Homotopy Theoretic Study of Higher Dimensional Categories

我尝试基于一般高度的形式群律，如Honda群律，对代数k理论进行新的建构。除加性群或乘法群外，没有给出代数群的形式群律，但可以看作是由Segal定义的集。由于集合是一个简单的集合，我们通过几何实现得到一个同伦类型。考虑局部域整数环上的满矩阵代数。然后得到的同伦类型是更高高度的代数k理论空间的候选类型。首先研究空间的上同群。在乘法群的情况下，空间是一般线性群的分类空间。上同调是最大环面上同调的对称多项式。如果形式群律的高度为h，则最大环面的维数为一般情况下的h倍。然后我们得到一个多对称多项式作为候选空间的上同调。我们接下来要做的是根据一个给定的高度较高的形式群律来定义或构建表征理论。我特别希望得到一个不使用Morava k理论的对Hokins-Kuhn-Ravenel特性的群论解释。

I tried to make a new construction of algebraic K-theory based on formal group laws of general heights, e.g., Honda group law. Formal group laws other than additive or multiplicative group, do not give algebraic groups, but can be regarded as a gamma set defined by Segal. Since a gamma set is a simplicial set, we have a homotopy type by taking the geometric realization. We consider the full matrix algebra over a ring of integers of a local field. Then the resulting homotopy type is a candidate for a space of algebraic K-theory of higher height. First I studied the cohomology group of the space. In the case of multiplicative group, the space is the classifying space of the general linear group. The cohomology is symmetric polynomials of the cohomology of the maximal torus. If the height of the formal group law is h, then the dimension of the maximal torus is h-times of the ordinary case. Then we obtain a multi-symmetric polynomials as cohomology of the space for the candidate. What we have to do next is to define or construct the representation theory based on a given formal group law of higher height. In particular I hope to get a group theoretic interpretation of Hokins-Kuhn-Ravenel character not using the Morava K-theory.