Motivic稳定同伦与环面拓扑中R-S谱序列的研究

To determine the stable homotopy groups of spectra by the classical Adams spectral sequence and by the Adams-Novikov spectral sequence is one of the most important problems in algebraic topology. Motivic stable homotopy theory comes from algebraic geometry. More relations between the classical Adams spectral sequence and algebraic Novikov spectral sequence can be seen from the Motivic Adams spectral sequence. By which we have more methods to determine the E2 term and higher Adams differentials. Rothenberg-Steenrod spectral sequence is a method to determine the homology, cohomology of the orbid space from that of the Lie group and of the total space. It was introduced many years ago, but very few people use it. In this project we will use Motivic Adams spectral sequence and many other spectral sequences to compute the E2-term of the Adams-Novikov spectral sequence and higher Adams differentials. By which we will compute the stable homotopy groups of sphere, MO<8>, T(n) etc. We also study the tours action on real and complex moment-angle manifolds and lifting problem in toric topology by the Rothenberg-Steenrod spectral sequence.

利用经典Adams谱序列和Adams-Novikov谱序列研究球面和其他一些谱的稳定同伦群是代数拓扑中的一个重要研究方向。Motivic 稳定同伦源自于代数几何，通过Motivic Adams谱序列建立了经典Adams谱序列和代数Novikov谱序列的直接联系。使得我们有更多的办法计算Adams谱序列的E2项和高价Adams微分。Rothenberg-Steenrod谱序列是利用紧Lie群的同调和全空间的同调研究轨道空间的同调、上同调的一种方法。已被提出许多年了，但至今很少利用这个谱序列具体计算。在本项目中我们将借鉴Motivic Adams谱序列利用各种谱序列计算Adams-Novikov谱序列的E2项和高价Adams微分并由此计算球面、MO<8>、T(n)等谱的稳定同伦群。利用Rothenberg-Steenrod谱序列研究环面拓扑中环面对moment-angle流形的作用及提升问题

Motivic Adams谱序列是近20来发展出来的一个方向。他到经典Adams谱序列和代数Novikov谱序列都有同态，通过这个同态可以建立起代数Novikov谱序列的微分与Adams谱序列的微分的联系。我们在项目执行期间学懂了这个谱序列，并做出了一些进展。. 上同调刚性和Gitler-Lopez猜想是环面拓扑中倍受关注的问题。在本项目中我们完全证明Gitler-Loez猜想；并证明了没有3-belt和4-belt的2维单纯球面（比如富勒烯的对偶）所对应的moment-angle流形和环面流形都是上同调刚性的. 对BPUn的上同调进行了一些计算，这里BPUn是n阶射影酉群PUn的分类空间。它的上同调在代数几何、代数拓扑和粒子物理中有着广泛的应用. 本项目无高国防应用。