ORIENTABILITY OF PSEUDOMANIFOLD AND GENERALIZATIONS OF SPERNER'S LEMMA
Yoshitsugu Yamamoto · Journal of the Operations Research Society of Japan · 1988
We propose a combinatorial framework for fixed point algorithms and constructive proofs of combinatorial lemmas in topology. The framework consists of two sets of pseudomanifolds and an operator relating them. They have lattice structures which are dual to each other. We show that the set of "joins" of pseudomanifolds related by the operator is a homogeneous and orientable pseudomanifold under several conditions. By exploiting this framework we generalize Sperner's lemma on convex polytope. Namely, let C be a convex polytope with m facets F_1,…,F_m, S be a finite triangulation of C and S^^- = {σ|σ is a face of some simplex of S}. Given a nondenerate vertex v of C and a labelling function l from the set of vertices of S to {1,…, m}, the set of indices of facets, there is an odd number of simplices σ of S^^- such that l(σ)∪{i|1≦i≦m, σ⊂F_i} strictly includes {i|1≦i≦m, v∈F_i} We also prove the generalization of Sperner's lemma by Fan and van der Laan-Talman-Van der Heyden's lemma as corollaries to the result.