Algebraic Topology

• Text: Algebraic Topology by Allan Hatcher

  Can be downloaded from Topology Text. This course will be strongly oriented toward the topology of CW complexes and/or simplicial sets.

Note: This course does not count towards computer science electives or numerical analysis. It is a pure mathematics course!

Fourth exam (due Fri, Mar 12, 1999)

• Topological games on the torus and Klein bottle
• Knot Theory Page
• Geometry Junkyard
• Abstract Algebra Home Page
• Homology computations

Course outline:

1. General concepts of topology
   1. Topology as a "geometry" in the sense of Klein
   2. Open sets and continuity
   3. Homeomorphism
   4. Some famous problems
2. Homotopy and Homotopy Type.
   1. Deformation Retractions. Homotopy of Maps. Homotopy Equivalent Spaces. Contractible Spaces.
3. Cell Complexes
   1. Definitions and Examples.
   2. Subcomplexes.
   3. Some Basic Constructions.
   4. Two Criteria for Homotopy Equivalence. The Homotopy Extension Property.
4. Fundamental Group and Covering Spaces
   1. Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms.
   2. Van Kampen's Theorem
   3. Free Products of Groups. Applications to Cell Complexes.
   4. Covering Spaces, Lifting Properties.
   5. The Classification of Covering Spaces.
   6. Deck Transformations and Group Actions.
5. Simplicial and Singular Homology
   1. Delta-Complexes. Simplicial Homology.
   2. Singular Homology. Homotopy Invariance.
   3. Exact Sequences and Excision.
   4. The Equivalence of Simplicial and Singular Homology.
6. Computations and Applications
   1. Degree.
   2. Cellular Homology.
   3. Euler Characteristic.
   4. Split Exact Sequences.
   5. Mayer-Vietoris Sequences.
7. Homology with Coefficients.
8. The Formal Viewpoint Axioms for Homology.
9. Categories and Functors
10. Cohomology
11. Cohomology Groups
    1. The Universal Coefficient Theorem. Cohomology of Spaces.
    2. Cup Product
    3. The Cohomology Ring. External Cup Product. Spaces with Polynomial Cohomology.
    4. Poincare Duality
    5. Orientations. Cap Product. Proof of Poincare Duality. Cup Product and Duality.
12. Homotopy Theory
    1. Homotopy Groups
    2. The Long Exact Sequence. Whitehead's Theorem. The Hurewicz Theorem. Eilenberg-MacLane Spaces.
    3. Homotopy Properties of CW Complexes
    4. Cellular Approximation. Cellular Models. Excision for Homotopy Groups. Stable Homotopy Groups.
    5. Fibrations
13. Obstruction Theory.
    1. The Homotopy Lifting Property. Fiber Bundles. Path Fibrations and Loopspaces. Postnikov Towers.