07081300/Class notes for Tuesday, April 1

Typed Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
First Hour
Theorem
If is an embbeding then
Proof:
By induction on k.
For k=0 this is easy, is a single point and  a point is just
Now suppose we know the theorem is true for k1, let be an embedding. Write the cube as where and
So,
where both sets in the union are open. Recall for sets where A and B are open we have the Mayer Vietoris Sequence.
Note: It is usefull to "open the black box" and think about the Mayer Vietoris Sequence from the veiwpoint of singular homology, ie using maps of simplexes and the like. This was briefly sketched in class.
We get a sequence
The first and last terms in this sequence (at least the small part of it we have written) vanish by induction hypothesis.
Note: Technically we need to check at Mayer Vietoris sequence also works for reduced homology.
We thus get the isomorphism:
Assume
As the above isormorphism is induced by the inclusion maps at the chain level, we get that either in or in
Now repeat, find a sequence of intervals such that
1)
2) in
Pictorially, we cut C in half, find which half is non zero, then cut this half in half and find the one the is non zero in and again cut it in half, etc...
But, in by induction. Hence, such that .
But, = { union of images of simplexes in U} is compact, so, for some j. But this contradicts assumption 2)
Q.E.D
We now prove an analogous theorem for spheres opposed to cubes
Theorem
For an embedding then if p = nk1 and zero otherwise.
Intuitively,
Recall we had seen this explicitly for when we wrote as the union of two tori (an inflated circle)
Proof:
As in the previous proof, we use Mayer Vietoris. Let
Then,
The first and last term written vanish by induction hypothesis and so have an isomorphism
Hence, it is enough to prove the theorem for k=0. Well in this case we have
Since , this is for p = n1 and 0 otherwise
Q.E.D
Take k=n1 so
Hence the compliment of an n1 sphere in has two connected components. This is the Jordan Curve Theorem.