Sperner, Tucker and Ky Fans lemmas for manifolds Oleg R. Musin - PowerPoint PPT Presentation

Sperner, Tucker and Ky Fan’s lemmas Sperner, Tucker and Ky Fan’s lemmas for manifolds Oleg R. Musin University of Texas at Brownsville IITP RAS SJTU, March 19, 2014

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma Theorem (Sperner, 1928) Every Sperner labelling of a triangulation of a d-dimensional simplex contains a cell labelled with a complete set of labels: { 1 , 2 , . . . , d + 1 } .

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma 1 2 1 1 2 2 1 3 1 2 3 3 Figure: A 2-dimensional illustration of Sperner’s lemma

Sperner, Tucker and Ky Fan’s lemmas Emanuel Sperner Emanuel Sperner (9 December 1905 – 31 January 1980) was a German math- ematician, best known for two the- orems. He was a student at Ham- burg University where his adviser was Wilhelm Blaschke. He was appointed Professor in K¨ onigsberg in 1934, and subsequently held posts in a number of universities until 1974.

Sperner, Tucker and Ky Fan’s lemmas Sperner’s theorems The Sperner theorem, from 1928, says that the size of an antichain in the power set of an n-set is at most the middle binomial coefficient. It has several proofs and numerous generalizations. Sperner’s lemma, from 1928, states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. It was proven by Sperner to provide an alternate proof of a theorem of Lebesgue characterizing dimensionality of Euclidean spaces. It was later noticed that this lemma provides a direct proof of the Brouwer fixed-point theorem without explicit use of homology.

Sperner, Tucker and Ky Fan’s lemmas Albert W. Tucker Albert William Tucker (28 November 1905 - 25 January 1995) completed his Ph.D. at the Princeton University under the supervision of Solomon Lefschetz.

Sperner, Tucker and Ky Fan’s lemmas Albert W. Tucker He was a faculty in Princeton from 1933 till 1974. He chaired the mathematics department for about twenty years, one of the longest tenures. His Ph.D. students include Michel Balinski, David Gale, Alan Goldman, John Isbell, Stephen Maurer, Marvin Minsky, Nobel Prize winner John Nash, Torrence Parsons, Lloyd Shapley, Robert Singleton, and Marjorie Stein. In 1945, Albert Tucker proved “Tucker’s lemma”. He is also well known for the Karush - Kuhn - Tucker conditions, a basic result in non-linear programming.

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma Theorem (Tucker, 1945) B d that is antipodally Let Λ be a triangulation of the ball symmetric on the boundary. Let L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + d , − d } be a labelling of the vertices of Λ that satisfies L ( − v ) = − L ( v ) for every vertex v on the boundary B d . Then there exists an edge in Λ that is “complementary”: i.e., its two vertices are labelled by opposite numbers.

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma

Sperner, Tucker and Ky Fan’s lemmas Ky Fan Ky Fan (September 19, 1914 - March 22, 2010) was an American mathematician and Emeritus Professor of Mathematics at the University of California, Santa Barbara (UCSB).

Sperner, Tucker and Ky Fan’s lemmas Fan’s lemma Theorem (Ky Fan, 1952) Let Λ be an antipodal triangulation of S d . Suppose that each vertex v of Λ is assigned a label L ( v ) from {± 1 , ± 2 , . . . , ± n } in such a way that L ( − v ) = − L ( v ) . Suppose this labelling does not have complementary edges. Then there are an odd number of d-simplices of Λ whose labels are of the form { k 0 , − k 1 , k 2 , . . . , ( − 1 ) d k d } , where 1 ≤ k 0 < k 1 < . . . < k d ≤ n. In particular, n ≥ d + 1 .

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma for manifolds Let T be a triangulation of a PL manifold M d . Suppose that each vertex of T is assigned a unique label from the set { 1 , 2 , . . . , n } . Such a labelling is called a n-labelling of T . We say that a d -simplex in T is a fully labelled simplex or simply a full cell if all its labels are distinct. Theorem Let T be a triangulation of a closed PL manifold M d . Any ( d + 1 ) -labelling of T must contain an even number of full cells.

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma for manifolds Let L : V ( T ) → { 1 , 2 , . . . , d + 1 } be a ( d + 1 ) -labelling of T . Then L defines a simplicial map f L : M → R d . Indeed, let v 1 , . . . , v d + 1 are vertices of a d -simplex s in R d . We define a piecewise linear map f L : T → R d for v ∈ V ( T ) by f L ( v ) = v i if L ( v ) = i . Therefore, for a d -simplex c of T we have f L ( c ) = s if and only if c is fully labelled.

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma for manifolds Corollary Let P denotes the boundary of a d-simplex. Let M d be a PL manifold with boundary S d − 1 = P. Any ( d + 1 ) -labelling of a triangulation of M which is a Sperner labelling on the boundary must contain an odd number of full cells; in particular, there is at least one.

