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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 515.164 DOI 10.22405/2226-8383-2020-21-2-207-227
Обобщённые шахматные комплексы и дискретная теория Морса1
Д. Иоич, Г. Панина, С. Т. Вречица, Р. Т. Живалевич
Йоич Душко — доктор наук, профессор, факультет математики и естественных наук, Баня-Лукский университет (г. Баня-Лука, Босния и Герцеговина). e-mail: ducci68@teol.net
Панина Гаянэ Юрьевна — доктор физико-математических наук, ведущий научный сотрудник, Санкт-Петербургский государственный университет, Санкт-Петербургское отделение Математического института им. В. А. Стеклова (г. Санкт-Петербург). e-mail: gaiane-panina@rambler.ru
Вречица Синиша Т. — доктор наук, профессор, математический факультет, Белградский университет (г. Белград, Сербия). e-mail: vrecica@matf.bg.ас.rs
Живалевич Раде Т. — доктор наук, профессор, математический факультет, Белградский университет, Математический институт, САНУ (г. Белград, Сербия). e-mail: rade@mi.sanu.ас.rs
Аннотация
Шахматные комплексы и их обобщения, как объекты, и дискретная теория Морса, как инструмент, представлены в виде объединяющей темы, связывающая различные области геометрии, топологии, алгебры и комбинаторики. Теорема Эдмондса и Фулкерсо-на о бутылочном горлышке (минимаксе) реализуется и интерпретируется как результат о критической точке дискретной функции Морса на сфере Вира Bier(K) ассоциированного симплициального комплекса К. Мы проиллюстрируем использование «стандартных дискретных функций Морса» на обобщенных шахматных комплексах, доказав результат связности для шахматных комплексов с кратностями. Приложения включают новые результаты типа Тверберга-Ван Кампена-Флореса для разбиений симплекса без j-кратных пересечений.
Ключевые слова: шахматные комплексы, дискретная теория Морса, теорема о бутылочном горлышке, теоремы Тверберга-ван Кампена-Флореса.
Библиография: 42 названия. Для цитирования:
Д. Иоич, Г. Панина, С. Т. Вречица, Р. Т. Живалевич. Обобщенные шахматные комплексы и дискретная теория Морса // Чебышевский сборник, 2020, т. 21, вып. 2, с. 207-227.
1Это исследование было поддержано Грантами 174020 и 174034 Министерства образования, Наука и технологическое развитие Республики Сербия. Авторы отмечают гостеприимство Математического института в Обервольфахе, где была завершена эта работа.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 515.164 DOI 10.22405/2226-8383-2020-21-2-207-227
Generalized chessboard complexes and discrete Morse theory
D. Jojic, G. Panina, S. T. Vrecica, R. T. Zivaljevic
Jojic Dusko — Doctor of Sciences, Professor, Faculty of Science, University of Banja Luka (Banja Luka, Bosnia and Herzegovina). e-mail: ducci68@teol.net
Panina Gaiane Yur'evna — doctor of physical and mathematical Sciences, Leading Researcher, St. Petersburg State University, St. Petersburg Department of Steklov Mathematical Institute (St. Petersburg).
e-mail: gaiane-panina@rambler.ru
Vrecica Sinisa T. — Doctor of Sciences, Professor, Faculty of Mathematics, University of Belgrade
(Belgrade, Serbia).
e-mail: vrecica@matf.bg.ac.rs
Zivaljevic Rade T. — Doctor of Sciences, Professor, Faculty of Mathematics, University of Belgrade, Mathematical Institute, SASA (Belgrade, Serbia). e-mail: rade@mi.sanu.ac.rs
Abstract
Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted as a result about a critical point of a discrete Morse function on the Bier sphere Bier(K) of an associated simplicial complex K. We illustrate the use of "standard discrete Morse functions" on generalized chessboard complexes by proving a connectivity result for chessboard complexes with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for j-wise disjoint partitions of a simplex.
Keywords: chessboard complexes, discrete Morse theorey, bottleneck theorem, Tverberg-Van Kampen-Flores theorems.
Bibliography: 42 titles. For citation:
D. Jojic, G. Panina, S. T. Vrecica, R. T. Zivaljevic, 2020, "Generalized chessboard complexes and discrete Morse theory" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 207-227.
1. Introduction
Chessboard complexes and their relatives have been for decades an important theme of topological combinatorics. They have found numerous and often unexpected applications in group theory, representation theory, commutative algebra, Lie theory, discrete and computational geometrv, algebraic topologv, and geometric and topological combinatorics, see fl], [2], [3] [5], [15], [16], [26], [28], [33], [35], [36], [38], [39], [40].
The books [25] and [31], as well as the review papers [38] and [41], cover selected topics of the theory of chessboard complexes and contain a more complete list of related publications.
Chessboard complexes and their generalizations are some of the most studied graph complexes [25]. From this point of view chessboard complexes can be interpreted as relatives of L. Lovasz Hom-complexes [29], matching complexes, clique complexes, and many other important classes of simplicial complexes.
