Learning Path — Algebraic Topology & Homotopy Theory
Suggested preparation for Algebraic Topology & Homotopy Theory. Choose an entry point that fits your preparation.
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Entry / bridge route · Additional preparation for research reading · All paths
Entry / bridge route
Proofs & Mathematical Writing
Read quantified statements; write direct, contrapositive, contradiction and induction arguments.
Book of Proof — Richard Hammack. Direct open PDF
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You can explain the quantifiers in a statement and write these forms of proof without relying on a worked solution.
Point-Set Topology
Understand open sets, continuity, products, quotients and compactness.
Topology Without Tears — Sidney A. Morris. Direct open PDF
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You can prove statements using these definitions and produce basic separation or compactness counterexamples.
Linear Algebra
Work with vector spaces, bases, linear maps, kernels, rank and eigenvalues.
Linear Algebra — Jim Hefferon. Open online text / archive
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You can prove rank–nullity, change bases and distinguish diagonalizable maps from general linear maps.
Abstract Algebra & Galois Theory
Use groups, rings, ideals, quotient constructions and field extensions.
Abstract Algebra: Theory and Applications — Thomas W. Judson. Open online text / archive
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You can construct quotient groups and rings correctly and state the assumptions of the isomorphism theorems.
Algebraic Topology & Homotopy Theory
Study fundamental groups, covering spaces, homology and cohomology.
Algebraic Topology — Allen Hatcher. Direct open PDF
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You can compute standard fundamental groups and homology groups; stable homotopy and higher categories require additional bridges.
Research reading: additional tools
Choose according to the particular text, not as an obligation to complete every foundation again.
Homological Algebra
Work with chain complexes, homology, exact sequences and resolutions.
Notes on Basic Homological Algebra — Andrew Baker. Direct open PDF
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You can compute elementary homology and explain kernels, cokernels and exactness; deeper derived-functor gaps may remain.
Category Theory
Understand functors, natural transformations, limits and adjunctions through examples.
Category Theory in Context — Emily Riehl. Direct open PDF
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You can express a universal property and verify a natural transformation or an adjunction in a concrete example.