# Math 427/527: Topics in topology

## Table of Contents

*This course took place Jan-Apr 2017*

## News

- Homeworks have been taken offline.

## 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:

- Allen Hatcher,
*Algebraic Topology*. Cambridge University Press, Cambridge, 2002. - Peter May,
*A Concise Course in Algebraic Topology*. The University of Chicago Press, 1999.

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:

- Emily Riehl,
*Category Theory in Context*. Aurora: Dover Modern Math Originals.

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

- Ronald Brown,
*Topology and Groupoids*. Printed and Distributed by Createspace, 2006.

## 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.

Cohomology

Again, I decided to follow Hatcher. Cohomology is Chapter 4.

## Assignments

Grades will be based on homework assignments.

^{1}

Plus Hatcher has pictures!