同伦群的结构与计算

This project aims to apply the motivic Ctau method, invented by the applicant and his collaborates, as well as the efficient algorithm designed by the applicant for computing the homology of Hopf algebroids, to compute the motivic homotopy groups and stable homotopy groups of spheres in a large range. Moreover, we will also use equivariant methods and unstable methods to determine some hard differential in the Adams spectral sequence and Adam-Novikov spectral sequence, These computations will also allow us to get informations on the structures of equivariant homotopy groups and unstable homotopy groups, such as the structure of equivariant Adams spectral sequences with the groups having higher order, and the homotopy groups of the Boudfield-Kuhn functor applied to spheres, and the monochromatic structure of the EHP sequence. .The project hopes that these computations of homotopy groups could solve some important problems in topology, such as the non-triviality of the Kervaire invariant in dimension 126, the Doomsday conjecture, and the finiteness of higher exponents of unstable homotopy groups of spheres.

项目计划利用申请人与合作者提出的motivic同伦论中的Ctau方法，以及申请人设计的计算Hopf代数胚同调的高效算法，深入的计算motivic同伦群和球面的稳定同伦群的结构。进一步的，项目将会结合等变同伦论，与不稳定同伦论的方法，来解决稳定同伦群计算中的一些困难的Adams微分与Adams-Novikov微分。同时这些计算也将得到等变同伦群与不稳定同伦群的结构的信息，例如高阶群作用下等变Adams谱序列的结构，Bousfield-Kuhn函子作用在球面后的同伦群，EHP序列在单分色下的结构等。.项目期望通过对这些同伦群的计算，可以解决诸如126维Kervaire不变量的非平凡性，审判日猜想，不稳定同伦群高阶指数有限性等拓扑学中的重要问题。

项目利用motivic同伦论中的Ctau方法，以及计算Hopf代数胚同调的高效算法，计算了前90个复数域上的motivic同伦群和球面的稳定同伦群的结构。进一步的，项目结合等变同伦论，与不稳定同伦论的方法，解决了稳定同伦群计算中的一些困难的Adams微分与Adams-Novikov微分。同时结合等变同伦伦的方法，得到了一些色展阶数等于4的同伦群的信息。同时在一般域上建立了周t-结构的理论，奠定了将前述计算推广到一般域上稳定同伦群的基础。