Mathematics 261: Algebraic Topology I (Spring 2003)

Instructor

Description

• Introductory ideas
  • Basic ideas of category theory
• Homotopy
  • Homotopy of maps
  • Fundemental group
  • Cell complexes
  • Van Kampen's Theorem
  • Covering spaces
  • Higher homotopy groups (very briefly)
• Homology
  • Chain complexes
  • Simplicial homology
  • Singular homology
  • Relative homology
  • Homotopy invariance
  • Excision
  • Mayer-Vietoris Sequence
  • Cellular Homology
  • Eilenberg-Steenrod Axioms

Algebraic topology studies topological spaces by associating to them algebraic invariants. The principal algebraic invariants considered in this course are the fundamental group (also known as the first homotopy group) and the homology groups. This course is a prerequisite for Math 262 (Algebraic Topology II). It is fundamental for students interested in research in Algebraic Geometry, Differential Geometry, Mathematical Physics, and Topology; it is also important for students in Algebra and in Number Theory.

Prerequisites

Text(s)

It may also be useful to refer to

1. M. Greenberg and J. Harper, Algebraic Topology: A First Course, Addison-Wesley 1981.
2. Edwin H. Spanier, Algebraic topology, Springer-Verlag 1966.
3. William S. Massey, Algebraic topology: an introduction, Springer-Verlag 1977.
4. William S. Massey, A basic course in algebraic topology, Springer-Verlag 1991.
5. William S. Massey, Introduction to homology theory, Yale 1977.
6. William S. Massey, Singular homology theory, Springer-Verlag 1980.
7. James R. Munkres, Topology: A First Course, Prentice Hall, 1974.
8. William Fulton, Algebraic Topology: A First Course, Srpinger-Verlag 1995.

Last modified: 17 October 2001