# Differential Topology Homework Solutions

## Syllabus

** Differentiable Manifolds: ** Differentiable structures. The tangent space. Submanifolds. Embeddings and immersions. Manifolds with boundary.

** Approximation and transversality: ** Topologies on spaces of smooth functions. Approximation by smooth functions. Existence and uniqueness of a smooth structure on a C^1 manifold. Sard's theorem. Transversality.

** Vector bundles and tubular neighborhoods: ** Vector bundles. Basic constructions. Orientations. Classification of vector bundles. Existence and uniqueness of tubular neighborhoods.

** Topological invariants: ** The degree of a map. The intersection number of submanifolds of complementary dimension. The Euler number of a vector bundle.

** Morse theory: ** Morse functions. The Morse lemma. Cell complexes. Morse inequalities. Morse-Bott theory.

** Cobordism: ** The Pontryagin-Thom construction. Thom's theorem.

## Bibliography

The course will follow Hirsch's book. Other recommended books are:- A. Banyaga and D. Hurtubise, Lectures on Morse homology. Kluwer Texts in the Mathematical Sciences, 29. Kluwer, 2004.
- R. Bott and L. Tu, Differential forms in algebraic topology, Springer GTM 82, 1982
- V. Guillemin and A. Pollack, Differential topology, Prentice Hall, 1974
**M. Hirsch, Differential Topology, Springer GTM 33, 1976**- J. Milnor, Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, PUP, 1997.
- J. Milnor, Characteristic Classes, Annals of Mathematics Studies, No. 76, PUP, 1974
- J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, PUP, 1963
- J. Milnor, Lectures on the h-cobordism theorem, PUP, 1965.
- J. Munkres, Elementary Differential Topology, Annals of Mathematics Studies, No. 54, PUP, 1963
- F. Warner, Foundations of differentiable manifolds and Lie groups, Springer GTM 94, 1983

And here is a classification of 1-manifolds which we used in class (this is the solution to a couple of exercises in Hirsch).

See also the Geometry of Manifolds course webpage of MIT's open course ware.

**Course webpages from previous years (in portuguese):**

## Grading policy:

There will be bi-weekly problem sets making up 30% of the grade. Late homework will not be accepted. There will be two tests each counting 35% towards the grade (dates to be arranged). You will be able to make up for one of these tests the week after classes end.## Homework and exams

- Homework due 3/10/08: 1.1.1 plus four of the following exercises: 1.1.4,7,8,9; 1.2.5,6,7,8,10,12,14; 1.3.2,3,4,6,7,11,16.
- Homework due 17/10/08: Five of the following exercises: 1.4.3,4,5,7,8,9,10,11; 2.1.1,2,3,5,7,9,11,14,15,16.
- Homework due 31/10/08: Five of the following exercises: 2.2.1,2,3,4; 2.3.1,4; 2.4.1,3,4,6,7,8,9,10,11,13,14,15,16
- Homework due 14/11/08: Five of the following exercises: 3.1.1,2,3,4; 3.2.1-11,13-15
- Here is a practice test with solutions.
- Here is the test of 18/11/08.
- Homework due 28/11/08: Three of the following exercises: 4.1.2,4; 4.2.2,3,5,6,7,8,9; 4.3.1,2,8,9,10.
- Homework due 12/12/08: Five of the following exercises: 4.4.2,5,6,8,9,11,12; 4.6.3,4,5,6,7,9; 5.1.1-12.
- Homework due 12/1/09: Eight of the following exercises: 5.2.1,2,3,6,9,10,11,17,18,21 ; 6.1,3,4,6; 6.2.3,10,11,13; 6.3.1,2,6,7,8,9; 6.4.1; 7.1.1,3,4; 7.2.1,3,4,5,6,8.
- Here is a second practice test with solutions.
- Here is the test of 15/01/09.
- Here is the make-up exam of 30/01/09.

Differential Topology

This page is always under construction, so you should check it regularly. Assignments with the word **homework** in **bold face** are set in stone. Other assignments are still tentative.

