(Weak) Homotopy Equivalences

Summary

Previously: Fibrations and Cofibrations. In topology, we say that two shapes are the same if there is a homeomorphism– an invertible continuous map– between them. Continuity means that nothing is broken and nothing is glued together. This is how we can turn a coffe cup into a torus. A homeomorphism, however, won’t let us shrink a torus to a circle. So if we are only interested in how many holes the shapes have, we have to relax our notion of equivalence. Let’s go back to the definition of homeomorphism. It is defined as a pair of continuous functions between two topological spaces, and , such that their compositions are identity maps, and , respectively. If we want to allow for extreme shrinkage, as with a torus shrinking down to a circle, we have to relax the identity conditions. A homotopy equivalence is a pair of continuous functions, but their compositions don’t have to be equal to identities. It’s enough that they are homotopic to identities. In other words, it’s possible to create an animation that transforms one to another. Take the example of a line and a point. The point is a trivial topological space where only the whole space (here, the singleton ) and the empty set are open. and are obviously not homeomorphic, but they don’t have any holes, so they should be homotopy equivalent. Indeed, let’s construct the equivalence as a pair of constant functions: and (the origin of ). Both are continuous: The pre-image is the whole real line, which is open. The pre-image of any open set in is , which is also open. The composition is equal to the identity on , so it’s automatically homotopic to it. The interesting part is the composition , which is emphatically not equal to identity on . We can however construct a homotpy between the identity and it. It’s a function that interpolates between them: ( is the unit interval .) Such a function exists: When a space is homotopy equivalent to a point, we say that it’s contractible. Thus is contractible. Similarly, n-dimension...