Chow rings and cobordism of some Chevalley groups

Steenrod operations on the Chow ring of a classifying space

We use the Steenrod algebra to study $CH^*BG$, the mod $p$ Chow ring of the classifying space of $G$. We describe a localization property which relates a given $G$ to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.

On the Chow ring of the classifying stack of PGL_3

We compute generators for the Chow ring of the classifying space of PGL_3 (over the complex numbers) as defined by Totaro. We also find enough relations after inverting 3. We show that this ring is not generated by Chern classes (this is the first example of this kind among classical groups) and prove that Totaro's refined cycle class map to a quotient of complex cobordism of BPGL_3 is surjective

The Chow rings of generalized Grassmannians

Based on the Basis theorem of Bruhat--Chevalley [C] and the formula for multiplying Schubert classes obtained in [D\QTR{group}{u}] and programed in [DZ$_{\QTR{group}{1}}$], we introduce a new method computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in [DZ$_{3}$, DZ$_{4}$] to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes.

Algebraic cobordism theory attached to algebraic equivalence

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordis…

二次定向丛的分类空间的Chow--Witt环

The Chow--Witt rings of the classifying space of quadratically oriented bundles

该研究计算了秩n二次定向向量丛分类空间${\ m BSL}_n^c$的Chow--Witt环，还证明对应二次定向配边谱${\ m MSL}^c$在对η取逆后等价于MSL，为相关代数拓扑研究提供了新结果。

Algebraic cobordism and flag varieties

Let $X$ be an algebraic variety over $k$ such that $\bar X=X\otimes_k\bar k$ is cellular. We study torsion elements in the Chow ring $CH^*(X)$ which corresponds to $v_iy$ in the algebraic cobordism $Ω^*(\bar X)$ where $0\not=y\in CH^*(\bar X)/p$ and $v_i$ is the generator of $BP^*$ with $|v_i|=-2(p^i-1).$ In particular, we try to compute $CH^*(X)$ from $Ω^*(\bar X)$ when $X$ are twisted complete flag varieties.