Theoretical / proof-based
Algebraic topology · Graduate · Math
Learning objectives from the Algebraic topology syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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
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)
Definitions and structure
- 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
Proofs and reasoning
- Applications to fixed-point — Applications to fixed-point
- Brouwer theorems (introduction) — Brouwer theorems (introduction)
- Singular homology — Singular homology
- Functoriality — Functoriality
- Eilenberg–Steenrod axioms (overview) — Eilenberg–Steenrod axioms (overview)
Abstraction and generalization
- 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
Modeling and computation
- Apply homotopy of maps — Apply homotopy of maps; homotopy equivalence in engineering contexts
- Apply fundamental group: computation for cw complexes and surfaces in... — Apply fundamental group: computation for cw complexes and surfaces in engineering contexts
- Apply covering spaces in engineering contexts — Apply covering spaces in engineering contexts
- Apply galois correspondence in engineering contexts — Apply galois correspondence in engineering contexts
- Apply seifert–van kampen theorem in engineering contexts — Apply seifert–van kampen theorem in engineering contexts
Data and technology
- Use software to explore algebraic topology problems numerically — Use software to explore algebraic topology problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Applications to fixed-point — Applications to fixed-point
- Brouwer theorems (introduction) — Brouwer theorems (introduction)
- Singular homology — Singular homology
- Functoriality — Functoriality
- Eilenberg–Steenrod axioms (overview) — Eilenberg–Steenrod axioms (overview)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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