Let \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{T}_{n}$\end{document} be the kernel of the natural map Out(Fn)→GLn(ℤ). We use combinatorial Morse theory to prove that \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{T}_{n}$\end{document} has an Eilenberg–MacLane space which is (2n-4)-dimensional and that \documentclass[12pt]{minimal}
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\begin{document}$H_{2n-4}(\mathcal{T}_{n},\mathbb{Z})$\end{document} is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{T}_{n}$\end{document} is equal to 2n-4 and recovers the result of Krstić–McCool that \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{T}_3$\end{document} is not finitely presented. We also give a new proof of the fact, due to Magnus, that \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{T}_{n}$\end{document} is finitely generated.
