CERF THEORY for GRAPHS Allen Hatcher and Karen Vogtmann
CERF THEORY FOR GRAPHS Allen Hatcher and Karen Vogtmann ABSTRACT. We develop a deformation theory for k-parameter families of pointed marked graphs with xed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), compu- tations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn). §1. Introduction It is standard practice to study groups by studying their actions on well-chosen spaces. In the case of the outer automorphism group Out(Fn) of the free group Fn on n generators, a good space on which this group acts was constructed in [3]. In the present paper we study an analogous space on which the automorphism group Aut(Fn) acts. This space, which we call An, is a sort of Teichmuller space of nite basepointed graphs with fundamental group Fn. As often happens, specifying basepoints has certain technical advantages. In particu- lar, basepoints give rise to a nice ltration An,0 An,1 of An by degree, where we dene the degree of a nite basepointed graph to be the number of non-basepoint vertices, counted “with multiplicity” (see section 3). The main technical result of the paper, the Degree Theorem, implies that the pair (An, An,k)isk-connected, so that An,k behaves something like the k-skeleton of An. From the Degree Theorem we derive these applications: (1) We show the natural stabilization map Hi(Aut(Fn); Z) → Hi(Aut(Fn+1); Z)isan isomorphism for n 2i + 3, a signicant improvement on the quadratic dimension range obtained in [5].
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