Algebraic topology
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Algebraic Topology — scope drawn from open algebraic topology notes and typical US graduate algebraic topology syllabi.
Homotopy and fundamental group
- Homotopy of maps — Homotopy of maps; homotopy equivalence
- Fundamental group: computation for CW complexes and surfaces — Fundamental group: computation for CW complexes and surfaces
- Covering spaces — Covering spaces
- Galois correspondence — Galois correspondence
- Seifert–van Kampen theorem — Seifert–van Kampen theorem
- Applications to fixed-point — Applications to fixed-point
- Brouwer theorems (introduction) — Brouwer theorems (introduction)
Homology theory
- Singular homology — Singular homology
- Functoriality — Functoriality
- Eilenberg–Steenrod axioms (overview) — Eilenberg–Steenrod axioms (overview)
- Mayer–Vietoris sequence — Mayer–Vietoris sequence
- Long exact sequences — Long exact sequences
- Cellular homology — Cellular homology
- Computations for CW complexes — Computations for CW complexes
- Degree theory — Degree theory
- Invariance of domain (introduction) — Invariance of domain (introduction)
Cohomology and duality
- Singular cohomology — Singular cohomology
- Universal coefficient theorem (statement) — Universal coefficient theorem (statement)
- Cup product — Cup product
- Ring structure on cohomology — Ring structure on cohomology
- Poincaré duality for closed oriented manifolds — Poincaré duality for closed oriented manifolds
- Künneth formula (introduction) — Künneth formula (introduction)
- Introduction to spectral sequences (optional) — Introduction to spectral sequences (optional)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Homotopy and fundamental group
Homotopy of maps; homotopy equivalence
Coming soon - Homology theory
Singular homology
Coming soon - Cohomology and duality
Singular cohomology
Coming soon
Notes
Graduate Algebraic Topology — scope drawn from open algebraic topology notes and typical US graduate algebraic topology syllabi. Topic outline: `content/topics/graduate/algebraic_topology.json`.