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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). Another popular book
is Algebraic
Topology by Allen Hatcher which could be
used as a reference. We will not have the time
to 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.
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