This course will discuss important concepts in algebraic topology such as homology, cohomology, and duality, and then introduce homology with local coefficients. The course will focus on examples from low-dimensional topology such as knots and 3-manifolds to illustrate the importance of these algebraic invariants.
If you decide to buy one book for this course, I would suggest buying Lecture Notes in Algebraic Topology by James F. Davis and Paul Kirk. It is available directly from the AMS or from the campus bookstore.
Another book which is a good reference for the basics of algebraic topology 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 from the campus bookstore.
If you want to learn more about knots and 3-manifolds, here are some good books to check out:
A goal for the end of this course is to discuss some of the results in the following papers:
Another (perhaps lofty) goal is to discuss some of the results in the following paper: