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Assembling homology classes in automorphism groups of free groups
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Conant, James, Hatcher, Allen, Kassabov, Martin and Vogtmann, Karen (2016) Assembling homology classes in automorphism groups of free groups. Commentarii Mathematici Helvetici, 91 (4). pp. 751-806. doi:10.4171/CMH/402
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Official URL: http://doi.org/10.4171/CMH/402
Abstract
The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out(Fn) and Aut(Fn) from classes for groups Γk,s, where the Γk,s generalize Out(Fk)=Γk,0 and Aut(Fk)=Γk,1. The symmetric group Gs acts on H∗(Γk,s) by permuting leaves, and for trivial rational coefficients we compute the Gs-module structure on H∗(Γk,s) completely for k≤2. Assembling these classes then produces all the known nontrivial rational homology classes for Aut(Fn) and Out(Fn) with the possible exception of classes for n=7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut(Fn) and Out(Fn) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Algebra, Homological, Graph theory | ||||||
| Journal or Publication Title: | Commentarii Mathematici Helvetici | ||||||
| Publisher: | European Mathematical Society Publishing House | ||||||
| ISSN: | 0010-2571 | ||||||
| Official Date: | 24 October 2016 | ||||||
| Dates: |
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| Volume: | 91 | ||||||
| Number: | 4 | ||||||
| Page Range: | pp. 751-806 | ||||||
| DOI: | 10.4171/CMH/402 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Funder: | Simons Foundation (SF), National Science Foundation (U.S.) (NSF), Royal Society (Great Britain). Wolfson Research Merit Award (RSWRMA) | ||||||
| Grant number: | 30518 (SF), DMS 0900932, 130311 and 1011857 (NSF) | ||||||
| Related URLs: | |||||||
| Open Access Version: |
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