The Atiyah–Janich theorem identifies the space of Fredholm operators on Hilbert space with the classifying space for topological complex K-theory. I will present recent work with Oliver Braunling and Michael Groechenig which establishes an analogous identification for algebraic K-theory. We relate this to the tame symbol of class field theory, and sketch a new proof of reciprocity laws for higher tame symbols and higher Contou-Carrere symbols in close analogy with the Weil reciprocity law for curves.