Computational Topology, Con guration Spaces, Equivariant Maps

Key words and phrases Rhombic dodecahedron Polyhedral curtain ∆-curtain ∆-curtains ∆-curtain theorem Flat polyhedral complexes and illumination complexes Illumination complex A(K,F, S) Proof of the polyhedral curtain theorem Applications of the polyhedral curtain theorem Computational Algebraic Topology of measure partitions A Borsuk-Ulam theorem for the dihedral group D8 G -manifold complexes D8-manifold complexes

Computational Topology,Configuration Spaces,

Equivariant Maps

Rade Zivaljevic

Mathematical Institute SASA, Belgrade

Applied Topology - Bedlewo 2013


CS/TM-scheme

Configuration space/test map scheme (CS/TM)

• Configuration spaceA space X that parameterizes a class of geometric objects or combinatorialstructures.Examples: Arrangements of geometric objects like points, lines, polytopes,fans, flags, etc. or combinatorial structures like trees, graphs, partitions, etc.Fundamental problem: Explore the main constructions arising ‘in nature’(classification problem).

• Test space V and test map f : X → VTest map(s) fi : X → Vi alow us to discriminate configurations, tell onefrom another, and search for configurations with a particular (desired)properties.

• Topological analysisExample: Group of symmetries G acting on both X and V such that thetest map f : X → V is equivariant (Borsuk-Ulam type theorems).


J. Matousek, Using the Borsuk-Ulam Theorem. Lectures on TopologicalMethods in Combinatorics and Geometry, 2nd edition, Springer, 2007.

R.Z. Topological methods. Chapter 14 in Handbook of Discrete andComputational Geometry, J.E. Goodman, J. O’Rourke, eds, Chapman &Hall/CRC 2004, 305–330.http://www.rade-zivaljevic.appspot.com/articles.html

R.Z. User’s guide to equivariant methods in combinatorics, I and II. Publ. Inst.Math. (Beograd) (N.S.), (I) 59(73), 1996, 114–130 and (II) 64(78), 1998,107–132. Electronic versions available at,http://elib.mi.sanu.ac.rs/pages/browse_publication.php?db=publ.

http://www.rade-zivaljevic.appspot.com/articles.html

http://elib.mi.sanu.ac.rs/pages/browse_publication.php?db=publ


Rhombic dodecahedron


Rhombic dodecahedron (∆-zonotope)as a Minkowski sum of line segments

∆ = conv{a0, a1, . . . , ad}; K∆ = [0, a0] + [0, a1] + . . .+ [0, ad ].

K∆ is referred to as ∆-zonotope.


Rhombic dodecahedron as the dualof the root system of type Ad

(∆−∆)◦ = (conv(Ad))◦ =⋂i 6=j

Hi,j = R∆.


Rhombic dodecahedron in toric topology


Polyhedral curtain

Definition

A conical, polyhedral hypersurface D = cone(a,Σ), where Σ is a(d − 2)-dimensional polyhedral sphere (PL-sphere) and a /∈ Σ, is called apolyhedral curtain in Rd .


∆-curtain

Definition

Let ∆ = conv{a0, a1, . . . , ad} ⊂ Rd be a non-degenerate simplex with thebarycenter at the origin. For each pair θ = (F1,F2) of complementary faces of∆ there is a join decomposition ∆ = F1 ∗ F2. Assuming that both F1 and F2 arenon-empty let Sd−2

θ = ∂(F1) ∗ ∂(F2) ⊂ ∂(∆) be an associated(d − 2)-dimensional, polyhedral sphere. A polyhedral hypersurface H ⊂ Rd iscalled a ∆-curtain if for some θ = (F1,F2) and x ∈ Rd

H = x + Cθ = x + cone(Sd−2θ ). (1)


∆-curtains


∆-curtain theorem

Theorem(Polyhedral curtain theorem) Suppose that ∆ ⊂ Rd is a simplex with thebarycenter at the origin. Let µ1, µ2, . . . , µd be a collection of continuous massdistributions (measures) on Rd . Then there exists a ∆-curtain H = H(x,θ) which

divides the space Rd into two ‘half-spaces’ H+ and H− such that for eachj ∈ {1, . . . , d},

µj(H+) = µj(H−).

R. Z, Illumination complexes, ∆-zonotopes, and the polyhedral curtain theorem(arXiv).


Comparison with the ‘Ham sandwich theorem’of Banach and Steinhaus

H. Steinhaus. A note on the ham sandwich theorem. Mathesis Polska 9 (1938),26-28.W. A. Beyer, A. Zardecki. The early history of the ham sandwich theorem,American Mathematical Monthly 111 (2004).

