 Oriented manifold retract for the Lefschetz theorem

A closed smooth manifold X embeds in Euclidean space. The double M of a closed tubular neighbourhood is a closed oriented manifold retracting onto X. If i is its inclusion and r its retraction, the fixed points of F correspond to those of f, and cyclicity of finite-dimensional trace gives $L(F)=L(f)$ from $i^{\ast }{r}^{\ast }=1$. Thus the oriented diagonal-class proof of the Lefschetz fixed-point theorem implies the theorem for possibly nonorientable X, without incorrectly treating its diagonal as cooriented.