Suppose you are walking your dog Spot. You put the leash on Spot and take him to his favorite tree to do his business, but he sees a squirrel on the tree and takes off after it. The squirrel is really tired of being chased, and decides to teach Spot a lesson. So, instead of climbing back up the tree, he runs in a counterclockwise direction around the trunk with Spot not far behind. As Spot follows him around the tree, the leash gets wound around the tree k times (assuming you stay in place while he is running). At this point Spot gives up and sits down near your feet to bark sullenly at the squirrel. We say that the winding number of the leash is k. No matter how you try to move the leash, unless you cut it, it will remain wound around the tree k times. That is, its winding number is a homotopy invariant. This prosaic example generalizes to higher dimensions and has interesting mathematical and physical applications.

We start with the stack of records theorem, so-called because it reveals that all smooth maps from a compact manifold to another manifold of the same dimension look like a smooth covering by a stack of records. (See Figure 9.1.)