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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $T$ be a triangulation of the [[closed ball|0Q0H13G8ZB]] $B^n \\subset \\R^n$ by a finite [[simplicial complex|V8GZ0W1ZZ5]].\nAssume that $T$ is antipodally symmetric on the boundary.\nThis means that $-\\sigma$ is a simplex of $T$ for every simplex $\\sigma$ of $T$ contained in the boundary sphere.\nLet $\\lambda$ be a labelling of the vertices of $T$ by elements of $\\set{\\pm 1, \\pm 2, \\ldots, \\pm n}$ such that $\\lambda(-v) = -\\lambda(v)$ for every vertex $v$ on the boundary.\nThen $T$ has a *complementary edge*: an edge $\\set{u, w}$ of $T$ with $\\lambda(u) = -\\lambda(w)$.\n\n## Remarks\n\nTucker's lemma is a combinatorial form of the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].\nTo derive the lemma from Borsuk–Ulam, send a vertex with label $\\pm i$ to $\\pm e_i$ and extend linearly on simplices to a map $f \\colon B^n \\to \\R^n$.\nWith no complementary edge, the image of $f$ avoids $0$.\nThen $f / \\norm{f}$ restricts to an odd map from the boundary sphere $S^{n-1}$ to $S^{n-1}$ that extends over the ball, which is impossible.\nConversely, applying the lemma to finer and finer triangulations proves the Borsuk–Ulam theorem.\nThere are also constructive path-following proofs, which give algorithms to find a complementary edge.\n\nExample: for $n = 1$, subdivide $[-1, 1]$ into intervals and label the vertices by $\\pm 1$ with $\\lambda(-1) = -\\lambda(1)$.\nMoving from $-1$ to $1$ the label changes sign, so some interval of the subdivision has endpoints labelled $1$ and $-1$.\n\nThe antipodal condition on the boundary is needed: if every vertex gets the label $1$, no edge is complementary.\n\n## Explanations\n\n### L1 human\n\nCut a disc into small triangles and write one of the labels $\\pm 1, \\pm 2$ at each corner, so that opposite points on the rim get opposite labels.\nThen somewhere two neighbouring corners carry opposite labels, such as $2$ and $-2$.\n\n### L2 human\n\nTucker's lemma is to the Borsuk–Ulam theorem what [[Sperner's lemma|HQG3V5Z6MR]] is to the Brouwer fixed-point theorem.\nBecause it is finite and combinatorial, it is the usual route to applications of Borsuk–Ulam in combinatorics, such as necklace splitting and Kneser's conjecture, and to algorithms for them.\n",
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