Homotopy theory, fall 2015

Last modified by elfving@helsinki_fi on 2024/03/27 10:46

Homotopy theory, fall 2015


Teacher: Erik Elfving 

Scope: 10 cr

Type: Advanced studies

Teaching: Lectures and exercises

Topics: Fundamental group, applications, covering spaces, higher homotopy groups, fibrations and cofibrations

Prerequisites: Topology II, a first course in abstract algebra

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  •  The course starts on Wednesday, 2nd of September (no lecture on Monday 31st of August). On Thursday, 3rd of September, there will be an extra lecture (12-14, B321).

Teaching schedule

Weeks 36-42 and 44-50, Monday 10-12 and Wednesday 14-16 in room C124. In addition, two hours of exercise classes per week. Guidance in Ratkomo (3rd floor corridor) on Wednesdays 10-12.


One exam at the end of the course, on Monday 21st of December, 10-14, auditorium A111.

Course material

Lecture notes (in Finnish):htteoria.pdf


Metsänkylä-Näätänen: Algebra, p. 92-98:D4.pdf

In English: W. S. Massey: Algebraic topology: an introduction; A. Hatcher: Algebraic topology; E. H. Spanier: Algebraic topology

Preliminary schedule:

  1. Homotopy of mappings; path homotopy; fundamental group; covering map; the fundamental group of a circle; examples.
  2. Applications: Brouwer's fixed point theorem and Borsuk-Ulam theorem (in dimension 2); fundamental theorem of algebra.
  3. More about covering spaces; monodromy theorems; connection with the fundamental group; classification of covering maps; universal covering space.
  4. Higher homotopy groups; examples of homotopy groups of spheres; the Hopf mapping S^3 -> S^2; Freudenthal suspension.
  5. Relative homotopy groups; exact homotopy sequences.


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Exercise classes










Erik Elfving

Course feedback

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