This week we derive the most important property of the fundamental group, the one that justifies the term *fundamental*: **homotopy invariance**. This means that homotopy equivalent spaces give rise to isomorphic fundamental groups. We will then list a few important applications, including the celebrated Brouwer Fixed Point Theorem, the No-Retract Theorem, and seemingly unrelated results about the eigenvalues of matrices!

**Learning outcomes**

After this week, you should be able to

- prove that the fundamental group is a homotopy invariant
- describe the effect of retractions and continuous maps on the fundamental group
- state and prove the Brouwer Fixed Point Theorem and the No-Retract Theorem

**Tasks and Materials**

- Have a look at the lecture notes when they appear.
- Read pages 31-34 and 36 in Hatcher’s book.
- Read Chapter 1 of Using Brouwerâ€™s fixed point theorem by Björner, Matoušek and Ziegler
- Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 6

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