Inventiones mathematicae

, Volume 170, Issue 1, pp 1–32 | Cite as

Dimension of the Torelli group for Out(Fn)

  • Mladen Bestvina
  • Kai-Uwe Bux
  • Dan Margalit


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


Homotopy Type Mapping Class Group Morse Function Label Graph Cohomological Dimension 
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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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