The ‘Polyhedral curtain theorem’ is more combinatorial since it involvesalternatives (like Radon’s and Tverberg’s theorem). The space of ∆-curtains isdisconnected.


Flat polyhedral complexes

A flat polyhedral complex arises if several copies of a convex polyhedron (convexbody) K are glued together along some of their common faces (closed convexsubsets of their boundaries). Examples: ‘small covers’ and other locally standardZ2-toric manifolds, A.D. Alexandrov’s ‘flattened convex surfaces’.

A class of flat polyhedral complexes, modelled on a product of two (or more)simplices K = ∆p ×∆q, was used (M. de Longueville, R.Z., Adv. Math., 2008)for a proof of a multidimensional generalization of Alon’s ‘splitting necklacetheorem’ (N. Alon, Adv. Math., 1987).


Flat complexes as configuration spaces

Flat polyhedral complexes arise as ‘configuration spaces’ which encode(parameterize) admissible divisions (illuminations) of the polyhedron. Theobjective is to establish new ‘fair division theorems’, the simplest example beingthe ‘polyhedral curtain division’ for two players.


Illumination complex A(K ,F , S)

Illumination system:

• K ⊂ Rd is a convex polytope (or more generally a convex body);

• F is finite complex of convex cones in Rd (a convex fan);

• S is finite set of ‘colors’.

The cell complex A(K ,F ,S) encodes all possible ways to illuminate the convexbody K by translates of the fan F .


A(K ,F , S) as a flat polyhedral complex

A(K ,F ,S) = K × Fun(Fmax ,S)/ ∼

where Fun(Fmax ,S) is the set of all possible labelings (or colorings) of maximalcones in F by S . Two colorings f , g : F → S are equivalent (f ∼ g) if and onlyif f (V ) = g(V ) for each x-active cone V ∈ F . A cone V ∈ F is active atx ∈ K (or x-active) if volume((x + V ) ∩ K ) > 0.


A(K∆,F∆, S)

TheoremSuppose that F is a ∆-fan associated to a simplex ∆ ⊂ Rd . Let K = K∆ bethe associated ∆-zonotope (higher dimensional rhombic dodecahedron). Moreexplicitly, K∆ is the zonotope K∆ = [0, a0] + . . .+ [0, ad ], where 0 is thebarycenter and ai the vertices of ∆. Then,

A(K∆,F ,S) ∼= S ∗ S ∗ . . . ∗ S = S∗(d+1).

In particular, A(K∆,F , [2]) is a Z2-space homeomorphic to the sphere Sd .


A(K∆,F∆, [2]) is a sphere


Polyhedral curtain theorem (the case d = 2)

Suppose that X and Y are two (bounded) measurable sets in the plane. LetK = K [F ] be a hexagon such that K ⊇ A ∪ B and let A(K ,F , [2]) be theassociated flat polyhedral complex. Apply the Borsuk-Ulam theorem to the testmap

φ : A(K ,F , [2])→ R2, φ((x , θ)) = (m(S∩X )−m(B∩X ),m(S∩Y )−m(B∩Y )).


Applications of the polyhedral curtain theorem

It may be interesting to review some standard applications of the ‘HamSandwich Theorem’ in order to check whether the ‘Polyhedral Curtain Theorem’has similar consequences.

Example

Quadratic spline equipartitions of several planar sets as an analogue of the‘Polynomial Ham Sandwich Theorem’ of Stone and Tukey.

Suppose that X1,X2, . . . ,X5 are five measurable sets in the plane. LetV : R2 → R5, V (x , y) = (x , y , xy , x2, y 2), be the Veronese embedding. Anapplication of the polyhedral curtain theorem in R5 yields a piecewise quadraticpolynomial curve (a quadratic spline) which divides each of the sets Xi in twoparts of equal measure.


Equipartition by circular splines

Example

Suppose that X1,X2,X3 are three measurable sets in the plane. LetW : R2 → R3 be the embedding defined by W (x , y) = (x , y , x2 + y 2). Anapplication of the polyhedral curtain theorem in R3 yields a piecewise circularcurve (a circular spline) which cuts each of the sets Xi in two parts of equalmeasure. The number of nodes of the spline is controlled by the number offaces in the polyhedral curtain.


