Below the video you will find accompanying notes and some pre-class questions.

- Previous video:
**7.01 Covering spaces**. - Next video:
**7.03 Path-lifting: uniqueness**. - Index of all lectures.

*(6.20)* We need to explain what it means for \(a\) to go around
a path that ends up at \(b\) (this will be justified by the
*path-lifting lemma*). We also need to explain why the monodromy
around a loop only depends on the homotopy class of that loop (which
will be justified by the *homotopy-lifting lemma*).

*(12.44)* We will construct \(\gamma\) by induction on \(k\).

\(k=0\): We need to have \(\gamma(0)=y\) in the end. Since \(p\) is a covering map, we have a local inverse \(q_0\colon U_0\to Y\) to \(p\) such that \(q_0(x)=y\) and \(p\circ q_0=id|_{U_0}\), so if we define \(\gamma_0=q_0\circ\delta_0\). Now \(\gamma_0\) is a lift of \(\delta_0\).

*(14.52)* Suppose we have constructed
\(\gamma_0,\ldots,\gamma_{k-1}\) and we wish to construct
\(\gamma_k\colon[t_k,t_{k+1}]\to Y\). In order for \(\gamma\) to be
continuous, we need \(\gamma_{k}(t_k)=\gamma_{k-1}(t_k)\). There
exists \(q_k\colon U_k\to Y\) such that
\(q_k(\delta(t_k))=\gamma_{k-1}(t_k)\), so define
\(\gamma_k=q_k\circ\delta_k\). This extends \(\gamma\) as a lift of
\(\delta\) continuously to the interval \([t_k,t_{k+1}]\).

*(16.35)* Define \(\gamma(t)=\gamma_k(t)\) if
\(t\in[t_k,t_{k+1}]\). The result is continuous because a
piecewise-defined function which is continuous on pieces and agrees
on overlaps is continuous. It is a lift because
\(p\circ\gamma=p\circ q_k\circ\delta_k=\delta_k\) on
\([t_k,t_{k+1}]\) (as \(p\circ q_k=id_{U_k}\)).

This gives existence of lifts. Uniqueness will be proved in the next video.

- Suppose that \(p\colon Y\to X\) is a covering map. Is it true that
for all \(x_0,x_1\in X\), there is a bijection \(p^{-1}(x_0)\to
p^{-1}(x_1)\)? Give a proof or a counterexample. What if \(x_0\) and
\(x_1\) are connected by a path?

- Previous video:
**7.01 Covering spaces**. - Next video:
**7.03 Path-lifting: uniqueness**. - Index of all lectures.