= Oriented manifold retract for the Lefschetz theorem
{title2=$r i=1_X,\quad F=i f r$}
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 $i^*r^*=1$. 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.
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