Last week we constructed a homomorphism from the natural numbers to the fundamental group of the circle. In Week 4 we will see that this homomorphism is bijective: every loop is homotopic to a loop in the image of this homomorphism, and if a loop is homotopic to the constant loop, then it is the constant loop. To prove this, we will have to study the Homotopy Lifting Property of coverings, the fact that one can lift homotopies from the base of a covering to the covering space.
After this week, you should be able to
- prove that the fundamental group of the circle is Z
- describe homotopy lifting of paths using the example of complex squaring
- show how to construct a homotopy lifting of a covering by pasting locally constructed liftings
Tasks and Materials