- The Fundamental Group
- Equivalent definitions of \(\pi_1(X)\)
- The van Kampen theorem
- Computing the fundamental group of CW complexes
- Covering spaces, fundamental theorem

- Homology
- Simplicial, singular, and cellular homology
- Simplicial approximation
- Homology theories
- Classical applications
- Brouwer fixed-point theorem
- Jordan curve theorem and generalization
- \(\pi_1(X)_{ab} \cong H_1(X) \)
- Borsuk-Ulam Theorem
- Invariance of dimension
- Invariance of domain
- Hairy ball theorem
- Degree theory
- Lefschetz fixed-point theorem

- Cohomology
- Simplicial, singular, cellular, de Rham cohomology
- Cohomology with local coefficients, presheaf cohomology
- Cup, cap, and cross products
- Thom isomorphism theorem
- Künneth Formulas
- Poincaré duality, Alexander duality, Lefschetz duality
- Universal Coefficients for Homology
- Cohomology theories

- Homotopy Theory
- Whitehead's theorem
- Cellular approximation theorem, CW approximation theorem
- The Hurewicz theorem
- Fibrations, HELP, long exact sequence
- Homotopy groups of spheres, stable structure, Freudenthal suspension theorem
- Homotopy construction of cohomology

- Generalized cohomology and homology theories
- Spectra, Eilenberg MacLane spectrum
- The Brown Representability Theorem
- Steenrod algebra, cohomology operations

- \(h\)-cobordism theorem
- \( (B,f) \) manifolds and cobordisms
- Computation of \(MO_*\), \(MSO_*\), \(MU_*\)
- Stiefel-Whitney classes, Chern classes, Pontrjagin classes
- Oriented cohomology theories

[DavKir] Davis, J.; Kirk, P.,Lecture Notes in Algebraic Topology[HatAT] Hatcher, A.,Algebraic Topology.[BotTu] Bott, R.; Tu, L.,Differential Forms in Algebraic Topology.[Vic] Vick, J.,Homology Theory.[MayAT] May, J. P.,A Concise Course in Algebraic Topology.[Wes] Weston, T.,An Introduction to Cobordism Theory.[MilSta] Milnor, J.; Stasheff, J.,Characteristic Classes.[Sto] Stong, R.,Notes on Cobordism Theory.[Ada] Adams, J. F.,Stable Homotopy and Generalised Homology.

- Chain complexes of modules
- Mapping cones and cylinders

- Derived functors
- Injective and projective objects for certain Abelian categories
- Injective and projective resolutions for certain Abelian categories
- \( \mathrm{Tor} \), \( \mathrm{Ext} \), \( \lim^1 \),
- Mittag-Leffler condition
- Acyclic assembly lemma

- Universal coefficient theorems
- Spectral sequences
- Serre-Leray spectral sequence for fibrations, ordinary homology and cohomology
- Gysin sequence, Wang sequence
- Exact sequence of low dimensional terms
- Serre exact sequence
- Computation of \( H^{*} \Omega ( \mathbb{S}^n ) \), \( \pi_4 (\mathbb{S}^3) \), \( \pi_4 (\mathbb{S}^2) \)
- Atiyah-Hirzebruch Spectral sequence
- K-theory computations using the AHSS
- Computations with \( MSO_* \)
- Exact couples and derivation of Serre-Leray spectral sequence
- Grothendieck spectral sequence

- Adjoint functors, adjoint functor theorems
- Adjoints and exactness of functors

- (Co-)limits
- Filtered limits and colimits
- (co-)homology of (co-)limits

- Abelian categories
- Examples and properties: \(R\mathrm{-\underline{Mod}}\), \(\mathrm{\underline{Sheaves}}(X)\), \(\mathrm{\underline{Presheaves}}(X)\), \(\mathrm{\underline{Ch}}\), functor categories \(\mathcal{A}^{\mathcal{C}}\)
- Freyd-Mitchell Embedding Theorem

- Simplicial sets
- Kan fibrations
- Yoneda lemma
- Geometric realization

- Monoidal categories
- Model categories

[HS] Hilton, P.; Stammbach, U.,A Course in Homological Algebra.[Hov] Hovey; Shipley; Smith,Symmetric Spectra.[MacCW] Mac Lane, S.,Categories for the Working Mathematician.[Wei] Weibel, C.,An Introduction to Homological Algebra.[MaySO] May, J. P.,Simplicial Objects in Algebraic Topology.[Mit] Mitchell, B.,Theory of Categories.