## MA4J7 Cohomology and Poincaré duality Term II 2019-2020

### Module Description

This module, MA4J7 (cohomology and Poincaré duality), has as its prerequisites MA3F1 (introduction to topology) and MA3H6 (algebraic topology).

This is the third module in the Warwick algebraic topology sequence. Cohomology is a theory dual to, but also deeper than, homology. This is because the collection of cohomology classes of a space, under formal addition and the cup product, forms a ring. This ring gives a new topological invariant. The cup product is also a key tool in a modern proof of "Poincaré duality" for manifolds. As an application, for any $$n$$-dimensional (closed, connected, oriented) manifold $$M$$, the homology groups $$H_k(M)$$ and $$H_{n-k}(M)$$ are isomorphic (ignoring torsion).

An understanding of cohomology is needed for advanced study in topology, differential geometry, algebraic geometry, algebraic number theory, and other areas in mathematics and physics.

### Schedule

The schedule has a planned list of topics, organized by week. We will update the schedule as needed, as we work through the material. Example sheets will be posted every two weeks.

### Lecturer

Name Building/Office E-mail Phone Office Hours
Saul Schleimer B2.14 Zeeman s dot schleimer at warwick dot ac dot uk 024 7652 3560 Fridays 15:00-16:00
Alessandro Bigazzi NA a dot bigazzi at warwick dot ac dot uk NA NA

### Class meetings

Activity Led by Time Building/Room
Lecture Saul Schleimer Monday 10:00-11:00 B3.02 Zeeman.
Lecture Saul Schleimer Tuesday 10:00-11:00 B3.01 Zeeman.
Lecture Saul Schleimer Thursday 11:00-12:00 B3.01 Zeeman.
Support class Alessandro Bigazzi Thursday 12:00-13:00 B1.12 Zeeman.

### Reference materials

We will closely follow the third chapter of Allan Hatcher's book. Other references include Spanier's book (a standard text), May's book (very concise), Rotman's book (gentle), tom Dieck's book (modern), ...

Here is a link to the discussion forum. You can also find a link to this on the on the module's Moodle page.

### Example sheets and assessed work

See the schedule for the example sheets.

The assessed work will be 15% of your total mark. Exercises will be due every other Friday (at noon) starting in week two. Every sheet will be worth four marks (for a total of 20); your assessed mark will be computed by taking the minimum of (a) your total marks and (b) 15. Please note that the exam questions will be modelled on problems on the example sheets.

There are also many excellent problems in Hatcher's book.

### Exam

The exam will be 85% of your mark. The exam will be closed book. Here are the exam papers for this module from the last five years. Here is last year's exam, written by myself, as well as various observations I made during marking.

And here is this year's exam, also written by myself, as well as various observations I made during marking.

### Mistakes

Please tell me in person, or via email, about any errors on this website or made in class. I am especially keen to hear about mathematical errors, gaffes, or typos made in lecture or in the example sheets.