A constructive proof of Ky Fanʹs generalization of Tuckerʹs lemma

We present a proof of Ky Fanʹs combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fanʹs lemma to allow for triangulations of S n that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tuckerʹs lemma that holds for a more general class of triangulations than the usual version.