# Homotopy theory, fall 2015

# 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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## News

- 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.

## Exams

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

## Course material

Lecture notes (in Finnish):

Liite:

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

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

Preliminary schedule:

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

## Registration

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## Exercises

### Assignments

- Exercise 1 (10.9.2015): ,
- Exercise 2 (17.9.2015):
- Exercise 3 (24.9.2015): ,
- Exercise 4 (1.10.2015): ,
- Exercise 5 (8.10.2015): ,
- Exercise 6 (15.10.2015):
- Exercise 7 (29.10.2015): ,
- Exercise 8 (5.11.2015): ,
- Exercise 9 (12.11.2015): ,
- Exercise 10 (19.11.2015): ,
- Exercise 11 (26.11.2015): ,
- Exercise 12 (3.12.2015): ,
- Exercise 13 (10.12.2015): ,

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Thursday | 12-14 | B321 | Erik Elfving |

## Course feedback

Course feedback can be given at any point during the course. Click here.