It is not known whether or not the stable rational cohomology groups H*(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that H5(Q'm; Z), H6(Q'm; Z), and H5(Qm; Z) never stabilize as m → ∞, where the moduli spaces Q'm and Qm are the quotients of the spines X'm and Xm of “outer space” and “auter space”, respectively, introduced in [3] by Culler and Vogtmann and [6] by Hatcher and Vogtmann.
@article{urn:eudml:doc:41445,
title = {Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs.},
journal = {Publicacions Matem\`atiques},
volume = {46},
year = {2002},
pages = {97-118},
mrnumber = {MR1904858},
zbl = {1061.20033},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41445}
}
Jensen, Craig A. Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs.. Publicacions Matemàtiques, Tome 46 (2002) pp. 97-118. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41445/