Equipartitions by hyperplanes

TheoremEach collection of j = 4 · 2k + 1 measures in Rd , where d = 6 · 2k + 2, admitsan equipartition by two hyperplanes.

(R. Z, Computational topology of equipartitions by hyperplanes (arXiv).)

Definition

A triple (d , j , k) is called admissible if for each collection M of j continuousmeasures on Rd there exists an equipartition of M by k hyperplanes.

After E.A. Ramos, Equipartitions of mass distributions by hyperplanes. DiscreteComput. Geom. (1996), one of benchmark problems for applying topologicalmethods in computational geometry is to evaluate/estimate

∆(j , k) = min{d | (d , j , k) is admissible }.

It follows from the theorem that,

∆(4 · 2k + 1, 2) = 6 · 2k + 2.


Borsuk-Ulam theorem for the group D8

TheoremThere does not exist a D8-equivariant map

f : Sd × Sd → S(W⊕j) (2)

where D8 is the dihedral group of order eight, d = 6 · 2k + 2, j = 4 · 2k + 1 forsome integer k ≥ 0, and W is a representation space of the real 3-dimensionalD8-representation.


The obstruction is twice the generator

The first obstruction to the existence of (2) lies in an equivariant cohomologygroup,

H2d−1D8

(Sd × Sd ,Z) ∼= Z/4

where,Z = H2d−2(S(W⊕j);Z) and 2d − 3j = 1.

The obstruction vanishes unless

d = 6 · 2k + 2 and j = 4 · 2k + 1

when it turns out to be equal to 2X where X is a generator of the group Z/4.


The first obstruction to chain maps

Proposition

C∗ := {Ck}n+1k=−1 and D∗ := {Dk}n+1

k=−1 are chain complexes of Λ = Z[G ]

modules where C−1∼= D−1

∼= Z. Fn−1 := (fj)n−1j=−1 : {Ck}n−1

k=−1 → {Dk}n−1k=−1

exists and is fixed in advance. Suppose that there exists a homomorphismfn : Cn → Dn such that ∂ ◦ fn = fn−1 ◦ ∂. Then the obstruction to the existenceof a chain map Fn+1 := (fj)

n+1j=−1 : {Ck}n+1

k=−1 → {Dk}n+1k=−1, extending the chain

map Fn−1, is a well defined element θ of the cohomology group

Hn+1(C∗; Hn(D∗)) = Hn+1(Hom(C∗,Hn(D∗))).

Cn+1∂ //

fn+1

��

Cn∂ //

fn��

Cn−1//

fn−1

��

. . . // C1∂ //

f1��

C0//

f0��

Z∼=��

// 0

Dn+1// Dn

// Dn−1// . . . // D1

// D0// Z // 0


The obstruction θ

Moreover, θ is represented by the cocycle

θ(fn) : Cn+1∂−→ Cn

fn−→ Zn(D∗)π−→ Hn(D∗). (3)

The vanishing of θ is not only necessary but also sufficient for the existence ofthe chain map Fn+1 if Cn and Cn+1 are free (or projective) modules.

Question: How to associate an economical chain complex to a G -space?

Answer: Use G -manifold complexes as an alternative to G -CW complexes.A natural G -manifold complex arises by iterating the

‘fundamental domain – geometric boundary’ procedure.


D8-manifold complex

D8-screens for S2 × S2 ⊂ R3 × R3.


Fundamental domain/geometric boundary procedure

D8-screens for Sn × Sn ⊂ Rn+1 × Rn+1 and the associated admissible filtration.


The tree of manifold ‘cells’

0 ≤ x1 ≤ 10 ≤ y1 ≤ 1y1 ≤ x1

∂

yy

∂

&&[y1 = 0]

�� 〈β〉

[x1 = y1]

��〈γ〉

[0 ≤ y2]

∂

��∂

��

[y2 ≤ 0] [x2 ≤ y2] [y2 ≤ x2]

∂

��∂

��x1 = 0y1 = 00 ≤ y2

y1 = 0y2 = 00 ≤ x1

x1 = 0y1 = 0y2 ≤ x2

x1 = y1

x2 = y2

0 ≤ x1


Banach-Steinhaus and Borsuk-Ulam theorems

• Stefan Banach (1892 − 1945)

• Karol Borsuk (1905 − 1982)

• Wladyslaw Hugo Dionizy Steinhaus (1887 − 1972)

• Stanislaw Marcin Ulam (1909 − 1984)

Computational Topology, Con guration Spaces, Equivariant Maps

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