Classical Homotopy Theory, Simplicial Groups, and Related Structures

DMS-0305094Frederick R.科恩PI打算研究与经典同伦理论的起源问题，以及它们与函数空间的重叠，群的上同调和经典配置空间的性质。具体问题如下：（1）扩大P.I. M. G.关于某些有限复形同伦群中挠率的增长。(2)通过P.I.开发的迭代循环空间的单纯模型，分析其他有限复合体的这些结构，和R.李维(3)在私家侦探的合作下，J. Berrick，Y. Wong和J. Wu继续分析辫子群和2-球面同伦群之间的相互作用。这些联系与D. Cohen的模形式的类似物从上同调的辫子群的系数在辛表示。(4)在与A。Adem和D.科恩，继续分析空间的结构表示的基本群体的补充复杂的超平面arrangements，及其相关的向量bundle.One的主要方向在这里是一个连接之间的artin的辫子群，和经典问题的numeratingways连续包装领域的任何层面aroundother几何对象。这些方法可以被认为是枚举许多粒子的拓扑性质，不允许碰撞，通过时间移动。该方法与组合给出的几种数学结构以及数论方法有很大的重叠。这些方法涉及“辫子的几何学”，以及这种结构如何通过与其他数学对象的连接来编码计数问题。

DMS-0305094Frederick R. CohenThe PI intends to study problems with origins in classicalhomotopy theory as well as their overlap with function spaces,cohomology of groups, and properties of classical configuration spaces. The specific problems are as follows:(1) Extend the P.I.'s recent solution of a 25 year oldconjecture of M.G. Barratt on the growth of torsion inthe homotopy groups of certain finite complexes.(2) Analyze these structures for other finite complexesthrough simplicial models for iterated loop spaces asdeveloped by the P.I., and R. Levi.(3) In joint work of the P.I., J. Berrick, Y. Wong, and J. Wu,continue to analyze the interplay between braid groups, andthe homotopy groups of the 2-sphere. These connections extendin joint work with D. Cohen to analogues of classical modularforms obtained from the cohomology of the braid groups withcoefficients in a symplectic representation.(4) In joint work with A. Adem, and D. Cohen, continue toanalyze the structure of spaces of representations offundamental groups of complements of complex hyperplanearrangements, and their associated vector bundles.One of the main directions here is a connection betweenArtin's braid groups, and the classical problem of enumeratingways of continuously wrapping spheres of any dimension aroundother geometric objects. The methods can be thought of asenumerating topological properties of many particles, notallowed to collide, moving through time. The methodsoverlap with several mathematical structures givencombinatorially as well as number theoretically. Thesemethods involve the "geometry of braids", and how thisstructure encodes counting problems through connections toother mathematical objects.