2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116354In this paper we give a proof of the Lefschetz fixed point formula of Freed$^{\rm [1]}$ for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced in [2] and then we generalize this formula under the noncommutative geometry framework.Differential GeometryK-Theory and Homologytext