MATH 6441 - Algebraic Topology I
Spring 2016 Course Information

Poincaré	No retraction	Brouwer

HW6 is due April 21. In 3.2 solve #1, 3, 11, 15. In #15 ignore the sentence that starts with "Compute the Poincare series".

HW5 is due April 7th. Solve #3, 27, 41 in section 2.2 and #10 in 2.B. For #10 you may use that S^infinity is contractible and S^infinity is the 2-sheeted covering space of RP^infinity. Here RP^infinity is given the usual CW structure with one cell in each dimension. Actually, the CW structure is irrelevant for #10. The transfer sequence does all the work.

HW4 is due March 17th. In section 2.1 (on homology) hand in the following problems: 16b (I did 16a in class), 17, 22, 31. Also let T_d denote a tree in which from every vertex emanates d edges. Suppose n > m > 2 are integers. Use local homology to show that the products of T_n and T_m with the real line are not homeomorphic.

HW3 is due March 3rd. In section 1.3 (on covering spaces) hand in the problem 20 and also show that the Klein bottle is not a covering space of the 2-torus. Other good problems in section 1.3 (not to be handed in) are 23, 25, 30, 32. In section 2.1 hand in problems 4, 7, and also fill details in the statement on page 110 that the H_0(X) is the direct sum of Z and the reduced zero-th homology of X.

HW2 is due February 11th.
In Section 1.2 (on van Kampen theorem) hand in problems 4, 7, 9. If you wish for more challenging problems try 2, 10, 16 but those are a bit harder, and you need NOT hand them in.
In section 1.3 (on covering spaces) hand in problems 5, 8, 14. Try (but do not hand in) 6, 9, 11.

HW1 is due January 28th.
In Chapter 0 hand in #5, 9, 14, 20 (pp18-19). Other practice problems in Chapter 0 (not to be handed in): 1, 3, 10, 16, 21, 23.
In Section 1.1 hand in #5, 18. Other practice problems in Section 1.1 (not to be handed in): 3, 4, 11, 12, 13, 16.

Instructor: Igor Belegradek

office: Skiles 240B

office hours: Thursday 12:05-12:55pm and Monday 1:05-1:55 in Skiles 240B, as well as by appointment. There will be no office hours January 18-22 and February 14-19.

phone: (404) 385-0053 (please do not leave voicemail as it is never checked).

Email: ib at math.gatech.edu (This is the best method of contact).

Lectures: TR 13:35-14:55 in Skiles 257.

Course homepage: www.math.gatech.edu/~ib/6441.html will contain homework, handouts (if any), and this syllabus. Grades will be posted on T-square. We shall not use T-square for any other purpose.

Grading: Homework 60%, Midterm 15%, Final 25%. Grades will be posted on T-square.

Grading scheme: A=80-100%, B=65-79%, C=50-64%, D=40-49%.

Tests: Midterm: February 16th (Tuesday) in class. Final: May 3rd (Tuesday) 6-8:50pm. Tests are closed book, closed notes. No electronic devices are allowed. The students are reminded of the Honor Code. If Midterm is missed due to a reason that the instructor finds valid, the Final will carry a higher weight; there will be no make up exam.

Homework:

• There will be 6 homeworks assigned bi-weekly, and due on Thursdays of January 28th, February 11th, March 3rd, March 17th, April 7th, April 21st.
• Homework will be collected in the beginning of the lecture; late homework will not be accepted unless in emergency, with instructor's permission.
• You may discuss homework with others and work in groups, but you must write up and submit your own solutions.
• Any outside help while doing the homework must be acknowledged. For example, if a student finds a solution somewhere (online, in a book, or from another person etc), the source must be mentioned to the instructor.

Objectives and Content:

• The primary goal of the course to help prepare a student for using algebraic topology in research.
• Another goal is to help students prepare for math comprehensive exams in topology.
• This course will cover basics of fundamental group, covering spaces, homology, and cohomology. The standard syllabus is both over-optimistic and outdated. We shall quickly go over the basic homotopy theory, fundamental group and covering spaces, and spend most of the time on homology. Eventually we will get to cohomology aiming towards to the Poincaré duality.
• This material has been completely understood by the mid 20th century, and it is quite far from the current research frontier in topology. Yet it represents the most widely-used and applicable part of algebraic topology both in mathematics and outside (e.g. in data analysis).
• Students in engineering, sciences, and discrete mathematics are very much welcome in the course but they should keep in mind that this is a math course, and it will be run as such. We will mostly stick to classical material above, so that those interested in applications would be well-equipped to pursue them.
• Some longer proofs will be omitted. The focus will be on ideas and techniques.

Textbooks. There are many excellent books on the subject and students are advised to find one that suits their learning style. The instructor would be happy to help with this task. Below are his recommendations:

• The primary text will be Allen Hatcher's Algebraic topology in which we plan to cover Chapters 0-3 (with substantial omissions). This is a popular graduate level textbook, and its style reflects how most topologists want their PhD students to think. The book is high on intuition and clarity, but some beginners feel that details are often omitted.
• For a more detailed slower exposition I recommend two texts by William S. Massey: Algebraic Topology: An Introduction treating fundamental group and covering spaces, and Singular Homology Theory .
• For those who find Hatcher too informal and Massey too slow a good choice could be Elements Of Algebraic Topology by James R. Munkres.
• Books on computational homology and applied topology which discuss algorithmic aspects and some applications to sciences and engineering:
  • Computational Topology: An Introduction Herbert Edelsbrunner and John L. Harer.
  • A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner.
  • Computational Homology by Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek.
  • Elementary Applied Topology by Robert Ghrist.
• Some fun articles on applications of homology:
• Homology: An idea whose time has come by Barry A. Cipra.
• Assembling geometric data using statistics for topology by Peter Bubenik.
• Barcodes: The Persistent Topology of Data by Robert Ghrist.
• Persistent Homology - a Survey by Herbert Edelsbrunner and John Harer.

Prerequisites: a student must have sufficient mathematical maturity needed to write and understand proofs, and to fill in details in certain semi-rigorous arguments. We shall rely on knowledge of basic properties of abelian groups, and point set topology as e.g. in the the handout written by John Etnyre; notably we shall need a good grip on quotient spaces.

Week	Dates	Tentative topics	Text Sections	Notes
1	Jan 12/14	Cell complexes/homotopy	Chapter 0
2	Jan 19/21	Fundamental group	Section 1.1	Substitute (Prof. Margalit)
3	Jan 26/28	Van Kampen's theorem	Section 1.2	HW due January 28
4	Feb 2/4	Covering spaces	Section 1.3
5	Feb 9/11	Covering spaces, Homology	Section 1.3	HW due February 11
6	Feb 16 (Tue)			Midterm in lecture room
6	Feb 18			No lecture (rescheduled to February 11, 4:35pm)
7	Feb 23/25	Homology	Section 2.1
8	Mar 1/3	Homology	Section 2.1	HW due March 3rd
9	Mar 8/10	Relative homology	Section 2.1
10	Mar 15/17	Excision / Mayer-Vietoris	Section 2.1-2.2	HW due March 17
11	Mar 22/24			Spring Break
12	Mar 29/31	Cohomology	Section 3.1
13	Apr 5/7	Cohomology	Section 3.2	HW due April 7
14	Apr 12/14	Products	Section 3.2
15	Apr 19/12	Poincaré duality	Section 3.3	HW due April 21
16	Apr 26	Poincaré duality	Section 3.3	Last day of classes
17	May 3 (Tue)			Final: 6-8:50pm in lecture room