Geometric Topology

Prof. Jo Nelson
Math 444/539
Fall 2019

Email: jo [dot] nelson [at] rice [dot] edu
Lectures: MWF 11-11.50am, HBH 453
Location: TBD
Office Hours:
Prof Jo: M 3-4pm and W 12-1pm in HBH 402
Tam: R 11-12pm in HBH 14

Textbooks

The official textbooks for the course are:
W. S. Massey, A Basic Course in Algebraic Topology, Springer Graduate Texts in Mathematics.
J. Lee, Introduction to Smooth Manifolds, Second Edition, Springer Graduate Texts in Mathematics.

Here are some other books which you may find helpful:

Munkres, Topology, second edition
Clearly and gently explains point set topology, if you need to review this. The assumed point-set material is: topological paces, open and closed sets, basis for a topology, special topologies such as the product topology, metric topologies and the quotient topology, continuous maps, compactness, connectedness, the countability and separation properties. Specifically see: Chapter 2 and Sections 23, 26, 30, 31, and 32.

Allen Hatcher, Algebraic Topology
This book is free and has numerous detailed examples.

Teaching Assistant

The teaching assistant for this course is Tam Cheetham-West. Tam will hold a discussion/example session weekly. They will grade homework and I will grade the exams (consistent with how the homework was graded).

Homework

Both undergraduate and graduate students should expect to spend 7-8 hours a week on homework in this course. Homework will count for 30% of your final grade, and you must upload your homework to gradescope by 5pm on Mondays. Clearly print your first and last name on your assignment and indicate those students that you worked with. Late homework will not be accepted. Your lowest homework score will be dropped. Math graduate students will have additional homework problems each week.

Exams

There will be one pledged take home midterm, worth 30% of your course grade, which you should spend no more than 5 hours/undergrad (7 hours/math grad) actively working on it. It will be made available on Monday September 30 and due by 5pm on Monday October 7. You are not permitted to work with other students and you are not permitted to consult the internet beyond the course Piazza page. You are allowed to refer to the course textbooks.

The final exam will count for 40% of your course grade. It will be a pledged take home exam given on TBA and you must turn it in TBA. As with the midterm you are not permitted to work with other students and you are not permitted to consult the internet beyond the course Piazza page. Math graduate students will be assigned course grades independent of other students.

In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from the dean and doctor must be provided to me.

Help

If you find yourself confused, please seek help sooner rather than later. I will be available to answer questions during my office hours and the course TA will hold office hours TBD. You should use Piazza to post questions about the course, including questions about topics covered in class or regarding the homework.

In order to receive disability-related academic accommodations, students must first be registered with the Disability Resource Center (DRC). Students who may need accommodations in this course should give me a written letter from the DRC within the first two weeks. More information on the DRC registration process is available online at https://drc.rice.edu/. Registered students must present an accommodation letter to the professor before exams or other accommodations can be provided. Students who have, or think they may have, a disability are invited to contact DRC for a confidential discussion.

Outline

The basic plan is to spend the first 6 weeks on the material in chapters 2, 4, and 5 of Massey and then spend 8 weeks on the material in chapters 1-6, 8, 11, 14-16 in Lee. My goal is for you to understand the basic concepts listed below and to be able to work with them. This material is all essential background for future graduate level coursework in geometry and topology. In class I will introduce the main ideas, explain where they come from, and demonstrate how to use them. I will leave most proofs and technical lemmas for you to read (or not). You will be expected to spend 1-2 hours reading the book to understand the proofs behind the concepts discussed in lectures.

The fundamental group (Massey, chapter 2)
Seifert and van Kampen Theorem (Massey, chapter 4)
Covering spaces (Massey, chapter 5)
Topological manifolds, smooth manifolds, smooth maps, diffeomorphisms, manifolds with boundary. (Lee, chapters 1-2).
Tangent vectors, tangent space, differential of a smooth map, tangent bundle. Calculations in coordinates. (Lee, chapter 3).
Immersions, submersions, and embeddings. Submanifolds. (Lee chapters 4 and 5).
Transversality and Sard's theorem. (Lee chapter 6 )
Vector fields. (Lee, chapter 8)
Cotangent Bundle (Lee, chapters 11)
Differential Forms (Lee, chapters 14-16)

Schedule & Assignments

Date	Material Covered	Homework (Tuesdays)
8/26	Introduction and Homotopy
8/28	The Fundamental group
8/30	Fundamental group of a circle	HW 1 LaTeX Due Wednesday 9/4
9/2	Labor Day
9/4	Fundamental group of a product space
9/6	Free groups and free products	HW 2 LaTeX Due 9/9
9/9	Seifert - van Kampen I
9/11	Seifert - van Kampen II
9/13	Surfaces	HW 3 LaTeX Due 9/17
9/16	Covering spaces
9/18	Fundamental group of a covering space
9/20	Examples	HW 4 LaTeX Due 9/24
9/23	Lifting of arbitrary maps
9/25	Deck transformations
9/27	Action of fundamental group on p^{-1}(x)	HW 5 LaTeX Due 10/1
9/30	Fun with group actions
10/2	MIDTERM ``HANDED OUT" Universal covers	Midterm LaTeX Due 10/7
10/4	CW complexes I
10/7	Definition of topological and smooth manifold. Diffeomorphisms
10/9	MIDTERM DUE Tangent vectors and the tangent space.
10/11	Differential of a smooth map between smooth manifolds
10/14	Fall Break
10/16	Computations in Coordinates
10/18	Tangent Bundle and Vector fields	HW 6 LaTeX Due 10/22
10/21	Vector fields and Lie Bracket
10/23	Immersions, embeddings, and submersions. Short movie ``Outside In".
10/25	Inverse function theorem. Implicit function theorem.	HW 7 LaTeX Due 10/29
10/28	Submanifolds and Level sets. Regular and critical values
10/30	Transversality.
11/1	Results that submanifolds "generically" intersect transversely.	HW 8 LaTeX Due 11/8
11/4	Sard and Sard-Smale.
11/6	Multilinear algebra and tensors.
11/8	The Cotangent Bundle	HW 9 LaTeX Due 11/15
11/11	Pullbacks. Symmetric and alternating tensors
11/13	Riemannian metrics and the musical isomorphisms
11/15	The algebra of alternating tensors.	HW 10 LaTeX Due 11/22
11/18	Differential forms on a manifold. Wedge product, pullback, and exterior derivative.
11/20	Scattered fun with Lie Derivatives
11/22	Orientations	HW 11 LaTeX Due 12/6
11/25	Integration of differential forms. Stokes' theorem.
11/27	NO CLASS
2/2	Riemannian manifolds and volume forms.
2/4	Definition of DeRham cohomology
2/6	Computations. Overflow.	Final Exam LaTeX Due Monday 12/16