The stable cohomology of automorphisms of free groups with coefficients in the homology representation
The stable cohomology of automorphisms of free groups with coefficients in the homology representation
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同调表示中带有系数的自由群自同构的稳定上同调
DOI：
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发表时间：
2010
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通讯作者：
O. Randal
中科院分区：
文献类型：
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作者：
O. Randal
本文研究了Aut（F_n）和Out（F_n）的上同调，其系数分别位于模wedge^q H，wedge H^*，Sym^q H或Sym^q H^* 中，其中H是Out（F_n）-模，通过交换自由群F_n得到。由于概念上不清楚的原因，取H及其相关模中的系数比取H^* 及其相关模中的系数要简单得多。基于单连通背景空间所标记的图空间的有理同调稳定性定理，给出了这类群在低度下的稳定积分计算，并进一步模猜想了这类群在所有度下的稳定有理计算.
We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules wedge^q H, wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not conceptually clear, taking coefficients in H and its related modules behaves in a far less trivial way than taking coefficients in H^* and its related modules. Based on a conjectural homology stability theorem for spaces of graphs labeled by a simply connected background space, we give a stable integral calculation of these groups in low degrees, and modulo a further conjecture a stable rational calculation in all degrees.