
620634
Algebraic Topology

Lecturer: Craig Westerland, 168 Richard Berry, phone: 8344 9712,
email: C.Westerland@ms.unimelb.edu.au
Time and Location: Mon 1112, Th 1011, Fri 1112, 215 Richard Berry.
Subject Outline
In topology, one explores the properties of geometric objects that are
left invariant under deformation. For instance, while the size
and curvature of a balloon can be changed with a puff of air, one
cannot turn it into two balloons without doing it a great deal of
violence. In this setting, the traditional means by which we
distinguish geometric objects (such as curvature, volume, and
geodesics) are less useful. Algebraic topology provides a
different set of tools for this purpose. A common aspect of these
invariants are that they are less numerical (as is the case in
geometry) and more algebraic  we find ourseleves working with groups,
rings, and categories.
In this course, students will become familiar with the basic notions of
algebraic topology: homotopy and homotopy groups, homology and
cohomology groups, categories, functors, and methods of
computation. Depending upon students' needs and interests, we
will review the fundamental group and covering spaces. If time
permits, we will explore fibre bundles and characteristic classes.
Main Topics
The material covered will be drawn from the following:
 Homotopy of maps
 The fundamental group and higher homotopy groups
 Seifertvan Kampen theorem
 Covering spaces
 Homology and cohomology groups
 Simplicial complexes
 Basic homological algebra
 Exact sequences and computations
 Fixedpoint theorems
 CW complexes and cellular homology
 Categories and functors
 EilenbergSteenrod axioms
 Cup product
 Manifolds: orientability and Poincare duality
 Fibre bundles and basics of characteristic classes.
References
 A. Hatcher, Algebraic
Topology, Cambridge University Press, 2002 ISBN: 0521795400,
available online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html.
Assessment
Up to 60 pages of written assignments (75%: three assignments worth
25% each, due early, mid and late in semester), a twohour written
examination (25%, in the examination period).
The plaigiarism declaration is available
here