2024/05/02 by Zachary Halladay, Halladay, Zachary, Yigal Kamel +1
Physics and Astronomy · #Algebraic Topology (math.AT) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2405.00963
openalex publication_date 2024/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a commutative orthogonal C2-ring spectrum, MSpincℝ, along with a C2-E∞-orientation MSpincℝ → KUℝ of Atiyah's Real K-theory. Further, we define E∞-maps MSpin → (MSpincℝ)C2 and MUℝ → MSpincℝ, which are used to recover the three well-known orientations of topological K-theory, MSpinc → KU, MSpin → KO, and MUℝ → KUℝ, from the map MSpincℝ → KUℝ. We also show that the integrality of the A-genus on spin manifolds provides an obstruction for the fixed points (MSpincℝ)C2 to be equivalent to MSpin, using the Mackey functor structure of \underlineπ_*MSpincℝ. In particular, the usual map MSpin → MSpinc does not arise as the inclusion of fixed points for any C2-E∞-ring spectrum.