Mathematical Sciences: Quillen-Type Homology and Homotopy Theory

首席研究者将尝试拓宽某些已被证明具有几何效用的代数技术的范围。在20世纪60年代，Daniel Quillen展示了如何在许多范畴中定义同源和上同源。其中包括简单集合，他在那里得到了拓扑空间的普通上同调，但是域上的简单代数的范畴和模p Steenrod代数上的简单不稳定代数都支持同伦理论中相关的上同调的思想。例如，简单不稳定代数的上同调是研究Bousfield- Kan谱序列E2项的正确背景。Bousfield- Kan谱序列是一种不稳定的adams型谱序列，由于Haynes Miller、Jean Lannes等人的工作而获得了突出的地位。在先前的NSF资助下，首席研究员已经成功地研究了各种代数类别的quillen型上同调。本项目的目的是采用这些结果并扩展和应用它们。特别是，他现在应该能够深入研究Barratt的悬浮光谱序列和他的“Lie ring分析仪”——这两者都是30年来猜想和推测的来源。其他项目包括将quillen型上同调技术与Chris Stover的方法相结合，以接近两个空间的楔形的同伦群，研究用于计算截面空间的同伦群的谱序列，以及使用Jean Lannes和A.K. Bousfield的工具探索函数复的同调。

The principal investigator will attempt to broaden the scope of certain algebraic techniques of proven geometric utility. In the 1960's Daniel Quillen showed how to define homology and cohomology in many categories. These included simplicial sets, where he obtained the ordinary cohomology of topological spaces, but the categories of simplicial algebras over a field and simplicial unstable algebras over the mod-p Steenrod algebra both support an idea of cohomology that is relevant in homotopy theory. Cohomology of simplicial unstable algebras, for example, is the correct context for studying the E2 term of the Bousfield- Kan spectral sequence, an unstable Adams-type spectral sequence which has gained prominence due to the work of Haynes Miller, Jean Lannes and others. Under prior NSF grants, the principal investigator has made a successful study of the Quillen-type cohomology in various categories of algebras. The purpose of this project is to take these results and to extend and apply them. In particular, he should now be able to make an in-depth study of Barratt's desuspension spectral sequence and his "Lie ring analyzer" - both a source of conjecture and speculation for 30 years. Other projects include combining the Quillen-type cohomology techniques with the methods of Chris Stover to approach the homotopy groups of a wedge of two spaces, the study of a spectral sequence for computing the homotopy groups of a space of sections, and an exploration of the homology of function complexes using tools of Jean Lannes and A.K. Bousfield.