Summary of Standard Topics

  • Algebraic topology topics:
    • Classification of compact surfaces
    • Euler characteristic
    • Connect sum
    • Homology and cohomology groups
    • Fundamental group
    • Singular/cellular/simplicial homology
    • Mayer-Vietoris long exact sequences for homology and cohomology
    • Diagram chasing
    • Degree of maps from \(S^n \to S^n\)
    • Orientability, compactness
    • Top-level homology and cohomology
    • Reduced homology and cohomology
    • Relative homology
    • Homotopy and homotopy invariance
    • Deformation retract
    • Retract
    • Excision
    • Kunneth formula
    • Factoring maps
    • Fundamental theorem of algebra
  • Algebraic topology theorems:
    • Brouwer fixed point theorem
    • Poincaré lemma
    • Poincaré duality
    • de Rham theorem
    • Seifert-van Kampen theorem
  • Covering space theory topics:
    • Covering maps
    • Free actions
    • Properly discontinuous action
    • Universal cover
    • Correspondence between covering spaces and subgroups of the fundamental group of the base.
    • Lifting paths
    • Homotopy lifting property
    • Deck transformations
    • The action of the fundamental group
    • Normal/regular cover