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We give some formulas for the ZZ pairing in KO theory using a long exactsequence for bivariant K theory which links real and complex theories. This isdiscussed under the framework of real structures given by antilinear operatorsverifying some symmetries. Topological phases protected by time reversalsymmetry from condensed matter physics will be discussed.

Exact sequence between real and complex bivariant K theories and application to the Z2 pairing