Stiefel-Whitney classes, united \(K\)-theory and real embeddings of number rings
The author studies the relations among the various Stiefel-Whitney classes associated to the real embeddings of a number ring \(A\), using a computation of the united \(K\)-theory of the algebraic \(K\)-theory spectrum \(KA\) of \(A\). More precisely, let \(F\) be a number field, let \(S\subset {\mathcal O}_F\) be a set of primes in its ring of integers \({\mathcal O}_F\), and let \(A= S^{-1}{\mathcal O}_F\) be its localization. In addition assume that \(1/2\in A\). Given a real embedding \(\alpha: F \subset {\mathbb R}\) there is an induced map \(BGL(A) \to BGL({\mathbb R}) \simeq BO\), which yields a sequence of Stiefel-Whitney classes \(w_*(\alpha)\). For a fixed real embedding \(\alpha\) the classes \(w_*(\alpha)\) are algebraically independent, but in general they do depend on the choice of \(\alpha\). The central matter of the paper is determining the relations between the various \(w_*(\alpha)\) for different \(\alpha\). This is equivalent to investigating the image of the map \(\varrho: H_*(BGL(A)) \to \prod_\alpha H_*(BO)\) in homology with coefficients in \(\mathbb{F}_2\), the field with two elements, which is induced by the set of the distinct real embeddings \(\alpha\). The motivation for computing \(\text{im}(\varrho)\) comes from the fact that it yields the relevant input data for a Serre spectral sequence that converges to \(H^*(BGL(A))\). To get a hand on the size of \(\text{im}(\varrho) \subset \prod_\alpha H_*BO\) the author makes use of the factorization \(BO \to BSC \to BU\) of the map \(BO \to BU\) induced by complexification, with \(BSC\) the classifying space of self-conjugate \(K\)-theory. More specifically, the author uses the fact that the image of a map \(H_*BGL(A) \to H_*BO\) of bicommutative Hopf algebras over \(\mathbb{F}_2\) whose composition to \(H_*BSC\) becomes trivial must lie in a suitable subgroup of \(H_*BO\). In order to get control on this vanishing property the author computes the united \(K\)-theory of \(KA\), which among other things yields relevant information on the map \(KO^0KA \to KSC^0KA\). As a result of his analysis he eventually gets an injection \(\text{im}(\varrho) \subset H_*BO \otimes H_*(U/O)^{r_1-b-1}\otimes H_*(Sp/U)^b\), with \(r_1\) the number of real embeddings of \(A\) and \(b\) the dimension of the \(\mathbb{F}_2\)-vector space \(\kappa A\), which is defined to be the quotient of the group of principal divisors of \(A\) modulo the totally positive principal divisors of \(A\). He then further shows that the injection is an isomorphism provided \(\kappa A\) is a direct summand in the bigger group \(\text{NPic}(A)\), the so-called narrow Picard group, which by definition is the quotient group of all divisors modulo the totally positive principal divisors. The united \(K\)-theory calculations for \(KA\), which describe the united \(K\)-theory of \(KA\) in terms of algebraic data of \(F\) and \(A\), also are of independent interest. In fact the author gains much more information than he actually needs in order to control the relations between the Stiefel-Whitney classes. E.g., he shows that the united \(K\)-theory of \(KA\) is \(CR\)-acyclic in the sense of \textit{A. K.~Bousfield} [J. Pure Appl. Algebra 66, 121--163 (1990; Zbl 0713.55007)]. For more information on united \(K\)-theory see \textit{A. K.~Bousfield} [ibid.].
scientific article; zbMATH DE number 5125365