Dimension of the Torelli group for Out(Fn)

摘要

Let $mathcal{T}_{n}$ be the kernel of the natural map Out(Fn)→GLn(?). 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 Krsti?–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.