[Paper Review] Around Sperner's lemma

Oleg R. Musin|arXiv (Cornell University)|May 29, 2014

Advanced Combinatorial Mathematics9 references10 citations

TL;DR

This paper generalizes Sperner's lemma by introducing a degree-based condition on boundary labelings that guarantees the existence of fully labeled simplices in triangulations of manifolds. It extends the result to oriented and non-orientable PL-manifolds, unifying and strengthening Sperner’s lemma, Tucker’s lemma, and Ky Fan’s lemma through a topological degree framework, with the key contribution being a lower bound of |deg(L, ∂T)| on the number of fully labeled simplices.

ABSTRACT

We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also shown that the main theorem implies Tucker's lemma and some other theorems.

Motivation & Objective

• To generalize Sperner’s lemma beyond standard colorings by introducing a degree-based condition on boundary labelings.
• To unify and extend Tucker’s lemma and Ky Fan’s lemma using a topological degree framework.
• To establish a lower bound on the number of fully labeled simplices in triangulations of PL-manifolds based on the degree of the labeling map.
• To extend results to both oriented and non-orientable manifolds using degree modulo 2.
• To demonstrate that the new condition generalizes Sperner’s lemma and implies the De Loera–Petersen–Su polytopal Sperner theorem.

Proposed method

• Define the degree deg(L, ∂T) as the difference between the number of (1,2) and (2,1) label transitions along the boundary of a triangulation.
• Use the degree of a labeling map to characterize the winding behavior of labels on the boundary of a manifold.
• Apply topological degree theory (mod 2 for non-orientable cases) to relate the degree to the existence of fully labeled simplices.
• Construct a labeling map f_{L,P} from the triangulation to a convex polytope P, ensuring the image of the boundary lies on ∂P.
• Prove that if deg(L, ∂T) ≠ 0, then there exists at least one simplex labeled as in the covering complex of a point y in P.
• Use the degree modulo 2 for non-orientable manifolds to guarantee the existence of at least one fully labeled simplex when deg₂(L, ∂T) ≠ 0.

Experimental results

Research questions

• RQ1 Can Sperner’s lemma be generalized beyond standard Sperner colorings using a topological invariant like degree?
• RQ2 Does the degree of the boundary labeling determine a lower bound on the number of fully labeled simplices in a triangulation?
• RQ3 Can Tucker’s lemma and Ky Fan’s lemma be derived as special cases of a unified topological principle?
• RQ4 How does the degree concept extend to non-orientable manifolds using degree modulo 2?
• RQ5 What is the minimal number of fully labeled simplices guaranteed when the labeling satisfies a non-vanishing degree condition?

Key findings

• The number of fully labeled d-simplices in a triangulation of a compact oriented PL-manifold M is at least |deg(L, ∂T)|.
• When the labeling satisfies f_{L,P}(∂M) ⊆ ∂P, the number of fully labeled d-simplices is at least (n−d)|deg(L, ∂T)|, where n is the number of vertices of the polytope P.
• For antipodal labelings on the boundary of a d-ball, the absence of complementary edges implies at least |deg(L, ∂T)| internal complementary edges.
• In the case of an ACS (n,d)-polytope, the number of d-simplices with labels {k₀, −k₁, ..., (−1)^d k_d} is at least |deg(L, ∂T)| when no complementary edges exist.
• For non-orientable manifolds, if deg₂(L, ∂T) ≠ 0, then there exists at least one fully labeled simplex.
• The degree-based condition generalizes Sperner’s lemma, as deg(L, ∂T) = 1 for any Sperner labeling, guaranteeing at least one fully labeled simplex.