Generalized Steenrod operations and equivariant geometry

拓扑学是一门研究形状的学科。1947年，诺曼·斯廷罗德介绍了斯廷罗德公司的业务，他彻底改变了这一主题。这些运算导致了几何学中一些最神秘的结果，这些结果在50年代和60年代普及了同伦理论，拓扑学的一个分支。这个项目将导致等变同伦理论的进步，同伦理论是对形状对称性敏感的同伦理论的增强。由此得到的Steenrod运算的等变类比将导致在等变几何中的新应用。该奖项还将用于刺激新墨西哥州立大学的研究文化和支持研究生。PI将建立一种新的方法，将经典Steenrod算子的构造推广到等变同伦理论，并构造出所有有限群的等变Steenrod算子。等变Steenrod运算直接导致Stiefel-Whitney类的等变相似，在等变向量丛和等变光滑流形的研究中有许多应用。Atiyah实向量丛是等变向量丛的重要例子，我们将对其进行研究。特别地，重言式Atiyah Real向量丛的James周期数将使用由Atiyah Real K-理论形成的亚当斯猜想的等变模拟来确定。PI还将继续研究色同伦理论的周期性自映射，并将其扩展到等变和动机设置。该项目由拓扑学计划和既定的激励竞争研究计划(EPSCoR)联合资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is a subject that studies shapes. This subject was revolutionized in 1947 by Norman Steenrod when he introduced the Steenrod operations. These operations led to some of the most enigmatic results in geometry which popularized homotopy theory, a subbranch of topology, in the fifties and the sixties. This project will lead to advances in equivariant homotopy theory, an enhancement of homotopy theory which is sensitive to the symmetries of shapes. The resulting equivariant analogs of Steenrod operations will lead to new applications in equivariant geometry. The award will also be used to stimulate the research culture and support graduate students at New Mexico State University. The PI will establish a new method that generalizes the construction of classical Steenrod operations to equivariant homotopy theory and constructs equivariant Steenrod operations for all finite groups. Equivariant Steenrod operations will immediately lead to equivariant analogs of Stiefel-Whitney classes which can find many applications in the study of equivariant vector bundles and equivariant smooth manifolds. Atiyah Real vector bundles, which are important examples of equivariant vector bundles, will be studied. In particular, the James periodicity number of tautological Atiyah Real vector bundles will be determined using an equivariant analog of the Adams conjecture formulated using Atiyah Real K-theory. The PI will also continue to study periodic self-maps of chromatic homotopy theory and extend them to equivariant as well as motivic settings.This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.