Information about the Final Exam for Math 6823-001

Mathematics 6823-001 - Algebraic Topology II - Spring 2009

Information about the Final Exam

The Final Exam will be in Room 1105 on Wednesday, May 13 from 1:30 to 3:45. As of this writing, it has eleven problems and 72 points possible. As on any exam, it is wise to start with the problems that you feel confident that you know how to do, before moving on to others.

Some of the exam is just definitions and verifying basic facts. It will not be necessary to know complicated proofs of theorems (although working through their proofs is a good way to absorb the definitions and basic facts, as well as seeing some great ideas). Of course it might be useful to understand how to work the problems that appeared on the midterm exam.

The following topics are likely to appear, although the exam is not limited to these topics:

  1. Homology and cohomology, calculation using cellular chains, calculation using the Mayer-Vietoris sequence.
  2. The Universal Coefficient Theorem and its use in calculating cohomology. Know the statement of the Universal Coefficient Theorem, but you do not need to be able to calculate Ext groups from the definition or to memorize specific Ext groups.
  3. Euler characteristic.
  4. Cup and cap product, their definitions and basic properties.
  5. Orientation, the two-fold orientation cover.
  6. Poincare Duality, its statement and application, fundamental classes.
  7. Higher homotopy groups, their definition from the Iⁿ viewpoint and from the Sⁿ viewpoint, the action of the fundamental group on the higher homotopy groups.

The following topics do not appear, at least not explicitly: invariance of simplicial homology from the triangulation, local degree, categories and Eilenberg-Steenrod axioms, free resolutions, the formulation of Poincare Duality using cup products.

1. Homology and cohomology, calculation using cellular chains, calculation using the Mayer-Vietoris sequence.
2. The Universal Coefficient Theorem and its use in calculating cohomology. Know the statement of the Universal Coefficient Theorem, but you do not need to be able to calculate Ext groups from the definition or to memorize specific Ext groups.
3. Euler characteristic.
4. Cup and cap product, their definitions and basic properties.
5. Orientation, the two-fold orientation cover.
6. Poincare Duality, its statement and application, fundamental classes.
7. Higher homotopy groups, their definition from the Iⁿ viewpoint and from the Sⁿ viewpoint, the action of the fundamental group on the higher homotopy groups.