Real spin bordism and orientations of topological K-theory

We construct a commutative orthogonal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 2"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -ring spectrum, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M normal upper S normal p normal i normal n Subscript double-struck upper R Superscript c"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathrm {MSpin}^c_{\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , along with a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 2"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -orientation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M normal upper S normal p normal i normal n Subscript double-struck upper R Superscript c Baseline right-arrow normal upper K normal upper U Subscript double-struck upper R"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msubsup> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">K</mml:mi> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {MSpin}^c_{\mathbb {R}} \to \mathrm {KU}_{\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Atiyah’s Real K-theory. Further, we define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M normal upper S normal p normal i normal n right-arrow left-parenthesis normal upper M normal upper S normal p normal i normal n Subscript double-struck upper R Superscript c Baseline right-parenthesis Superscript upper C 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msubsup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {MSpin} \to (\mathrm {MSpin}^c_{\mathbb {R}})^{C_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <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