Math 109a: Introduction to Geometry and Topology
Instructor: Alexander Gorodnik
Office: 172 Sloan
E-mail: gorodnik "at" caltech "dot" edu
Office Hours: MW 2-3pm and by appointment
TA: Dongping Zhuang
E-mail: dongping "at" caltech "dot" edu
Math 109a is the first of three courses in the 109 sequence, and is an introduction
to topology. Topology studies geometrical properties of objects that are
preserved under continuous deformations.
We will cover the following topics:
- Point-set topology: topological spaces and basic operations with them,
seperation axioms, continuous maps, compact and connected spaces.
- Homotopy theory: covering spaces, fundamental group and its computation,
retracts, presentations of groups, Van Kampen theorem.
- Homology theories: simplicial and cell complexes, elements of simplicial and singular homology.
Prerequisites: Ma 2 or
equivalent, and Ma 108 must be taken previously or concurrently.
Book: The main textbook for this course is
Our approach will deviate slightly from the one in this book.
Other useful references are
- M. A. Armstrong, Basic Topology.
There is also a number of online lecture notes:
- J. Munkres, Topology. (2nd Edition)
- A. Vassiliev, Introduction to Topology.
Grading Policy: Weekly homework (30%), a midterm (30%) and a final (40%).
Late work will only be accepted with a valid and well-documented explanation.
Please, contact Dongping Zhuang to submit late homework.
Collaboration Policy: You may discuss homework problems
with other students, but solutions should be written up individually in
your own words. Take-home exams must be your own work, with outside
references properly attributed.
- Homework 1 (due Wednesday, Oct. 4, 12pm)
- Homework 2 (due Wednesday, Oct. 11, 12pm)
- Homework 3 (due Wednesday, Oct. 18, 12pm)
- Homework 4 (due Wednesday, Oct. 25, 12pm)
- Midterm, pick up at the HW box (due Wednesday, Nov. 1, 12pm)
- Homework 5 (due Wednesday, Nov. 8, 12pm)
- Homework 6 (due Wednesday, Nov. 15, 12pm)
- Homework 7 (due Wednesday, Nov. 22, 12pm)
- Homework 8 (due Friday, Dec. 1, 12pm)
- Final Exam, pick up at the HW box (due Friday, Dec. 8, 12pm)