This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about at all.
In algebraic topology you (for example) associate every topological space with a group, like or . All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.
The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup” work in algebraic topology.
Note that is a functor
i.e. to every space we can associate a group . (Of course, replace by integer of your choice.) Recall that:
Thus for a map we can take its homotopy class (the equivalence class under this relationship). This has the nice property that and so on.
In light of this, we can define
Then the big result is that:
The proof of this is geometric, using the so-called prism operators. In any case, as with all functors we deduce
In particular, the contractible spaces are those spaces which are homotopy equivalent to a point. In which case, for all .
2. Relative Homology,
In fact, we also defined homology groups
for . We will now show this is functorial too.
Thus, we naturally arrive at two categories:
- , the category of pairs of toplogical spaces, and
- , the same category except with maps only equivalent up to homotopy.
Then, the prism operators now let us derive
The usual corollaries apply.
Now, we want an analog of contractible spaces for our pairs: i.e. pairs of spaces such that for . The correct definition is:
Of course in this situation we have that
As a special case of the above, we define
Now again we mod out:
And lo and behold:
Same corollaries as before.