page_title: Tucker's lemma
hierarchical_section_title: Tucker's lemma
caption_attribution_description: English: Example, in 2 dimensions, of Tucker's lemma.
context_page_description: 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. Assume T is antipodally symmetric on the boundary sphere. That means that the subset of simplices of T which are in provides a triangulation of where if σ is a simplex then so is −σ.
  Let be a labeling of the vertices of T which is an odd function on, i.e, for every vertex.
  Then Tucker's lemma states that T contains a complementary edge - an edge whose vertices are labelled by the same number but with opposite signs.
context_section_description: 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 . Assume T is antipodally symmetric on the boundary sphere . That means that the subset of simplices of T which are in  provides a triangulation of  where if σ is a simplex then so is −σ.
  Let  be a labeling of the vertices of T which is an odd function on , i.e,  for every vertex .
  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.
caption: In this example, where n=2, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist.
exif: {}