not arbitrary but it is picked among a basis of neighbourhoods, so it is enough. That it is enough is a point-set topology basic fact, that we may for instance state as
Let $f:X\to Y$ be a function between two topological spaces; let $x\in X$ and let $\mathcal{V}$ be a basis of neighbourhoods of $f(x)$. Then, if for each $U\in \mathcal{V}$ there is $V\subset X$ a neighbourhood of $x$ such that $f(V)\subset U$, $f$ is continuous at $x$.
For your last question, Hatcher is not being very precise here, in the sense that $\widetilde{\gamma}$ in this last section is not the same as the one at the beginning which specifically denoted a lift with starting point $\widetilde{x_0}$; here it denotes some lift, with the starting point "obvious from context", as is often the case.
In particular, a claim that is true is that if $\widetilde{\alpha},\widetilde{\beta}$ are lifts of $\alpha,\beta$ that can be put after another, then so can $\alpha, \beta$, and $\widetilde{\alpha}\widetilde\beta$ is a lift of $\alpha\beta$. It is then your job here to figure out which specific lifts are meant here, but that shouldn't be a problem.
You have to show that for all $\gamma \in \pi_1(B,b_0)$ and all $\chi \in H_0$ one has $\gamma ^{-1} \chi \gamma \in H_0$ (not $\gamma \chi = \chi \gamma$ as you write).
With your notation this means $$[g]^{-1} * [p \circ f] * [g] \in H_0. $$
Let us lift the loop $g^{-1} * (p \circ f) * g$ in $B$ to $E$. Take the unique lift $\tilde g$ of $g$ such that $\tilde g(0) = e_0$. This is a path in $E$, but not necessarily a loop in $E$. Let $e_1 = \tilde g(1)$. The path $$F = \tilde g^{-1} * f * \tilde g$$ is a loop based at $e_1$.
Let $h : E \to E$ be a deck transformation such that $h(e_1) = e_0$. Then $h \circ F$ is a loop based at $e_0$, i.e. $[h \circ F] \in \pi_1(E,e_0)$. We get $$p_*([h \circ F]) = [p \circ h \circ F] = [p \circ F] = [(p \circ \tilde g^{-1}) * (p \circ f) * (p \circ \tilde g)] = [g^{-1} * (p \circ f) * g] \\= [g]^{-1} * [p \circ f] * [g]$$ which means that $$ [g]^{-1} * [p \circ f] * [g] \in H_0 .$$
Best Answer
This is basically just by definition.
Let $\gamma$ be a loop on $x_0$ that represents $p_*([\vartheta])\,\in\pi_1(X,x_0) $ with $[\vartheta] \in\pi_1(E,e)$.
This means $[\gamma] =p_*([\vartheta]) =[p\circ\vartheta]$, that is, $\gamma$ is homotopic to $p\circ\vartheta$, which lifted to $e$ obviously gives $\vartheta$.