Rational homology of $\text{\rm Aut}$($F_n$)

Cohomological structure of the mapping class group and beyond

“…On the other hand, the first non-trivial rational cohomology of the group Aut F n was given by Hatcher and Vogtmann [36]. They showed that, up to cohomology degree 6, the only non-trivial rational cohomology is…”

“…Responding to an inquiry of the author, Vogtmann communicated us that she modified the argument in [36] to obtain an isomorphism H 4 (Out F 4 ; Q) ∼ = Q. Thus we could conclude that µ 1 is the generator of this group (see [80][106]).…”

Cutkosky Rules from Outer Space

“…We follow [3]. Consider a pair (Γ, T ) of a bridge free graph Γ and a chosen spanning tree T for it.…”

“…For each of the three spanning trees we get a cell as in (5.1). The boundary operator for such a cell in the cubical cell complex of [3] is the obvious one stemming from co-dimension one hypersurfaces at 0 or 1 with suitable signs. So the square populated by the triangle ∆ in (5.1) has four boundary components, the edges populated by the four graphs as indicated.…”

“…Its further study in the conext needed for physics will inform our understanding of amplitudes considerably. Future work will be dedicated in understanding the relation between the monodromy of physical singularities and the fundamental group underlying Outer Space as used in [3].…”