432/532 - Class Outlines, Homework, and Handouts

Math 432/532 - Class Outlines, Homework, and Handouts

You have a take-home final exam, due at noon on Wednesday March 16.

For further information on the class including Prof. Sinha's contact information, please read the syllabus.
In the second half of the class, all references to a text are implied to be to "Topology from the Differentiable Viewpoint" by Milnor.

Pay close attention to the class day-by-day descriptions over the next few weeks. On occasion, especially when there is technical material to cover, you will be asked to read ahead and then submit a response to the material - questions you have, your view of the overall structure of the argument, examples you thought about, comments on for example whether the argument was predictable or surprising. This response will be due either in my box across from the math office or to me personally at least ten minutes before class. It can be e-mailed to dps@noether. I will use these comments to lead more of a discussion of, rather than lecture about, the material.

Homework due on Friday, March 11. We will have a review session to discuss it after class on Wednesday March 9.

Week Ten, Mar 7-11

On Monday we will give a(n incomplete) development of the Brouwer degree for maps between oriented manifolds, and use it to show that a sphere has a non-vanishing vector field if and only if its dimension is odd.
On Wednesday we will use the degree to construct invariants of immersions and embeddings. As a warm up, we ask you to think about immersions of the circle in the plane (doodles)
On Friday we will discuss linking invariants of maps between spheres, explaining why homotopy groups are difficult to understand in general, and we might also talk about one of Prof. Sinha's recent theorems.

Week One, Jan 3-7:
We will begin to explore the notion of homotopy between maps and homotopy equivalence of spaces.

On Monday we will define what it means for maps to be homotopic and show that it is an equivalence relation.
For Wednesday, we will discuss some of the origins of the notion of homotopy from multivariable calculus (which will tie in to manifold theory, which we will do during the second half of the term).
For Friday, we will introduce the notion of homotopy equivalence and look at some examples. Mini-assignment, due Monday: make a conjectural classification of the letters of the alphabet up to homeomorphism and homotopy equivalence. Supply proofs where you can. Section 58 of "Topology" might be useful.

Week Two, Jan 10-14:

On Monday we will discuss the mini-assignement and then show that the the set of homotopy classes of maps from a space Z to a space X is a potentially interesting invariant of the homotopy type of X.
On Wednesday we introduce the fundamental group. See sections 51 and 52. We will focus on the fact that the fundamental group is in fact a group and that a map of spaces induces a homomorphism of groups. Mini assignment, due Friday: show that the fundamental group is associative (start with a schematic picture).
On Friday we will discuss associativity and show that the fundamental group is independent of basepoint for path connected spaces. We will also show that a map of based spaces induces a homomorphism of fundamental groups.

Week Three, Jan 17-21:

There is no class on Monday because of the Martin Luther King holiday.
On Wednesday we gather what we've done so far to show that homotopy equivalent spaces have isomorphic fundamental groups. We then start on a conceptual treatment of the fundamental group of the circle.
On Friday we will proceed with a formal treatment of the fundamental group of the circle. Relevant reading is in sections 53 and 54.

Homework, due Wednesday Jan 26 - Section 52, problems 3-7. A number of these could be difficult, so we will have a problem session on Monday at 3.

Week Four, Jan 24-28:

On Monday we finish the computation of the fundamental group of the circle, giving a treatment at a moderate level of detail to complement the intuitive and finely detailed approaches we have taken in previous classes. We will also start on applications.
On Wednesday we follow Section 55, giving applications of our development.
On Friday we give additional applications which include the Borsuk-Ulam theorem (in a manner different from the book) and the theorem that for any three closed subsets which cover S², one must contain a pair of antipodal points.will finish applications and then discuss the fundamental group of products (in particular the torus).

Week Five, Jan 31- Feb 4:

On Monday we treat the fundamental group of the projective plane. We end with a discussion of the compact-open topology and the relationship between homotopy of maps and paths in mapping spaces.
On Wednesday groups will be presenting the Van Kampen theorem and the Jordan theorems.
On Friday groups will be presenting embeddings of graphs and classification of covering spaces.

Instead of a mid-term, you will have team presentations and individual write-ups of additional topics.

Week Six, Feb 7 - 11:

On Monday the last group will present the classification of surfaces. This topic will naturally lead into the study of manifolds more generally, which is our focus for the second half of the term. Make sure to get your copies of Milnor's book.
On Wednesday we will have an informal overview of the study of manifolds.
On Friday we will start with a formal treatment of smooth submanifolds of Euclidean space, which is the class of manifolds we will be focusing on. We will indicate why the boundary of a square in the plane is not smooth.

Week Seven, Feb 14 - 18:

On Monday we will complete our development of the tangent bundle and start to define smooth maps and regular values.
On Wednesday we will formally define regular values and then give some intuition behind the theorems of Sard and Brown.
On Friday we will have our first "preparation assignment" on Sard's Theorem. You are to focus on the proof, which occupies pages 16-19 of Milnor's book. You should also address the question: why does it suffice to consider a map f from U open in Rⁿ to R^p when we want the result for general manifolds? Also, since some of you may be having trouble with the unfamiliarity of arbitrary manifolds, we may also spend some discussion time on the topics we have covered this week, in particular the definition of tangent spaces and the derivative of a smooth map. Feel free to share some of your questions on these topics in your write-up.

Week Eight, Feb 21 - 25:

On Monday and Wednesday we will continue our slow, careful discussion of the proof of Sard's theorem. There are many small statements in the proof which require some details to be filled in; they should be taken as exercises rather than straightforward reading. Our focus will go back and forth between such details and the big picture of the argument.
On Friday we will work towards developing the degree modulo two of a mapping between manifolds. We need Lemma 1 on page 11 and the discussion of manifolds with boundary on page 12. Then we will follow the material of section 4, which starts on page 20.

Week Nine, Feb 28 - Mar 04

On Monday we will develop and apply the mod 2 degree, using it to show that the only compact manifold (without boundary) which is contractible is the singleton point.
On Wednesday we will construct degree one maps from any manifold to a sphere, and finish showing that the mod 2 degree at a point y is well defined.
On Friday we will have a preparation assignment for which you are to look at the homogeneity lemma starting on page 22.