The topics of this course will be homology, cohomology, and duality. The course will focus on examples from low-dimensional topology such as knots and 3-manifolds to illustrate the importance of these algebraic invariants.
The textbook for this course is Algebraic Topology by Allen Hatcher. It is available to download here, however it's probably worth the price to buy it already printed and bound. We will cover chapters 2 and 3 of this text.
If you are interested in supplementing the textbook, two additional relevant texts are Lecture Notes in Algebraic Topology by James F. Davis and Paul Kirk (which is available directly from the AMS) and A Course in Homological Algebra by P.J. Hilton and U. Stammach.