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620-634
Algebraic Topology

Lecturer: Craig Westerland, 168 Richard Berry, phone: 8344 9712, email: C.Westerland@ms.unimelb.edu.au

Time and Location: Mon 11-12, Th 10-11, Fri 11-12, 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
  • Seifert-van Kampen theorem
  • Covering spaces
  • Homology and cohomology groups
  • Simplicial complexes
  • Basic homological algebra
  • Exact sequences and computations
  • Fixed-point theorems
  • CW complexes and cellular homology
  • Categories and functors
  • Eilenberg-Steenrod 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: 0-521-79540-0, 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 two-hour written examination (25%, in the examination period).

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