§ Algebraic Topology

Topics

Geometric and algebraic preliminaries

• Cell attachments and complexes
• Topological categories and functors

Coverings and the fundamental group

• Homotopies
  • Homotopy equivalence
• Covering maps
  • Lifting properties of covering spaces
• Group actions on topological spaces
• Fundamental groups
  • Induced homomorphism between fundamental groups

Theorems:

• (Theorem) Path and homotopy lifting
• (Theorem) The fundamental group of the circle is isomorphic to the additive group of integers
• (Theorem) The fundamental group of the circle is isomorphic to the additive group of integers
• (Theorem) A group acting on a simply connected space is isomorphic to the fundamental group of its orbits
• (Theorem) Classification of covering maps

Homology

• Free abelian group
• Exact sequences
• Singular chains and singular homology
• Homology of general chain complexes
• Reduced singular chains and reduced homology
• Simplicial chains and simplicial homology
  • Delta-complexes
• Euler characteristic
• Homology with coefficients
• Calculating homology groups
  • Mayer–Vietoris sequences

Theorems:

• (Theorem) The first homology group is isomorphic to the abelianization of the fundamental group for path-connected spaces
• (Theorem) A short exact sequence of chain complexes induces a long exact sequence of homology groups
• (Theorem) The relative homology of a subset is isomorphic to the subset with excision when the closure of the excision is in the interior of the subset
• (Theorem) The inclusion of simplicial chains into singular chains induces an isomorphism in homology

Cohomology

• Cohomology of spaces
• The cup product of cochains