Sperner, Tucker and Ky Fan’s lemmas M¨ obius band Figure: M¨ obius band. Diametrically opposite points of the inner boundary circle are to be identified. The outer circle is the boundary of the M¨ obius band.

Sperner, Tucker and Ky Fan’s lemmas Sperner’s lemma for the M¨ obius band 1 2 1 1 2 2 1 3 1 2 3 3

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma for manifolds Theorem Let Λ be a triangulation of a PL manifold M d with the boundary S d − 1 . Let Λ be antipodally symmetric on the boundary. Let L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + d , − d } be a labelling of the vertices of Λ that satisfies L ( u ) = − L ( v ) for all antipodal vertices u , v on the boundary. Then Λ contains a complimentary edge.

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma for the M¨ obius strip +2 +1 +2 +1 +2 +2 -1 -1 -1 -2 -1 -1 -2 -1 -2

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma for spheres Theorem Let Λ be an antipodal triangulation of S d . Let L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + d , − d } be an antipodal labelling of the vertices of Λ that satisfies L ( − v ) = − L ( v ) for all vertices. Then Λ contains a complimentary edge.

Sperner, Tucker and Ky Fan’s lemmas BUT manifolds Theorem (M., 2012) Let M n be a closed connected manifold with a free involution T. Then the following statements are equivalent: (a) For any antipodal (i.e. f ( T ( p )) = − f ( p ) ) continuous map f : M n → R n the set Z f := { f − 1 ( 0 ) } is not empty. (b) M admits an antipodal continuous transversal map h : M n → R n with | Z h | = 4 k + 2 , k ∈ Z . (c) [ M n , T ] = [ S n , A ] + [ V 1 ][ S n − 1 , A ] + . . . + [ V n ][ S 0 , A ] in N n ( Z 2 ) .

Sperner, Tucker and Ky Fan’s lemmas BUT manifolds (d) M is a Lusternik-Shnirelman type manifold, i.e. for any cover F 1 , . . . , F n + 1 of M n by n + 1 closed (respectively, by n + 1 open) sets, there is at least one set containing a pair ( x , T ( x )) . (e) cat ( M / T ) = cat ( RP n ) = n , where cat ( X ) is the Lusternik-Shnirelman category of a space X , i.e. the least m such that there exists open covering U 1 , . . . , U m + 1 of X with each U i contractible to a point in X .

Sperner, Tucker and Ky Fan’s lemmas BUT manifolds: examples S d M 2 2 g P 2 2 m M # M

Sperner, Tucker and Ky Fan’s lemmas A polytopal Tucker’s lemma for manifolds Theorem Let P be a centrally symmetric set in R d with 2 n points. Let points of P are equivariantly labelled by { + 1 , − 1 , + 2 , − 2 , . . . , + n , − n } . Let M d be a closed PL manifold with a free involution. Let Λ be any equivariant triangulation of M. Let L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + n , − n } be an equivariant labelling. Then M d is a BUT manifold if and only if there exists a k-simplex s in Λ with labels such that the simplex which is formed by points of P with the same labels contains 0 .

Sperner, Tucker and Ky Fan’s lemmas Proof f L , P : M d → R d f − 1 L , P ( 0 ) ∈ s

Sperner, Tucker and Ky Fan’s lemmas Tucker’s lemma for manifolds Corollary A closed PL free Z 2 -manifold M d is BUT if and only if for any equivariant labelling L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + d , − d } of any equivariant triangulation Λ of M there exists a complementary edge.

Sperner, Tucker and Ky Fan’s lemmas Fan’s lemma for manifolds Theorem Let M d be a closed PL BUT manifold with a free involution. Let Λ be any equivariant triangulation of M. Let L : V (Λ) → { + 1 , − 1 , + 2 , − 2 , . . . , + n , − n } be an equivariant labelling. Then there is a complementary edge or an odd number of d-simplices whose labels are of the form { k 0 , − k 1 , k 2 , . . . , ( − 1 ) d k d } , where 1 ≤ k 0 < k 1 < . . . < k d ≤ n.

Sperner, Tucker and Ky Fan’s lemmas ACS polytope Definition. Let P be a convex polytope in R d with 2 n centrally symmetric vertices { p 1 , − p 1 , . . . , p n , − p n } . We say that P is ACS (Alternating Centrally Symmetric) ( n , d ) -polytope if the set of all simplices in cov P ( 0 ) , that contain the origin 0 of R d inside, consists of edges ( p i , − p i ) and d -simplices with vertices { p k 0 , − p k 1 , . . . , ( − 1 ) d p k d } and { − p k 0 , p k 1 , . . . , ( − 1 ) d + 1 p k d }, where 1 ≤ k 0 < k 1 < . . . < k d ≤ n .