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