Quasirationality and prounipotent crossed modules

Abstract

We study quasirational presentations (QR-presentations) of (pro-p)groups, which contain aspherical presentations and their subpresentations, and also still mysterious prop-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to certain prounipotent crossed module, we study cohomological properties of pro-p-groups with a single defining relation (a question of J.-P. Serre).

Rational and $p$-adic analogs of J.H.C. Whitehead's conjecture

“…[18, A.2. ], [34,2] Let us fix a group G and a field k of zero characteristics, define the prounipotent completion of G as the following universal diagram, in which ρ is a Zariski dense homomorphism from G to the group of k-points of a prounipotent affine group scheme G ∧ u :…”

“…Bousfield and Kan viewed the rational case as "often take care of itself" [8, p.2], in contrast, recent results from [34], [36] show that this is not the case in twodimensional homotopy. For example, one-relator presentations over fields of zero characteristics have prounipotent cohomological dimention two i.e.…”

“…In contrast to our previous works [36], [34] we use right topological modules, as they are better compatible with the existing topological literature [24], [9].…”

“…Besides homotopy theory importance prounipotent groups are especially cute since they have presentations theory perfectly similar to the discrete case, modules over such affine group schemes are simple enough to get deep results of cohomological nature. We refer the reader to [29], [30], [31] for pioneering papers on prounipotent groups presentations to classical results on presentations of pro-pgroups [49], [27], [44], [40], recent result of the author scattered across the articles [34], [36], [33], [35].…”

“…In [34], [36] the notion of a prounipotent presentation of finite type was extended to prounipotent groups over non-algebraically closed fields of characteristic zero. Their simplicial forms also appeared in [34], [36] as free simplicial prounipotent groups of finite type, degenerate in dimensions greater than one. In this section we define the notion of a simplicial prounipotent presentation over punctured profinite spaces.…”