# Proposition 1.33 in Hatcher's Book

Recall the following proposition in Hatcher's Algebraic Topology:

Proposition 1.31 The map $$p_* : \pi_1(\widetilde{X},\widetilde{x}_0) \to \pi_1(X,x_0)$$ induced by a covering space $$p : (\widetilde{X}, \widetilde{x}_0) \to (X,x_0)$$ is injective. The image subgroup $$p_*(\pi_1(\widetilde{X},\widetilde{x}_0))$$ in $$\pi_1(X,x_0)$$ consists of the homotopy classes of loops in $$X$$ based at $$x_0$$ whose lifts to $$\widetilde{X}$$ starting at $$\widetilde{x}_0$$ are loops.

I am currently working through the proof of following proposition:

Proposition 1.33 Suppose a given covering space $$p : (\widetilde{X},\widetilde{x}_0) \to (X,x_0)$$ and a map $$f : (Y,y_0) \to (X,x_0)$$ with $$Y$$ path-connected and locally path-connected. Then a lift $$\widetilde{f} : (Y,y_0) \to (\widetilde{X},\widetilde{x}_0)$$ of $$f$$ exists iff $$f_*(\pi_1(Y,y_0)) \subseteq p_*(\pi_1(\widetilde{X},\widetilde{x}_0))$$

Here is the proof Hatcher gives in his book:

In the first paragraph, why does Hatcher say that $$h_0$$ is path homotopic to a loop at $$x_0$$ which lifts to a loop in $$\widetilde{X}$$ at $$\widetilde{x}_0$$? This seems unnecessary. If $$[h_0] \in p_*(\pi_1(\widetilde{X},\widetilde{x}_0)$$, then $$h_0$$ itself lifts to a loop in $$\widetilde{X}$$ at $$\widetilde{x}_0$$. It seems unnecessary to talk of a homotopy between $$h_0$$ and $$h_1$$.

My next two questions concern the second paragraph. I'm having trouble following the proof that $$\widetilde{f}$$ is continuous. It appears that he does this by proving continuity at an arbitrary point $$y \in Y$$. Usually these sorts of proof start out as follows:

Let $$y \in Y$$ be arbitrary, and let $$\mathcal{O}_1$$ be an open set about $$\widetilde{f}(y)$$. We need to find an open set $$\mathcal{O}_2$$ about $$y$$ such that $$\widetilde{f}(\mathcal{O}_2) \subseteq \mathcal{O}_1$$

As you may see, he shows that we can find an open set $$V$$ about $$y$$ such that $$\widetilde{f}(V) \subseteq \widetilde{U}$$. But $$\widetilde{U}$$ was not arbitrarily selected at the beginning. How do we know we can fit it in an arbitrarily chosen neighborhood about $$y$$?

My next question is, why is $$\widetilde{(f \gamma )} \cdot \widetilde{(f \eta)}$$ a lift of $$(f \gamma ) \cdot(f \eta)$$? Is he implicitly using the following: if $$\alpha$$ is a path in $$X$$ from $$x_0$$ to $$x_1$$, $$\beta$$ a path from $$x_1$$ to $$x_2$$, then $$\widetilde{\alpha \cdot \beta} = \widetilde{\alpha} \cdot \widetilde{\beta}$$? I tried proving this, but I don't think it is true since the end point of $$\widetilde{\alpha}$$ may be different from the starting point of $$\widetilde{\beta}$$.

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.

• But $[h_0] \in p_*(\pi_1(\widetilde{X},\widetilde{x}_0))$, so why doesn't proposition 1.31 tell us directly that $h_0$ lifts to a loop? – user193319 Mar 4 at 19:07
• Proposition 1.31 tells us that $[h_0]$ is the class of a loop that lifts to a loop, the sentence that comes afterwards tells you why it follows that all elements of that class actually lift to a loop – Max Mar 4 at 19:13