Aut（F_n）的Birman exact sequence
A Birman exact sequence for Aut(F_n)
摘要：
Birman正合列描述了具有胶合圆盘边界的曲面对边界分量的映射类群的影响。我们构造了自由群的自同构群的一个类似的正合列。对于映射类群，Birman正合列的核是曲面辫群.我们证明了在自由群的自同构群的上下文中，自然核是非线性生成的。然而，它是不可分的，事实上，我们证明了它的第二个合理的同调群有无限的秩，通过构建一个明确的无限集合的线性无关阿贝尔循环。我们还确定了内核的阿贝尔化并为其构建了一个简单的无限表示。我们许多证明的关键是约翰逊同态的几个新推广。
英文摘要：
The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The key to many of our proofs are several new generalizations of the Johnson homomorphisms.
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