Math 570  Topology



Renzo's class



MWF 10 – 10:50 am



ENGRG  E 206








Office Hours







Algebraic Topology

Allen Hatcher

Available online here




For the part of general topology I will not be following any book in particular. I believe any reasonable book could be a good reference if you would like one. For example Armstrong’s book is well written and pretty essential. Munkries is a bit more hard core and dry, contains more information (probably more than you care about). Schick is quite elementary and a bit funky, but will be the book for you if you are fascinated by exotic topologies...


Bill Fulton’s Algebraic Topology is a beautifully written book. We will probably be picking and choosing things from that book from time to time.



Project 1: Making New Spaces from Old Ones (LaTeX source)

Project 1 with solutions (Part 1) (Part 2)

Compactifications (LaTeX source)

Cool Theorems (LaTeX Source)




PDF file

LaTeX source file



Office hours : there are no official office hours for this class. However, you are very welcome to make an appointment and come ask questions, make comments, or just chat. You can also try showing up at my door anytime. But I might tell you to come back at another time if I am immersed into something else.

The tables of the law for this class are contained in the following document:

Once you get settled, you are warmly invited to tell me what you
do and don't like about me and this class. I  try my best to take criticism very well.

Homework:  OK, this is a graduate course, and so we can be pretty flexible about this. However homework is important stuff to make sure that things sink in and you are not just spending a semester assisting to my creative rambling. Here is some general guidelines I would like to follow:

1.      Grad students who are already working on a thesis problem with an adviser are not required to turn in homework.

2.      For other students, if the homework load is incompatible with other grad school priorities (especially for students not specifically working in this area) please talk to me and we can agree on an appropriate fraction  of the workload.

3.      Homework will be collected and checked, but not necessarily graded. Work with the buddy system to make sure you are doing things right, and, in doubt, come ask me.



Friday Aug 28th

Life with a metric (LaTeX source)

This worksheet is intended to present  the intuitive version of topology when a space is endowed with a notion of distance. Although in this class we will not always have such a luxury, it is a good thing to anchor your intuition to.

Friday’s class, if you wish, can be devoted to discussion about this homework and the relation between the abstract definitions of  the first couple days of class and those in this worksheet.  But this will happen ONLY if you ask questions, otherwise I’ll assume all is good and move on.

Friday Sep 11th

Some more basic concepts (LaTeX source)

We continue exploring with the basic concepts in topology. We define some more  unusual topology (like the half line and the Zariski topology) and explore on them the concepts illustrated in class. We also introduce some new notions (dense sets, separation axioms). If you are unfamiliar with these concepts pay special attention to those exercises!

Friday Sep 18th

Writeups for projects due. Please LaTeX your answer in such a way that I can compile them in a unique document!

Friday Sep 25th

Product Spaces, Connectedness, Compactness (LaTeX source)

Friday Oct 9th

Write up a (art most) three page account of the classification theorem for compact surfaces. Give good outlines, choose carefully what details to give. Make it pleasant to read!

Friday, Oct 23rd

Exercises 5, 8, 12, 18, 20 pages 38-39 of Hatcher (current webpage version)

Exercise 9 page 53

Friday Nov 20th

Exercises 3, 4 (hint, pull the diameter outside the sphere!), 14, 18, 27 on pages 79 and following of Hatcher.














Exams: I am not extremely fond of exams either, so we will have some alternative ways of evaluating your performance that hopefully can be more productive. These could be in class presentations, compiling good notes etc…I will let you know when I figure out the precise details.