Tucker's lemma

In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.

Let T be a triangulation of the closed n-dimensional ball ${\displaystyle B_{n}}$. Assume T is antipodally symmetric on the boundary sphere ${\displaystyle S_{n-1}}$. That means that the subset of simplices of T which are in ${\displaystyle S_{n-1}}$ provides a triangulation of ${\displaystyle S_{n-1}}$ where if σ is a simplex then so is −σ. Let ${\displaystyle L:V(T)\to \{+1,-1,+2,-2,...,+n,-n\}}$ be a labeling of the vertices of T which is an odd function on ${\displaystyle S_{n-1}}$, i.e., ${\displaystyle L(-v)=-L(v)}$ for every vertex ${\displaystyle v\in S_{n-1}}$. Then Tucker's lemma states that T contains a complementary edge - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs.^[1]