Classifying spaces of algebraic groups and A1-homotopy theory.

我们建议从基元同伦理论的角度系统地研究代数群G的分类空间，以及G-等变基元同伦理论的一些自然相关问题。例如，在我们证明Friedlander-Milnor猜想之后，我们可以攻击关于不一定是代数闭域F的点群G(F)的同调的新结果。另一个(相关的)方向与Galois上同调中的Serre猜想有关。我们确实有一个精确的猜想，它暗示了大多数这类猜想，最重要的是，到目前为止还没有被触及的猜想，以及解决它的新方法。更广泛地说，我们感兴趣的是将(等变)同伦理论的经典问题转化为动机同伦理论。就像亚当斯猜想、西格尔猜想一样，各种结构的余边线的行为首先类似于正交边边线。

We propose to study in a systematic way the classifyings spaces of algebraic groups G from the point of view of motivic homotopy theory, as well as some naturally related problems of G-equivariant motivic homotopy theory. For instance following our proof of the Friedlander-Milnor conjecture, we may attack new results on the homology of groups of points G(F) for not necessarily algebraically closed fields F. Another (related) direction is concerned with Serre's conjectures in Galois cohomology. We do have a precise conjecture that implies most of these type of conjectures, most importantly the one which is up to now untouched, and a new approach to settle it. More generally we are interested in transporting classical problems of (equivariant) homotopy theory to motivic homotopy theory. Like the Adams conjecture, the Segal conjecture, the behaviour of various structured cobordism like, to start with, the orthogonal cobordism.