If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class group to the induced outer automorphism of $\pi_1$?

I naturally looked in Farb-Margalit’s Primer on Mapping Class Groups for something like this. But I do not understand their argument. They appear to assert (Chapter 8, p. 220) that since $S$ is a $K(\pi,1)$, there is a bijection between homotopy classes of continuous maps from $S$ to itself and homomorphisms from $\pi_1(S)$ to itself, up to homotopy.

I do not understand why this should be true. There is a common belief that if two maps between CW complexes induce the same map on all homotopy groups, then they are homotopic; this is false. See, eg, here: Maps inducing zero on homotopy groups but are not null-homotopic . But maybe something like this is true for homotopy equivalences?

I do find it asserted several other places that if a manifold $M$ is a $K(\pi,1)$, then its homotopy mapping class group is the same as $\mathrm{Out}(\pi_1(M))$, for instance:

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    $\begingroup$ @DylanThurston: The generally false belief is true for based homotopy classes of maps $K(\pi, 1) \to K(\pi',1)$, which are in bijection with homomorphisms $\pi \to \pi'$. The proof Ryan alluded to is Hatcher's Theorem 1B.9. This theorem immediately implies that the homomorphisms induced by homotopy equivalences are precisely the automorphisms, and we pass to $\text{Out}(\pi,1)$ when we forget about the basepoint. $\endgroup$ – Mike Miller May 27 '15 at 17:51
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    $\begingroup$ Thanks! I was looking in completely the wrong section of Hatcher. I guess the generally false belief is true more generally as long as the domain is a CW complex and the target is a $K(\pi, n)$, right? $\endgroup$ – Dylan Thurston May 27 '15 at 18:00
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    $\begingroup$ I think you need some connectivity conditions on the domain - it seems to me there is a nontrivial map $K(\Bbb Z_2,1) \to K(\Bbb Z_2, 2)$ given by the cup square. (It is a theorem that there is a bijection between homotopy classes of maps $K(\pi, n) \to K(\pi', m)$ and natural transformations between the Set-valued functors $H^n(-, \pi) \to H^m(-,\pi')$. The cup square represents a nontrivial natural transformation because it does so on $\Bbb{RP}^2$, say.) If you assume the domain is $(n-1)$-connected you can prove what you want by exploiting the isomorphisms... $\endgroup$ – Mike Miller May 27 '15 at 18:27
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    $\begingroup$ $[X,K(G,n)] = H^n(X;G) = \text{Hom}(H_n(X;\Bbb Z);G) = \text{Hom}(\pi_n(X);G)$. For a more down-to-earth example of a map $X \to K(\pi,n)$ that's not null-homotopic but kills $\pi_n$ we could take the fibration $S^1 \to \Bbb{RP}^\infty \to \Bbb{CP}^\infty$ obtained by considering $\Bbb{RP}^\infty$ as a quotient of the infinite "odd-dimensional sphere" and extending this to a quotient by $S^1$. If this map was null-homotopic so would be the identity map of $\Bbb{RP}^\infty$. This should correspond to the Bockstein $H^1(-,\Bbb Z/2) \to H^2(-,\Bbb Z)$. $\endgroup$ – Mike Miller May 27 '15 at 18:29
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    $\begingroup$ Thanks, everyone. I wish authors would spell this out a bit more. $\endgroup$ – Dylan Thurston May 27 '15 at 18:32

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