E-infinity Algebras and Homotopy Theory

michael Mandell a.本拨款提出使用e -∞微分梯度代数研究空间的同伦理论，并提出研究e -∞微分梯度代数的同伦理论和e -∞环谱。e -∞微分渐变代数的概念是对交换微分渐变代数概念的推广；交换性只适用于同伦，而e -无穷微分梯度代数在其上同伦上允许Steenrod运算，这在一定程度上度量了与实际交换性的偏差。e -无穷环谱是e -无穷微分梯度代数的稳定同伦理论推广。最常研究的具有交换环结构的广义上同调理论是用e -无穷环谱表示的。这种外结构提供了关于理论的重要计算信息，对相关理论的各种构造很有用。代数拓扑学试图将拓扑或几何分类问题简化为代数。因为代数通常是离散的，它在连续变形或同伦的变化下通常是不变的。正因为如此，代数拓扑中的基本问题通常被表述为理解空间的同伦等价类或计算空间间映射的同伦类的数量（或更经常的是“群”或“模”）。一般来说，这类问题可以重新表述为代数中的等价问题。本提案的目的是开发工具来研究在这种情况下出现的代数。

DMS-0203980Michael Mandell A.This grant proposes to use E-infinity differential graded algebras to study the homotopy theory of spaces and proposes to study the homotopy theory of E-infinity differential graded algebras and E-infinity ring spectra. The concept of E-infinity differential graded algebra is a generalization of the concept of commutative differential graded algebra; commutativity holds only up to homotopy, and E-infinity differential graded algebras admit Steenrod operations on their cohomology, which measure to some extent the deviation from actual commutativity. E-infinity ring spectra are a stable homotopy theory generalization of E-infinity differential graded algebras. Most commonly studied generalized cohomology theories that have commutative ring structures are represented by E-infinity ring spectra. This extrastructure yields important calculational information about the theory and is useful for various constructions of related theories.Algebraic topology tries to reduce topological or geometricclassification problems into algebra. Because the algebra is usually discrete, it is typically invariant under changes by continuous deformations, or ``homotopies.'' Because of this, fundamental problems in algebraic topology are often phrased in terms of understanding the homotopy equivalence classes of spaces or in terms of computing the number (or more often the ``group'' or ``module'') of homotopy classes of maps between spaces. Quite generally this kind of question can be reformulated as an equivalent question in algebra.The purpose of this proposal is to develop tools to study the sort of algebra that arises in this context.