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 \(H_{nk}(M)\) 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.

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.

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.
