0708-1300/Class notes for Tuesday, January 15
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
We begin by reformulating our previous lemma into a more general form:
Let be a covering map. Then every family of paths uch that has a unique lift such that
Claim 1: is well defined, hence
Proof of Claim 1
We construct the homotopy H between two such gamma's which, schematically, is a square with along the bottom, along the top and along the side the parameter Y where the homotopy is just horizontal lines between and
Then, the lemma implies the existence of a homotopy which is schematically a square with along the bottom, along the top, x=0 on the left and side and the right hand side taking values in
Proof of Lemma 1
Recall the proof for the version of the lemma when we were dealing with a single path, not a family. We now just need to extend this proof to the case of having a family of paths. We know that for each we can get a
We need to show that in fact the result is continuous. As continuity is a local property, we simply need to prove this in a neighborhood about .
Now by a "good" open set in B, we mean that the preimage under p CAN be written as a product.
Hence, we choose a neighborhood about the first interval in extending from (see proof of one path case for explanation) and this gets mapped to a "good" open set in B. As it is the image of an interval, it is compact in an open set, so can put a small neighborhood about the interval such that the image of the neighborhood is in the "good" open set in B.
Then, duplicating the proof of the earlier version of the lemma, this establishes continuity in the neighborhood.
1) We get again the proof of non existence of a retract
Indeed, assume we DID have such a retract. We would have the following commuting diagram.
Applying the functor would yield the diagram
But clearly this does not exist, as the only map from is the trivial one.
Recall the non existence of such retracts implies Brouwer's Theorem
2) Fundamental Theorem of Algebra
if is a polynomial with degree greater than zero then there such that
Suppose not, i.e., there exists a polynomial p that has no roots. dividing by the coefficient of the highest order term, lower order terms.
Define a homotopy of paths by first setting
Now the homotopy starts with a base of q on the point 0. During the first half of the homotopy we grow the size of the loop on which q is acting on. At the halfway point, the loop is so large such that the first term in the polynomial dominates the lower order terms. Then in the second half of the homotopy, the lower order terms are shut off. More precisely we consider the polynomial (lower order terms) and let tend from 1 to 0 in the second half of the homotopy.
Hence, at the end of the homotopy we are left with on a large loop. This is the generator we had previously, and is not the identity. Hence, we have a constant homotopic to something non trivial and this establishes the contradiction.
If is continuous then such that
1) is not a subset of (cannot be embedded) as f is not 1:1
2) If , with closed then at least one contains a pair of antipodes.