0708-1300/Class notes for Thursday, October 18
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
Today we stated the Whitney Embedding Theorem and began to discuss its proof. Along the way, we also encountered some related notions that will serve us well in the future. We begin by stating the theorem:
Theorem (Whitney Embedding)
Let be a smooth -manifold. Then there exists an embedding .
We will break the proof of the theorem into three parts:
- Find an embedding of a compact into for some .
- Use Sard's Theorem to reduce to .
- Use the "Zebra Trick" to prove the theorem for non-compact .
Parts two and three shall be left to the next lecture.
Suppose that is compact. Let be an atlas for , and note that is an open cover of . Hence it possesses a finite subcover , and the corresponding collection of charts is an atlas.
Choose smooth functions with the following properties:
where for is the support of , ie. the closure of The existence of such functions follows from the existence of smooth partitions of unity for manifolds---a concept that will be discussed later on.
Now define by , where is defined by and .
We claim that is an embedding. That is smooth follows immediately from its construction (the s have been used to smear out the to smooth functions on all of ). That is injective is also clear. It takes a bit of work to show that is an immersion, but this is left as an exercise. It remains to see that is a homeomorphism, but this fact follows from the following topological lemma.
Let and be topological spaces. Suppose that is compact, is Hausdorff, and that is continuous and injective. Then is a homeomorphism.
Since is an injection onto its image, it is a bijection. Since is continuous, it remains to show that is continuous. Thus, it suffices to see that takes closed sets to closed sets. Let be closed. Since is compact, so is . Hence is compact since is continuous. But is Hausdorff, and every compact subset of a Hausdorff space is closed. Hence is closed.
The smearing functions we used in Part 1 of the proof of the Whitney Embedding Theorem are very similar to partitions of unity---collections of functions that break the constant function into a bunch of bump functions. We will now formalize this notion and show that such collections of functions exist for smooth manifolds.
Let be an open cover of a topological space (manifold) . A partition of unity subordinate to is a collection of continuous (smooth) functions such that
- For every there is an such that
- is locally finite, ie. for every , there exists a neighbourhood of such that intersects for only finitely many .
A local refinement of is an open cover of such that for every , for some . is called paracompact if every open cover of has a locally finite refinement.
For further information on paracompactness, we refer the reader to the corresponding Wikipedia entry. Note, in particular, that locally compact, second-countable topological spaces---such as manifolds---are paracompact, and that paracompact spaces are shrinking spaces. The following result (which follows immediately from these facts) will be useful for constructing partitions of unity on manifolds:
Manifolds are paracompact. In particular, if is locally finite then there is an open cover such that for every .