We compute the Chern subgroup of the 4-th integral cohomology group of a certain classifying space and show that it is a proper subgroup. Such a classifying space gives us new counterexamples for the integral Hodge and Tate conjectures modulo torsion.
Let p be an odd prime number. We show that there exists a finite group of order p^{p+3} whose the mod p cycle map from the mod p Chow ring of its classifying space to its ordinary mod p cohomology is not injective.
For a smooth projective variety $X$ of dimension $2n-1$, Zhao defined topological Abel-Jacobi map, which sends vanishing cycles on a smooth hyperplane section $Y$ of $X$ to the middle dimensional primitive intermediate Jacobian of $X$. When the vanishing cycles are algebraic, it agrees with Griffiths Abel-Jacobi map. On the other hand, Schnell defined a topological Abel-Jacobi map using the $mathbb R$-splitting property of the mixed Hodge structure on $H^{2n-1}(Xsetminus Y)$. We show that the two definitions coincide, which answers a question of Schnell.
For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${mathbb A}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${mathbb A}^1$-homotopy theory paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${mathbb A}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${mathbb A}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${mathbb A}^n setminus 0$ is rationally equivalent to a suitable motivic Eilenberg--Mac Lane space, and the special linear group decomposes as a product of motivic spheres.
We study generically split octonion algebras over schemes using techniques of ${mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another mod $3$ invariant. We review Zorns vector matrix construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gilles analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.