We present implementations of the topological K-theory spectra $KU$ and $KO$ in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We also introduce a geometric definition of weakly interacting time evolution operators, and show how associated spectra $KU^{ wi}$ and $KO^{ wi}$ deformation retract to $KU$ and $KO$. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.
Categories: math-ph math.AT math.KT math.MP quant-ph