以下是一次Math 141: Differential Topology代考案例
The following sequence of problems explains the idea of how to produce a proper map from a manifold to $\mathbb{R}$.
(a) Recall the “bump function” defined in problem 18 of GP $1.1 .$ In that problem, you showed that for any $0<a<b$, there exists a smooth function on $\mathbb{R}^{k}$ that is 1 on the ball of radius $a$, zero outside the ball of radius $b$, and strictly between 0 and 1 at intermediate points. Let $\mu_{j}: \mathbb{R} \rightarrow \mathbb{R}$ be a bump function that is 0 outside of $(j-2, j+2)$ and 1 on $(j-1, j+1)$. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function
$$
f(x)=\sum_{j \in \mathbb{Z}}|j| \mu_{j}(x)
$$
Show that this sum is finite for every $x$ (so that $f$ is well-defined), that $f$ is smooth, and that $f$ is proper.
(b) Now we generalize to higher dimensions. Let $p_{1}, p_{2}, p_{3}, \ldots$ be a list of all the points in $\mathbb{R}^{k}$ with integer coordinates. For example, the point $(1,2,3,-4) \in \mathbb{R}^{4}$ counts, but not $(0.37,2,3,5)$. Note that there are countably many such points, so it is possible to make such a list. Let $\mu_{j}: \mathbb{R}^{k} \rightarrow \mathbb{R}$ be a bump function that is 0 outside of the ball of radius $k+1$ centered at $p_{j}$, and 1 on the ball of radius $k$ centered at $p_{j} .$ Show that
$$
f(x)=\sum_{j=1}^{\infty} j \mu_{j}(x)
$$
is well defined, smooth, proper, and never zero.
(c) (partitions of unity) Using the functions $\mu_{j}$ from the previous part, define
$$
\theta_{j}(x)=\frac{\mu_{j}(x)}{\sum_{j=1}^{\infty} \mu_{j}(x)}
$$
Show that $\theta_{j}$ has the following properties (these should be easy)
– $0 \leq \theta_{j}(x) \leq 1$ for all $x$
– At each point $x$, all but finitely many $\theta_{j}$ are zero. In fact, $\theta_{j}$ is zero outside the ball of radius $k+1$ about $p_{j}$
– For each $x, \sum_{j=1}^{\infty} \theta_{j}(x)=1$
Remark: A collection of functions $\theta_{j}$ on a manifold $X$ with these properties is called a “partition of unity”. On page 52 of GP, it is proved that such a collection exists for any manifold, and instead of “ball of radius 2 about $p_{j} “$ one can choose any collection of open sets $B_{1}, B_{2}, \ldots$ that cover $X$ and have $\theta_{j}$ be zero outside of $B_{j}$
(d) (bonus question, not to hand in) Now let $X$ be an arbitrary manifold. Generalize our strategy for $\mathbb{R}^{k}$ to produce a proper map from $X$ to $\mathbb{R}$. The solution proposed in GP uses partitions of unity.
(a) (3 points) Suppose $f$ and $g$ are smooth functions from $X$ to $Y$. What does it mean for $f$ to be homotopic to $g ?$
(b) ( 6 points) Let $Y$ be a manifold with the property that every smooth function $f: S^{1} \rightarrow Y$ is homotopic to a constant function (in other words, $Y$ is “simply connected”). Prove that $Y \times \mathbb{R}$ also has this property.
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Essential Prerequisites:
Students should be very comfortable with the following concepts:
i. Continuity of functions (from R^n to R^n, and on general metric spaces)
ii. Metric spaces
iii. Topology: open and closed sets, compactness, covers, homeomorphisms.
iv. Multivariable calculus: Differentiability of functions from R^n to R^n. Linear maps.
Suggested reading for i-iii: A. Hatcher’s Notes on introductory point-set topology or Chapter 2 of C. Pugh’s book Real Mathematical Analysis.
General information for students
- My office hours: Tuesday 12:30-1:30, Wednesday 1-2, Thursday 10-11.
- Course syllabus and rough schedule.
- DSP students should speak to the instructor as soon as possible, even if you don’t have a letter yet.
- Guidelines on what to do if you think you may have a conflict between this class and your extracurricular activities.
In particular, you must speak to the instructor before the end of the second week of classes. - Academic honesty in mathematics courses. A statement on cheating and plagiarism, courtesy of M. Hutchings.
- Policy on absences for tests and midterms.
- How to get an A in this class
Textbook
The textbook for this course is Differential Topology by Guillemin and Pollack. We will cover most of chapters 1, 2 and 3.
