# 1.07 Induced maps

## Video

Below the video you will find accompanying notes and some pre-class questions.

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## Notes

### Induced maps

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*.

*(1.16)* Given a continuous map \(F\colon X\to Y\), we get a map
\(\Omega_xX\to\Omega_{F(x)}Y\) which sends a loop \(\gamma\) based
at \(x\) to the loop \(F\circ\gamma\) based at \(F(x)\). (Recall
that \(\Omega_xX\) is the set of loops in \(X\) based at
\(x\)).

*(2.10)* This map \(\gamma\mapsto F\circ\gamma\) descends to a
well-defined homomorphism
\(F_*\colon\pi_1(X,x)\to\pi_1(Y,F(x))\). Moreover, if \(G\colon Y\to
Z\) is another continuous map then \((G\circ F)_*=G_*\circ F_*\).

*(3.50)* The identity \((G\circ F)_*[\gamma]=G_*(F_*(\gamma))\) is
clear on the level of loops: it simply says \[(G\circ
F)\circ\gamma=G\circ(F\circ\gamma).\]

*(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 \(F_*\)

*(5.08)* If \(F\colon X\to Y\) is a *homeomorphism* (continuous
bijection with continuous inverse) then \(F_*\) is an isomorphism.

*(5.58)* The homomorphism \((F^{-1})_*\) is an inverse for \(F_*\),
because \[(F^{-1})_*\circ F_*=(F^{-1}\circ F)_*=id_*,\] which is the
identity on \(\pi_1(X,x)\).

*(6.30)* In fact, the fundamental group is invariant under a much
wider set of equivalences called *homotopy equivalences*. See the
videos on homotopy equivalence (1.09) and homotopy invariance (1.10).

## Pre-class questions

- Suppose that \(\gamma_s\) is a homotopy. Show that \(F_*([\gamma_0])=F_*([\gamma_1])\) (i.e. that \(F_*\) is well-defined).

## Navigation

- Previous video:
**1.06 Fundamental theorem of algebra: reprise**. - Next video:
**1.08 Brouwer's fixed point theorem**. - Index of all lectures.