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

• f labels: {1, 2, . . . , d + 1}.

Sperner, Tucker and Ky Fan’s lemmas

Sperner’s lemma

1 1 1 3 2 2 1 3 2 2 1 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-

• rems.

He was a student at Ham- burg University where his adviser was Wilhelm Blaschke. He was appointed Professor in K¨

• nigsberg in 1934, and

subsequently held posts in a number

• f 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) Let Λ be a triangulation of the ball Bd that is 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(−v) = −L(v) for every vertex v on the boundary Bd. Then there exists an edge in Λ that is “complementary”: i.e., its two vertices are labelled by

• pposite 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 Sd. 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 {k0, −k1, k2, . . . , (−1)dkd}, where 1 ≤ k0 < k1 < . . . < kd ≤ 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 Md. 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 Md. 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 fL : M → Rd. Indeed, let v1, . . . , vd+1 are vertices of a d-simplex s in Rd. We define a piecewise linear map fL : T → Rd for v ∈ V (T) by fL(v) = vi if L(v) = i. Therefore, for a d-simplex c of T we have fL(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 Md be a PL manifold with boundary Sd−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¨

• bius band

Figure: M¨

• bius band. Diametrically opposite points of the inner

boundary circle are to be identified. The outer circle is the boundary of the M¨

• bius band.

Sperner, Tucker and Ky Fan’s lemmas

Sperner’s lemma for the M¨

• bius band

1 1 1 3 2 2 1 3 2 2 1 3

Sperner, Tucker and Ky Fan’s lemmas

Tucker’s lemma for manifolds

Theorem Let Λ be a triangulation of a PL manifold Md with the boundary Sd−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¨

• bius strip

+1 +1 +2 +2

• 1
• 1
• 2
• 2
• 1
• 1
• 1
• 1
• 2

+2 +2

Sperner, Tucker and Ky Fan’s lemmas

Tucker’s lemma for spheres

Theorem Let Λ be an antipodal triangulation of Sd. 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 Mn 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 : Mn → Rn the set Zf := {f −1(0)} is not empty. (b) M admits an antipodal continuous transversal map h : Mn → Rn with |Zh| = 4k + 2, k ∈ Z. (c) [Mn, T] = [Sn, A] + [V 1][Sn−1, A] + . . . + [V n][S0, A] in Nn(Z2).

Sperner, Tucker and Ky Fan’s lemmas

BUT manifolds

(d) M is a Lusternik-Shnirelman type manifold, i.e. for any cover F1, . . . , Fn+1 of Mn 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(RPn) = n, where cat(X) is the Lusternik-Shnirelman category of a space X, i.e. the least m such that there exists open covering U1, . . . , Um+1 of X with each Ui contractible to a point in X.

Sperner, Tucker and Ky Fan’s lemmas

BUT manifolds: examples

Sd M2

2g

P2

2m

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 Rd with 2n points. Let points

• f P are equivariantly labelled by {+1, −1, +2, −2, . . . , +n, −n}.

Let Md 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 Md 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

fL,P : Md → Rd f −1

L,P(0) ∈ s

Sperner, Tucker and Ky Fan’s lemmas

Tucker’s lemma for manifolds

Corollary A closed PL free Z2-manifold Md is BUT if and only if for any equivariant labelling L : V (Λ) → {+1, −1, +2, −2, . . . , +d, −d}

• f 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 Md 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 {k0, −k1, k2, . . . , (−1)dkd}, where 1 ≤ k0 < k1 < . . . < kd ≤ n.

Sperner, Tucker and Ky Fan’s lemmas

ACS polytope

• Definition. Let P be a convex polytope in Rd with 2n centrally

symmetric vertices {p1, −p1, . . . , pn, −pn}. We say that P is ACS (Alternating Centrally Symmetric) (n, d)-polytope if the set of all simplices in covP(0), that contain the origin 0 of Rd inside, consists of edges (pi, −pi) and d-simplices with vertices {pk0, −pk1, . . . , (−1)dpkd} and {−pk0, pk1, . . . , (−1)d+1pkd}, where 1 ≤ k0 < k1 < . . . < kd ≤ n.

Sperner, Tucker and Ky Fan’s lemmas

ACS polytope

Theorem For any integer d ≥ 2 and n ≥ d there exists ASC (n, d)-polytope.

Sperner, Tucker and Ky Fan’s lemmas

Convex curves in Rd

• Definition. We say that a curve K in Rd, which is an Euclidean

space of dimension d, is convex if it intersects any hyperplane at no more than d points We consider the cases of a closed and a nonclosed curve, i.e., the cases where the curve K is an embedding of a circle, σ : S1 → Rd, and an interval, σ : (a, b) → Rd. Note that in Rd there exist closed convex curves if only if d is even.

Sperner, Tucker and Ky Fan’s lemmas

Examples of convex curves in Rd

Example 1. Planar convex curves. Example 2. Let the curve K in Rd be defined as x(t) = (t, t2, . . . , td), t ∈ (a, b). A hyperplane H : a0 + a1x1 + . . . + adxd = 0 intersects K in t which are roots of the polynomial Pd(t) = a0 + a1t + . . . + adtd. Thus the curve K is convex.

Sperner, Tucker and Ky Fan’s lemmas

Examples of convex curves in Rd

Example 3. Let x(t) = (cos(t), . . . , cos(kt), sin(t), . . . , sin(kt)), t ∈ [0, 2π], where d = 2k. It is a classical trigonometric Chebyshev system. Then K is a convex closed curve in Rd. Example 4. It is well known that the collection of arbitrary power functions {1, tα1, . . . , tαd}, where 0 < α1 < · · · < αd, forms a Chebyshev system on an arbitrary interval. If we consider the curve K on Rd: x(t) = (tα1, tα2, . . . , tαd), t ∈ (a, b), a > 0, we see that it is convex.

Sperner, Tucker and Ky Fan’s lemmas

Radon partitions

Radon’s theorem on convex sets states that any set S of d + 2 points in Rd can be partitioned into two (disjoint) sets A and B whose convex hulls intersect. Moreover, if rank(S) = d, then this partition is unique. The partition S = A B is called the Radon partition of S.

Sperner, Tucker and Ky Fan’s lemmas

Radon partitions and convex curves

Theorem A curve K in Rd is convex if and only if for any (d + 2)-subset S of K its Radon partition sets A and B alternate along K.

Sperner, Tucker and Ky Fan’s lemmas

ASC (n, d)-polytope

Corollary Let q1, . . . , qn be points on a convex curve K in Rd−1. Let pi = (qi, 1) ∈ Rd. Denote by P(n, d) a convex polytope with vertices {p1, −p1, . . . , pn, −pn}. Then P(n, d) is an ASC (n, d)-polytope.

Sperner, Tucker and Ky Fan’s lemmas

Thank you