Real spin bordism and orientations of topological K-theory

We construct a commutative orthogonal C 2 C_2 -ring spectrum, M S p i n R c MSpin^cR , along with a C 2 C_2 - E ∞ E∞ -orientation M S p i n R c → K U R MSpin^cR → KUR 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 → (MSpin^cR)C_2 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