0708-1300/Class notes for Thursday, January 24
INCOMPLETE AND UNEDITED: Completion coming soon.
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
Proof of Van Kampen
We aim to show that
Hence, we want to define two maps:
such that they are inverses of each other.
Now, recall the commuting diagram:
Further, let alternate between 1 and 2 for successive i's.
Hence we define via for ,
Clearly this is well defined. We need to check the relations in indeed hold. Well, the identity element corresponds to the identity path so the relation that removes identities holds. Furthermore, the concatenation of paths is a sum after and the definition necessitates the third relation holds.
Now for :
Elements in G correspond with paths in
On consider such a . The Lesbegue Lemma let us break up so such that each is in just
Now, does not go from base point to base point, so we can't consider it as a loop itself. Let denote the endpoint of (and hence is the beginning point) and further, the base point.
Choose paths connecting to b such that if then
where each section is entirely in . This above mapping is
We need to show that is well defined. I.e.,
1) is independent of the subdivision
2) is independent of choice of the 's
3) If then
For 1) it is enough to show that one can add or remove a single subdivision point. Suppose you add a new subdivision point c in between two others a and b. Traveling between a and b is the same as going from a to c then to the basepoint and back and then continuing on to b. But this is precisely what happens when you add a subdivision point. Likewise for removing points.
For 2), Suppose goes from the basepoint to . Add a subdivision point y right beside and remove the one at . Then add again the original point . The trick is that the new is the one used when adding y. The map from the new to the basepoint is this with the infinitesimal connection between y and added. The subdivision point y is then removed leaving our original configuration of basepoints, only with the path from to b now being instead of the original
For 3) we consider the homotopy between and thought of as a square with on the bottom and on the top. We use the Lesbegue Lemma to subdivide the square into many subsquares such that each one is entirely in
We further modify the homotopy H to a new homotopy such that each grid point gets mapped to b.
This can be thought of as "pinching" each gird point and pulling it to b or, alternatively, as tossing a "handkerchief on a bed of nails"
If we let be broken in to section along the bottom then first lifts to the top and right sides of the subsquare such that the gridpoint goes to b. then moves the next square up in an analogous manner until we are at the top with .