Algebraic Topology I: Homology (Fall 2015)

This course is taught jointly by Ieke Moerdijk and Javier J. Gutiérrez.
The teaching assistant for this course is Joost Nuiten.

  • Time and place: Fridays 12:45-15:30, Radboud Universiteit Nijmegen, room HG03.085 (Huygensgebouw).
    Week 39: The lecture will take place on Wednesday 23rd, 10:30-12:30, room HG00.108
  • Prerequisites: Point set topology and algebra.

Course description >

This course offers an introduction to algebraic topology, that is, the study of topological spaces by means of algebra. The first part of the course focuses on homology theory. Singular homology groups are algebraic invariants of spaces: for every space there are such groups and every map of spaces induces a map between the corresponding groups. These invariants turn out to be rather computable, and they allow for some immediate geometric applications. We will establish some key properties of these homology groups like the homotopy invariance and the excision theorem.

A convenient variant is provided by singular homology with coefficients --a framework which makes necessary a short discussion of basic homological algebra including tensor and torsion products. For CW-complexes, there is also the more combinatorial cellular homology theory. The course culminates in a proof that singular homology and cellular homology agree on CW-complexes. This allows for more explicit calculations in examples of interest (e.g., projective spaces).

Lecture notes >

Exercise sheets >

Evaluation >

Wednesday 20th January, room HG03.054 (Huygensgebouw)
  • 10:45-11:45 Eilenberg-Steenrod axioms. Uniqueness of singular homology (Tamira Hoeks and Lisa Noorlander)
  • 12:00-13:00 Homology of product spaces. Künneth theorem (Fokke de Haan and Luuk Verhoeven)
  • 14:00-15:00 Simplicial homology (Erik Bosch and Sandra van Dijk)
  • 15:15-16:15 Cohomology and the universal coefficients theorem (Krijn Reijnders and Carmen van Schoubroeck)

Oral exam
Friday 22nd January, room HG03.722 (Huygensgebouw)
  • 09:30-10:00 Tamira Hoeks
  • 10:10-10:40 Carmen van Schoubroeck
  • 10:50-11:20 Erik Bosch
  • 11:30-12:00 Sandra van Dijk
  • 13:00-13:30 Fokke de Haan
  • 13:40-14:10 Luuk Verhoeven
  • 14:20-14:50 Krijn Reijnders
  • 15:00-15:30 Lisa Noorlander

References >

  • A. Hatcher, Algebraic Topology. Cambridge University Press, 2002.
  • W.S. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, 1997.
  • J.R. Munkres, Elements of Algebraic Topology. Westview Press, 1993.
  • J.J. Rotman, An Introduction to Algebraic Topopogy. Springer-Verlag, 1998.
  • E.H. Spanier, Algebraic Topopogy. Springer-Verlag, 1966.

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