Real structures and Morava K-theories
Let \(F\) be a formal group law over a ring \(R\). A \(k\)-structure over an \(R\)-algebra, \(S\), is a power series \(f(x_1 \ldots x_k)\) over \(S\) which is symmetric in the \(x_i\)'s and satisfies the conditions \(f(0, x_2 \ldots x_k)=1\) and \(f(x_1 \ldots x_k) \cdot f(x_0, F(x_1, x_2), x_3 \ldots x_k) =f(F(x_0,x _1), x_2 \ldots x_k) \cdot f(x_0, x_1, x_3 \ldots x_k)\). For \(k =1\) or \(2\) a \(k\)-structure is said to be real if \(f(x) = f(-_F x)\) or \(f(x,y) =f(-_F x, -_F y)\). Denote the universal rings for real \(k\)-structures by \(C^r_k(F)\). The paper under review proves that the map \(C^r_2 F \to C^r_1 F\) induced by the map of series \(i_*(f)(x_1,x_2)=f(F(x_1,x_2))/f(x_1)f(x_2)\) is an isomorphism. This result has topological significance because of work of Ando, Hopkins and Strickland relating \(C^r_k(F)\) to \(K(n)_* BO \langle 2k \rangle\) for \(K(n)\) the Morava K-theory at the prime \(2\). Using this result the authors construct polynomial generators for \(K(n)_* B\operatorname {Spin}\) and \(K(n)_* BSO\) for \(n=1, 2\). They also show that a similar algebraic result holds for any formal group law for which multiplication by \(2\) is an epimorphism.
scientific article; zbMATH DE number 1786890