2023/03/17 by Prasit Bhattacharya, Bhattacharya, Prasit, Foling Zou +1
Mathematics · Medicine · Pharmacology, Toxicology and Pharmaceutics · #19L20 #55P91 #55R40 #55R50 #55R91 #55S91 #Algebraic Topology (math.AT) #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2303.10259
openalex publication_date 2023/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group G, a precise condition under which an R-orientation of a G-equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel-Whitney class of a "homogeneous" G-equivariant bundle with respect to an 𝔼_∞G-ring spectrum R. As an application, we show that the 2-fold direct sum of any homogeneous bundle is H\underlineAG-orientable, where \underlineAG is the Burnside Mackey functor. We notice that H\underlineAG-orientability is equivalent to H\underlineℤ-orientability when the order of G is odd. When the order of G is even, we show that a G-equivariant analog of the tautological line bundle over \mathbbRP^∞ is H\underlineℤ-orientable but not H\underlineAG-orientable.