Supplementary reading (not required)
– Chapter 1 (two lectures on manifolds) from A mathematical gift III. Highly recommended as motivation for the content of this class.
– L. Christine Kinsey, Topology of Surfaces (alternate text suggested by a previous instructor)
– V. V. Prasolov, Intuitive topology. (not as closely related to this class, but a good perspective on aspects of topology)
Also of interest
You may enjoy doing an independent study through the Directed Reading Program .
Problem sets and reading
- Week 1: Basic topology review.
Reading: A. Hatcher’s Notes on introductory point-set topology, this is more than you need to know, but a good reference!
Homework: Problem set 1. Notes/solutions
Optional just-for-fun homework: from V. Praslov’s “Intuitive topology” - Week 2: Smooth maps, manifolds, diffeomorphisms
Reading: GP sections 1.1 and 1.2
Homework: Problem set 2. Some comments .
Information on bump functions (related to problem 18), written by J. Loftin, here. - Week 3: The inverse function theorem, immersions and submersions.
Reading: GP sections 1.3 and 1.4.
Optional: notes on the implicit function theorem for R^n by T. Austin. Alternatively, you can refer to Chapter 5.5 in Pugh’s book.
Homework: Problem set 3 . Some comments . - Week 4: More on submersions, transversality
Reading: GP sections 1.4 and 1.5
No hand-in homework. Here is some information on the test next week, and extra problems - Week 5: (30-minute test), Homotopy and stability
Reading: GP section 1.6
Homework: Problem set 5.
Here are SOLUTIONS to the test. The average was 10.4/16. - Week 6: Measure and Sard’s theorem
Reading: GP section 1.7 — first part only. We will not discuss Morse theory yet. Beginning of 1.8.
Also, some notes on measure zero by H. Chan that you might find helpful.
Homework: Problem set 6. - Week 7: Whitney embedding. Intro to manifolds with boundary
Reading: GP section 1.8, 2.1
I will give you the idea of the proof of Whitney for noncompact manifolds, but you are not required to learn all the details.
Homework: Problem set 7. - Week 8: Manifolds with boundary and related topics.
Reading: GP section 2.1, 2.2.
Suggested review problems and information on the midterm.
Extra office hours: I will have office hours 5-6pm on Thursday, and 3-4pm next Monday. - Week 9: Midterm exam (Tuesday, in class). Thursday topic: the classification of 1-manifolds.
Homework: Problem set 8, due Tuesday, March 29.
(this is shorter than usual and certainly possible to finish before the spring break)Just for fun: The classification of 1-manifolds, using only topological techniques not differential topology or smooth maps!
Presented as a series of exercises (a “take-home exam”) by David Gale.
Those of you who feel very comfortable with topology might want to give this a try for fun.
(Note that you need to be on campus to access the link above.) - Week 10: Transversaily and the epsilon-neighborhood theorem
Reading: GP section 2.3
Solutions to the midterm. The average on the midterm was 29.9/40
Homework: Problem set 9. - Week 11: Intersection theory mod 2
Reading: GP section 2.4
Homework: Problem set 10. - Week 12: Winding number, Jordan-Brouwer separation, and Borsuk-Ulam
Reading: GP section 2.5 and 2.6
Here is a solution to GP 2.3 problem 10 from the homework handed back this week!
Homework: Problem set 11. Here is the .tex fileJust for fun:
Outside in is a video that describes something like the winding number (here called “turning number”, not exactly the same thing, but close!), then describes how to turn a sphere inside out via a family of immersions. - Week 13: End of Borsuk-Ulam, Orientation
Reading: GP section 3.1 and 3.2
Some remarks on the ham sandwich theorem from last week.
Final homework set: Problem set 12. Here is the .tex file
NOTE: this homework is due on the last day of class, Thursday April 28. (of course you can hand it in early if you want) - Week 14: Oriented intersection number
Reading: GP section 3.2 and 3.3 (up to the fundamental theorem of algebra) - Review materials and final exam info:
For reference: solutions to all the exercises for the proof of Jordan-Brouwer Separation, from Prof. Kozai’s 2014 class.Information on the final exam including suggested practice problems.
Practice final and solutions (to be posted)I will have office hours at the following times: Friday, May 6, 3-5pm. Monday, May 9, 3-5pm,
You may pick up the last homework set on Monday.Final Exam: Your final exam is Thursday, May 12, 8-11am in 234 HEARST GYMNote: Unusual location!!! And extremely early start time!!
