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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. $\endgroup$
    – mme
    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? $\endgroup$ 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... $\endgroup$
    – mme
    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 $H^1(-,\Bbb Z/2) \to H^2(-,\Bbb Z)$. $\endgroup$
    – mme
    May 27 '15 at 18:29
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    $\begingroup$ The mapping class group is usually defined as diffeos. modulo isotopies. To show equivalence with $Out(\pi_1(M))$, first one may show equivalence with self-homotopy equivalences up to homotopy, and then show that homotopy equivalences up to homotopy are equivalent to diffeomorphisms up to isotopy. $\endgroup$
    – Ian Agol
    May 27 '15 at 21:43
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There is a simple direct proof of the injectivity, which I find extremely elucidating, given in the lecture notes of:

Au, Thomas Kwok-Keung; Luo, Feng; Yang, Tian. Lectures on the mapping class group of a surface. In "Transformation groups and moduli spaces of curves", 21–61, Adv. Lect. Math. (ALM), 16, Int. Press, Somerville, MA, 2011.

available from the page of one of the authors (see section 5.1 there): https://www.math.tamu.edu/~tianyang/lecture.pdf

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    $\begingroup$ This paper uses the definition of the mapping class group as homotopy classes of homeomorphisms (or diffeomorphisms) rather than isotopy classes. Injectivity is fairly easy with this definition. More work is needed to get injectivity for isotopy classes. $\endgroup$ Feb 7 at 15:39

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