The common layman’s definition of topology generally involves rubber sheets or clay, with the idea that things are “the same” if they can be stretched, squeezed, or bent from one shape into the other. But the notions of topological equivalence we’ve been using up until now don’t really match up to this picture. Homeomorphism — or diffeomorphism, for differentiable manifolds — is about having continuous maps in either direction, but there’s nothing at all to correspond to the whole stretching and squeezing idea.
Instead, we have homotopy. But instead of saying that spaces are homotopic, we say that two maps are homotopic if the one can be “stretched and squeezed” into the other. And since this stretching and squeezing is a process to take place over time, we will view it sort of like a movie.
We say that a continuous function is a continuous homotopy from to if and for all . For any time , the map is a continuous map from to , which is sort of like a “frame” in the movie that takes us from to . As time passes over the interval, we highlight one frame at a time to watch the one function transform into the other.
To flip this around, imagine starting with a process of stretching and squeezing to turn one shape into another. In this case, when we say “shape” we really mean a subspace or submanifold of some outside space we occupy, like the three-dimensional space that contains our idiomatic doughnuts and coffee mugs. The maps in this case are the inclusions of the subspaces into the larger space.
Anyway, next imagine carrying out this process, but with a camera recording it at each step. Then cut out all the frames from the movie and stack them up. We see in each layer of this flipbook how the shape at that time is included into the larger space . That is, we have a homotopy.
Now, for an example: we say that a space is “contractible” if its inclusion into itself is homotopic to a map of the whole space to a single point within the space. As a particular example, the unit ball is contractible. Explicitly, we define a homotopy latex H(p,t)=(1-t)p$, which is certainly smooth; we can check that and , so at one end we have the identity map of into itself, while at the other we have the constant map sending all of to the single point at the origin.
We should be careful to point out that homotopy only requires that the function be continuous, and not invertible in any sense. In particular, there’s no guarantee that the frame for some fixed is a homeomorphism from onto its image. If it turns out that each frame is a homeomorphism of onto its image, then we say that is an “isotopy”.