Classical Homotopy Theory, Simplicial Groups, and Related Structures

DMS-0305094 Frederick R.Cohenthe PI打算研究经典同伦理论中的起源问题，以及它们与函数空间的重叠、群的上同调以及经典配置空间的性质。具体问题如下：(1)推广P.I.S关于某些有限复形的同伦群的挠率增长的猜想已有25年之久.(2)通过P.I.和R.Levi建立的迭代循环空间的单纯模型来分析其他有限复形的这些结构.(3)P.I.，J.Berrick，Y.Wong，和J.Wu共同工作，继续分析辫子群和2-球面的同伦群之间的相互作用.(4)在与A.Adem和D.Cohen的联合工作中，继续分析了复超平面排列的基本补群及其相关向量丛的表示空间的结构。这里的主要方向之一是Artin辫群之间的联系，以及关于任意维球面围绕其他几何对象连续包络的计数方法的经典问题。这些方法可以被认为是列举了许多粒子的拓扑性质，这些粒子不允许碰撞，在时间中移动。该方法具有多种数学结构，在组合上和理论上都给出了数值。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.