(9.06) The strategy to construct the lifted homotopy \(\tilde{H}\) (such that \(\tilde{H}(s,t)=\tilde{\gamma}_s(t)\)) is to pick local inverses \(q_{ij}\colon U_{ij}\to Y\) for \(p\) and set \(\tilde{H}|_{R_{ij}}=q_{ij}\circ H|_{R_{ij}}\). Certainly this defines a lift of \(H\) because \(p\circ\tilde{H}=p\circ q_{ij}\circ H=H\), however, we need to pick \(q_{ij}\) carefully to ensure that the map \(\tilde{H}\) constructed in this way is continuous. We need to ensure that \(\tilde{H}|_{R_{ij}}\) and \(\tilde{H}|_{R_{kl}}\) agree along the overlap \(R_{ij}\cap R_{kl}\).

(11.13) For simplicity, let's imagine that we only have a two-by-two grid of squares. We will start by constructing \(q_{1j}\). To begin with, we know that \(\tilde{H}(0,t)=\tilde{\gamma}_0(t)\) for a given lift \(\tilde{\gamma}_0\). This tells us which \(q_{ij}\) to use on the rectangles \(R_{11},R_{12}\): namely, \(q_{11}\colon U_{11}\to Y\) is the unique local inverse with \(q_{11}(\gamma_0(0))=\tilde{\gamma}_0(0)\); \(q_{12}\colon U_{12}\to Y\) is the unique local inverse with \(q_{12}(\gamma_0(t_1))=\tilde{\gamma}_0(t_1)\) where \(t_1\) is the value of the \(t\)-coordinate at the top of the rectangle \(R_{11}\).

(14.15) We need to check that the results \(\tilde{H}_{11}:=q_{11}\circ H\) and \(\tilde{H}_{12}:=q_{12}\circ H\) agree along the overlap \(R_{11}\cap R_{12}\) (which is an edge comprising points of the form \((s,t_1)\) with \(s\in[0,s_1]\)). By construction, we have \(\tilde{H}_{11}(0,t_1)=\tilde{H}_{12}(0,t_1)=\tilde{\gamma}_0(t_1)\) and the restrictions \(\tilde{H}_{1j}(s,t_1)\) to the edge \(R_{11}\cap R_{12}\) define paths; these paths both lift \(H(s,t_1)\) and have the same initial condition \(\tilde{H}_{11}(0,t_1)=\tilde{H}_{12}(0,t_1)\). Therefore, by uniqueness of path-lifting, we have \(\tilde{H}_{11}(s,t_1)=\tilde{H}_{12}(s,t_1)\) for all \(s\in[0,s_1]\).

(18.02) We can do the same trick starting with \(\tilde{\gamma}_{s_1}\) and extend our homotopy over \(R_{2j}\) and, by induction, over the whole square.

(19.06) We need to check that \(\tilde{H}\) is a homotopy rel endpoints, i.e. \(\tilde{H}(s,0)\) and \(\tilde{H}(s,1)\) should be constant. But \(\tilde{H}(s,0)\) lifts \(H(s,0)\), which is constant. The constant lift is also a lift, so by uniqueness of path-lifting, it must agree with \(\tilde{H}(s,0)\), which must therefore be constant. Similarly for \(\tilde{H}(s,1)\).