On the EO-orientability of vector bundles

2020/03/08 by Prasit Bhattacharya, Bhattacharya, Prasit, Hood Chatham +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2003.03795

openalex publication_date 2020/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the orientability of vector bundles with respect to a family of cohomology theories called EO-theories. The EO-theories are higher height analogues of real K-theory KO. For each EO-theory, we prove that the direct sum of i copies of any vector bundle is EO-orientable for some specific integer i. Using a splitting principal, we reduce to the case of the canonical line bundle over \mathbbCP^∞. Our method involves understanding the action of an order p subgroup of the Morava stabilizer group on the Morava E-theory of \mathbbCP^∞. Our calculations have another application: We determine the homotopy type of the S¹-Tate spectrum associated to the trivial action of S¹ on all EO-theories.

Related

• An Orientation Map for Height p-1 Real E Theory
• There aren't that many Morava E-theories
• Equivariant orientation of vector bundles over disconnected base spaces
• The Lubin-Tate stack and Gross-Hopkins duality
• The Tate spectrum of the higher real K-theories at height n=p-1