Math 581-1 – Algebraic Topology
Course Description
Algebraic
topology uses techniques from abstract algebra to study
how (topological) spaces are connected. Most often,
the algebraic structures used are groups
(but more elaborate structures such as rings or modules also
arise). A typical approach projects continuous maps between
topological spaces onto homomorphisms between the
corresponding groups. This course will introduce basic
concepts of algebraic topology at the first-year graduate
level.
We will follow mostly the book Elements
of Algebraic Topology by James R.~Munkres, and cover in
a fair bit of detail the topics on homology of simplicial
complexes, relative homology, cohomology, and the basics of
duality in manifolds (selected Sections from Chapters
1–5 and 8). We will also refer to the
book Algebraic
Topology by Allen Hatcher for some of the topics. But
unless there is strong interest, we will not cover topics
related to the fundamental group. We will stress geometric
motivations as well as applications (where relevant)
throughout the course.
Prerequisites: Some background in general topology as
well as abstract algebra, both at the undergraduate level,
will be assumed. In particular, familiarity with the
concepts of continuous functions, connectedness, and
compactness, as well as with the concepts of groups,
homomorphisms, fields, and vector spaces will be helpful to
follow the course. But no particularly deep theorems from
these topics will be needed. Some flexibility could be
afforded as far as this background is concerned –
please contact the instructor if you have doubts.
Syllabus (Updated 09/07/2017)
Announcements
Tue, Aug 08: | The class will meet TTh 1:25-2:40 pm in Spark 233 (Pullman) and VECS 120 in Vancouver. |
Wed, Sep 13: | Homework 3 is now due on Tuesday, Sep 19. |