Index

This illustrates the process of lifting a homotopy $h\colon [0,1]^2\to Y$ over a covering map $p\colon X\to Y$. We divide $[0,1]^2$ into small rectangles $Q$ such that $h(Q)$ is contained in an open set that is trivially covered by $p$. We then work through all these rectangles in a series of stripes, lifting $h$ over each rectangle in turn. For each rectangle $Q$, there are an infinite number of possible lifts of $h|_Q$. At the first stage, we can choose any one of those lifts. At every subsequenct stage, there is only one possible lift that matches up with what we have already done.