Let A be a fixed subring of a number field F that contains the ring of integers. Each complex embedding β: F --> C induces group homomorphisms GLm(A) --> GLm(C), and therefore also maps of classifying spaces BGLm(A) --> BGLm(C). Pulling back the universal Chern classes we obtain Chern classes cn(β) in H2n(BGLA). It turns out that for any such A, the mod 2 Chern classes are independent of the complex embedding β, and for fixed β these classes are algebraically independent (taking m = infinity).
Now suppose that F admits a real embedding α: F --> R. Then Stiefel-Whitney classes wn(α) in Hn(BGLA; Z/2) are defined in the analogous way. These classes, however, denitely will depend on the choice of real embedding. Furthermore, although the wn (α) are again algebraically independent for fixed α, various relations hold as α varies. This raises the question:
How are the Stiefel-Whitney classes of the various real embeddings related?
The answer depends on classical number-theoretic invariants, such as the "totally positive units" and the "narrow ideal class group". The methods depend on a computation of the complex, real and self-conjugate K-theory of the algebraic K-theory spectrum KA.
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John H. Palmieri, Department of Mathematics, University of Washington, palmieri@math.washington.edu