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Algebraic Topology

This course is taken in sequence, part 1 in the fall, and part 2 in the spring.


  • Basic notions of differentiable manifolds (see Chapter 0 and 1 of do Carmo’s Riemannian Geometry).
  • Fundamental group, van Kampen’s theorem, covering spaces.
  • Commutative algebra of PIDs (free modules, classification of finitely generated modules).

Algebraic Topology I


Part I: Foundations of differential topology and homotopy theory

  • Basic properties of higher homotopy groups
  • Smooth manifolds, regular values and transversality
  • Vector fields and the Poincaré-Hopf theorem
  • Pontryagin construction
  • Morse functions
  • Topology of CW complexes
  • Whitehead theorem

Part II: Homology and cohomology

  • Homological algebra of chain complexes
  • Singular and cellular homology
  • The universal coefficient theorem
  • Hurewicz theorem
  • Cohomology and cup products
  • Poincaré duality


Part I: Topology of fibrations

  • Loopspaces
  • Homology with local coefficients
  • Obstruction theory
  • Eilenberg MacLane spaces
  • Filtered chain complexes and the associated spectral sequence
  • The Serre spectral sequence of a fibration
  • Multiplicative structures
  • Applications to the computation of homotopy groups of spheres

Part II: Characteristic classes

  • Classifying spaces of vector bundles
  • Cohomology of Grassmannians
  • Stiefel-Whitney, Chern, Pontryagin and Euler classes
  • Thom spaces
  • Computations of bordism groups
  • Hirzebruch’s signature formula and Milnor’s exotic 7-spheres

Part III (time permitting): Further topics in Geometric Topology

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