We construct a commutative orthogonal C 2 C₂ -ring spectrum, M S p i n R c MSpinᶜ ₑ, along with a C 2 C₂ - E ∞ E -orientation M S p i n R c → K U R MSpinᶜ ₑ KU ₑ of Atiyah’s Real K-theory. Further, we define E ∞ E -maps M S p i n → (M S p i n R c) C 2 MSpin (MSpinᶜ ₑ) ^C₂ and <mml: math xmlns: mml="http: //www. w3. org/1998/Math/MathML" alttext="normal upper M normal upper U Subscript double-struck upper R Baseline right-arrow normal upper M normal upper S normal p normal i nor
Halladay et al. (Wed,) studied this question.