1.07 Induced maps
Below the video you will find accompanying notes and some pre-class questions.
- Previous video: 1.06 Fundamental theorem of algebra: reprise.
- Next video: 1.08 Brouwer's fixed point theorem.
- Index of all lectures.
Given a continuous map \(F\colon X\to Y\), we get a homomorphism \[F_*\colon\pi_1(X,x)\to\pi_1(Y,F(x)),\] called the induced map or pushforward map.
(4.19) This lemma expresses the fact that \(\pi_1\) is a functor: not only does it give us a group for each space, it also gives us a homomorphism for each continuous map, and composition of continuous maps corresponds to composition of homomorphisms. This allows us to translate many topological problems into pure algebra.
We will prove the lemma in class and in the pre-class questions.
Properties of \(F_*\)
- Suppose that \(\gamma_s\) is a homotopy. Show that
\(F_*([\gamma_0])=F_*([\gamma_1])\) (i.e. that \(F_*\) is