Non-Formality of Planar Configuration Spaces in Characteristic 2

Abstract

We prove that the ordered configuration space of 4 or more points in the plane has a non-formal singular cochain algebra in characteristic two. This is proved by constructing an explicit non trivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt-Eccles-Smith simplicial model. We also show that if the number of points does not exceed its dimension, then an euclidean configuration space is intrinsically formal over any ring. *

On the formality of the little disks operad in positive characteristic

“…Two related formality results can be found in [41,Theorem 3.10]. On the one hand, Salvatore shows that the space Conf n (R d ) is formal with any ring of coefficients if d n (and in fact is even intrinsically formal), on the other hand he shows that C * (Conf n (R 2 ), F 2 ) is not a formal dg-algebra if n 4.…”

Planar non-formality of the little discs operad in characteristic two

“…It should be mentioned that our theorem does not disprove Vassiliev's conjecture. The non-formality of D 2 as a planar operad, together with the non-formality of the single spaces D 2 (k) in characteristic 2 for k > 3, that we proved in [13], show that the behaviour in The author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. characteristic 2 is opposite to that in characteristic 0, where instead Hopf operad formality holds [3] (both at the level of spaces and operads).…”

“…By the universal property f defines an operad mapf : F (W ≤1 ) → H * (C). Then ∂(f ) is the composition The proof is similar to that of Lemma 6.7 in [13].…”

On the multisimplicial cup product

“…Therefore the multisimplicial approach using χ is much more efficient than the simplicial approach using BE, when performing computations as in [6].…”

“…As an example we consider a family of multisimplicial sets Sur(k) defined by McClure and Smith, see [5], modelling euclidean configuration spaces. The proof by the second author of the non-formality of the cochain algebra of planar configuration spaces in [6] used the Barratt-Eccles simplicial model and the classical cup product. Our new product on the multisimplicial McClure-Smith models makes the computation much simpler and faster, paving the way for an extension to higher dimensions.…”