Subtle characteristic classes for Spin-torsors

Abstract

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with $${{\,\mathrm{\mathbb {Z}}\,}}/2$$ Z / 2 -coefficients of the Nisnevich classifying space of the spin group $$Spin_n$$ S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $$I^3$$ I 3 , which are closely related to $$Spin_n$$ S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between $$Spin_7$$ S p i n 7 and $$G_2$$ G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of $$G_2$$ G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

Motivic cohomology of the Nisnevich classifying space of even Clifford groups

“…

In this paper, we consider the split even Clifford group Γ + n and compute the mod 2 motivic cohomology ring of its Nisnevich classifying space. The description we obtain is quite similar to the one provided for spin groups in [10]. The fundamental difference resides in the behaviour of the second subtle Stiefel-Whitney class that is non-trivial for even Clifford groups, while it vanished in the spin-case.

…”

“…Following [7], we studied the motivic cohomology rings of the Nisnevich classifying spaces of unitary groups in [9], of spin groups in [10] and of projective general linear groups in [8]. This paper is a natural follow-up of [10].…”

A Serre-type spectral sequence for motivic cohomology

“…Recall that H 0,0 (Y •,• , R) is the free R-module with rank equal to the number of connected components of Y •,• and, analogously, H 0,0 (Y 0,• , R) is the free R-module with rank equal to the number of connected components of Y 0,• . Since, as in the argument at the end of [18,Proposition 3.4], the homomorphism…”

“…As in topology, our Serre spectral sequence reconstructs the motivic cohomology of the total space out of the motivic cohomology of the base and the cellular structure of the fiber. It is a natural generalisation of the Gysin long exact sequence that was first introduced by Smirnov and Vishik in [16] for computing the motivic cohomology of the Nisnevich classifying space of othogonal groups, and subsequently studied by the author in [19] for the general case of morphisms of simplicial schemes with reduced Tate fiber (the motivic analogues of sphere bundles). For constructing our spectral sequence, we need to work with bisimplicial schemes instead of simplicial ones.…”

“…More precisely, the Gysin sequence was used for orthogonal groups in [16], for unitary groups in [18] and for spin groups in [19]. Unfortunately, some algebraic groups are out of its range of application, e.g.…”