**Homework # 1**: (due January 29) Available here. Solutions available here.

**Homework # 2**: (due February 5) Available here. Solutions available here.

**Homework # 3**: (due February 12) Just Guillemin and Pollack this week:

Page 18, #2, 3, 4, and 9.

Page 25, #1, 2, 6, 7, 12 and 13.

Page 32, #1, 2, 4. (We'll do some more from this section next week).

Solutions available here.

**Homework # 4**: (due February 19) All but one problem from Guillemin and Pollack this week:

Page 32, #5, 7, 10

Page 38, #7, 8, 9, 11. On problem (8), either prove part (e) of the theorem or make sure you understand the proof in the book, since we did not do this part in class.

Page 45, #5, 16, 17, 18.

Extra problem a) Let X be a k-manifold and Y be the n-sphere (i.e. the unit sphere in R^{n+1}), and suppose that n>k. Show that any smooth map from X to Y is homotopic to a constant map.

b) Give an example of a k-manifold X and an n-manifold Y (not a sphere!) with n>k, and a map X to Y that is not homotopic to a constant map. (You may need facts from other courses to show that the map isn't null-homotopic. We'll develop our own tools later in the semester.)

Solutions available here.

**Homework # 5**: (due February 26)

Page 54, problems 6, 7, 8, 9.

Page 62, problems 1, 2, 6, 7, 8.

Page 66, problems 6 and 7.

Let X be an arbitrary manifold, Y=S^{2}, and Z a closed submanifold of Y. Let f: X ⇒ Y be a smooth map. Show that, for almost every rotation R in SO(3), the map f_{R} = R ° f is transversal to Z.

Solutions available here.

**Homework # 6**: (due March 4) Available here.

Solutions available here.

**Homework # 7**: (due March 21) Note that this set, which wraps up Chapter 2, is due the Monday AFTER Spring Break.

Page 74, problems 1, 16, 17, 18

Page 82, problems 3, 4, 5, 6, 9, 10. On Problem 5, assume that dim(X)>0. Contrary to the parenthetical comment, there IS a zero dimensional anomaly!

Write a 1-paragraph description of what you intend to write your term paper about.

If you're feeling REALLY energetic, or bored over Spring Break, work through the full proof of the Jordan-Brouwer separation theorem on pages 87-89. That's the best way to really get a feel for mod-2 intersection theory.

Solutions available here.

**Homework #8** (due WEDNESDAY, March 30).

Page 103, problems 10, 11, 14, 17, 22. Note that there are some typos on problem 11. The formulas for v_{1}, v_{2} and n should involve partial derivatives of f(x,y), not of F(x,y) (which doesn't even make sense, since F is a function of (x,y,z)).

Page 116, problems 3, 4, 6, 7, 9, 10, 11.

Solutions available here.

**Homework # 9**: (due FRIDAY April 8) Available here.

Solutions available here.

**Homework #10** (due Friday, April 15) is a series of exercises embedded in part 1 of the lecture notes on differential forms. Do **all** of the exercises.

Solutions available here.

**Homework #11** (due Friday, April 22). Do all the exercises from part 2 and the first 6 exercises from part 3 of the lecture notes on differential forms.

Solutions to the problems on part 2 of the notes available here. Solutions to the questions from part 3 are available here.

**Homework #12** (due Friday, April 29). Do the remaining exercises from part 3 of the lecture notes on differential forms, and all of the exercises from part 4. Here are solutions to the questions from part 3 and part 4.

**Homework #13** (due Friday, May 6). Do all of the exercises from sections 1 through 6 of part 5. Solutions to the questions from part 5 are available here.

**Extra homework for your long-term benefit** (not to be turned in). Do exercises 12 and 13 from part 5 of the notes, and all the exercises from the final part 6. These go beyond what you will be expected to know for the prelim exam, but are Very Good Things To Know. Solutions to the questions from part 6 are available here.

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