Integral Euler characteristic of Out F_{11}

Morita, Shigeyuki; Sakasai, Takuya; Suzuki, Masaaki

doi:10.48550/arxiv.1405.4063

Cited by 1 publication

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“…However, even if they are nonzero it seems that they account for only a small fraction of the homology. The Euler characteristic for H * Out(F n ) was computed for n ≤ 11 by Morita, Sakasai, and Suzuki in [35,36], and after starting with the values 1 and 2 for n ≤ 8, it becomes −21, −124, −1202 for n = 9, 10, 11. If this trend continues for larger n, it would say there are many odd-dimensional classes for Out(F n ), though the only one discovered to date is the 11-dimensional class in Out(F 7 ) recently found by Bartholdi [1].…”

Section: Introductionmentioning

confidence: 99%

Assembling homology classes in automorphism groups of free groups

Conant¹,

Hatcher²,

Kassabov³

et al. 2016

Comment. Math. Helv.

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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 Γ k,s , where the Γ k,s generalize Out(F k ) = Γ k,0 and Aut(F k ) = Γ k,1 . The symmetric group Ss acts on H * (Γ k,s ) by permuting leaves, and for trivial rational coefficients we compute the Ss-module structure on H * (Γ k,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.

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Section: Introductionmentioning

confidence: 99%