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Math 427/527: Topics in topology

Table of Contents

News

  • Solutions for homework 2 are now available.
  • Homework 4 is now available.
  • Office hours are suspended for the next two weeks (until [Fri Mar 24]). Instead I encourage people to ask me questions right before or after class, or via email.
  • The due date for homework 3 has been changed to [Wed Mar 08].

About

This is an introduction to Algebraic Topology. We will cover homology and cohomology theory including the ring structure on cohomology and Poincaré duality for orientable manifolds. We will mention (but not thoroughly cover) other topics such as the fundamental group, covering spaces and higher homotopy groups.

Prerequisites: For people taking this as MATH 427, MATH 426 is a prerequisite. Generally speaking, some knowledge of topological spaces (say, compactness, connectedness and quotient topologies) and of linear algebra (or even better, rings and modules) is recommended.

  • Classes: Monday, Wednesday and Fridays from 10am to 11am in ESB 4133 (note the room change!).
  • Office Hours: These will be held in LSK 300 on:

    Mondays 1pm-2pm
    Thursdays 4pm-5pm

    I’m willing to reschedule if these times are inconvenient.

Recommended books

We are fortunate to live in a time when many excellent books are freely available online from the author’s web page. In the following list, all book titles are links to electronic versions!

For the main content of the course I recommend:

Those books are very different in style, so if one doesn’t suit you, definitely try the other!

We will use some basic Category Theory throughout, for which I recommend:

To review the fundamental groupoid and covering spaces, besides the above books I also recommend:

Topics

  • The fundamental groupoid of a topological space.

    We just need the definitions. If you know about the fundamental group but not the fundamental groupoid, you should pick up the groupoid version pretty quick. If you want something more leisurely to read, I have old notes on groupoids in general, and the fundamental groupoid in particular.

  • The van Kampen theorem and applications (PDF).

    I’ve put up notes for this topic because I want to present a proof of the Jordan Curve Theorem that isn’t concisely written up in a single place: it’s a modified version of Ronnie Brown’s proof.

  • Covering space theory

    For a development using fundamental groupoids see chapter 3 of May or chapter 10 of Brown. For a more traditional group-only view see section 1.3 of Hatcher, which more than makes up for the lack of groupoids by having lots of great pictures!

  • Higher homotopy groups

    We just gave the definitions, showed they are abelian and discussed the difficulty in computing the homotopy groups of spheres. For the small amount we talked about, the Wikipedia page is probably enough, but of course, check out Hatcher, Chapter 4.

  • Homology

    There is a real choice of how to do things here: we need to discuss the Eilenberg-Steenrod axioms for homology and the definitions of simplicial, singular and cellular homology in some order. After some soul searching, I decided to follow Hatcher’s order (simplicial, singular, axioms, cellular) so we’ll follow Chapter 3 of his book. For those more homotopy theoretic at heart, I recommend looking at May’s book too (cellular, axioms, singular); the main reason I decided against that approach is that it requires more homotopy theory than we have time for.1

    You can use the Smith normal form to compute homology of a chain complex, and you can use a computer to compute the Smith normal form! This explained in this short note.

Assignments

Grades will be based on homework assignments, which will appear here.

Assignment Due date Solutions
HW1 (PDF) [Wed Jan 18] HW1 (PDF)
HW2 (PDF) [Wed Feb 08] HW2 (PDF)
HW3 (PDF) [Wed Mar 08]  
HW4 (PDF) [Wed Mar 29]  

1
Plus Hatcher has pictures!

by Omar Antolín Camarena