The index.- Real representations.- Extension and reduction.- The functor O * (H; F, F).- Index as a bordism invariant.- The trace invariant.- The local invariant.- Periodic maps on a riemann surface.- The atiyah-bott formula.- Weakly complex involutions.- The ring F.- The ring R(Z2).- The ring C (Z2) of constructions.- Applications.- Dimension of the fixed point set.- A local invariant for O * U (Z2).