1.07 Induced maps

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


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

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


Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

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