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.

**Learning outcomes**

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**

- Have a look at the lecture notes when they appear.
- Read pages 29-30 and 60 in Hatcher’s book.
- Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 5

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