More recently new classes of generalized chessboard complexes have emerged and new methods, based on novel shelling techniques and ideas from Forman's discrete Morse theory, were introduced. Examples include multiple and symmetric multiple chessboard complexes [22, 23], Bier complexes [18], and deleted joins of collectively unavoidable complexes, see [18] and [20, 21]. Among applications are the resolution of the balanced case of the "admissible/prescribable partitions conjecture" [23], general Van Kampen-Flores type theorem for balanced, collectively unavoidable complexes [20], and "balanced splitting necklace theorem" [21].
This paper is both a leisurely introduction and an invitation to this part of topological combinatorics, and a succinct overview of some of the ideas of discrete Morse theory, combinatorics and equivariant topology, used in our earlier papers.
New results are in Sections 5, 6 and 7. They include an alternative treatment of Edmonds and Fulkerson bottleneck (minmax) theorem (Section 5) and the construction of "standard discrete Morse functions" on generalized chessboard complexes with multiplicities (Section 6). This leads to a frequently optimal connectivity result for generalized chessboard complexes with multiplicities (Theorem 2 in Section 6), which is used in Section 7 for the proof of new Tverberg-Van Kampen-Flores type results for j-wise disjoint partitions of a simplex.
2. Chessboard complexes
Chessboard complexes naturally arise in the study of the geometry of admissible rook configurations on a general (m x n)-chessboard. An admissible configuration is any non-taking placement of rooks, i.e., a placement which does not allow any two of them to be in the same row or in the same column. The collection of all these placements forms a simplicial complex which is called the chessboard complex and denoted by Am,n.
More formally, the set of vertices of ATO,ra is Vert(ATO,ra) = [m] x [n] and S C [m] x [n] is a simplex of Am,n if and only if to each two distinct elements (i1,j1), (i2,j2) G S neither i1 = i2 nor ji = 322.1. An example
Let us take a closer look at one of the simplest chessboard complexes, the complex A4,3, based on the 4 x 3 chessboard (see Figure 1).
The /-vector of A4,3 is f (A4,3) = (12,36,24) so its Euler characteristics is %(A4,3) = 0. Moreover, the geometric realization of A4,3 is an orient able 2-dimensional manifold.
Indeed, the link of each vertex is isomorphic to A3,2 (= hexagonal triangulation of the circle S1) while the link of each edge is the circle S0. Each 2-dimensional simplex a = [Ai,Bj,Ck} is uniquely completed to a permutation n = (i,j, k, I) of the set [4] = {1, 2, 3, 4} and Sign(a) := Sign(^) defines
A4,3
A4,3 2
T2. The universal covering of A4,3 is identified as the honeycomb tiling of the plane and the corresponding fundamental domain is exhibited in Figure 1. From here we can easily read off the generators of the group fli(A4,3; Z) = Z2 as the geodesic edge-paths connecting the three copies of vertex C3, shown in Figure 1.
12 3 4
A B C
'3 C4
C-.
B2 B1
Figure 1: Chessboard complex A4,3
Bi
2.2. Graph complexes
Let G be a finite graph with vertex set V = Vg and edge set E = Eg- A graph complex on G is an abstract simplicial complex consisting of subsets of E. We usually interpret such a complex as a family of subgraphs of G. The study of graph complexes, with the emphasis on their homology, homotopv type, connectivity degree, Cohen-Maeaulayness, etc., has been an active area of study in topological combinatorics, see [25].
The chessboard complex Am,n can be interpreted as a graph complex of the complete bipartite graph Km,n, where the simplices S c [m] x [n] are interpreted as "matchings" in Km,n. Recall that r C Eg is a matching on the graph G if each v £ Vg is incident to at most one edge in r.
All "generalized chessboard complexes", introduced in Section 3, can be also described as graph complexes of the graph Km,n.
C
2
B
2.3. Chessboard complexes as Tits coset complexes
Perhaps the first appearance of chessboard complexes was in the thesis of Garst [16], as Tits coset complexes. Recall that a Tits coset complex A(G; G1,..., Gn), associated to a group G and a family [G\,..., Gn] of its subgroups is the nerve Nerve(F) of the associated family of cosets T = [gGi | g G G,i G [n]}. More explicitly vertices of A(G; G\,..., Gn) are cosets gGi and a colection S = [gjGi}(¿j)eI, for some I ç [n] x G, is a simplex of A(G; G\,..., Gm) if and only if
n 9i = 0.
(i,j)ei
If G = Sm is the symmetric group and Gi := [^ G Sm | -iï(i) = i} for i = 1,... ,n, the associated Tits coset complex is the chessboard complex Am,n.
2.4. Chessboard complexes in discrete geometry
Chessboard complexes made their first appearance in discrete geometry in [40], in the context of the so called colored Tverberg problem.
For illustration, an instance of the type. B colored Tverberg theorem [35, 41] claims that for each collection C C R3 of fifteen points in the 3-space, evenly colored bv three colors, there exist three vertex disjoint triangles Ai, A2, A3, formed by the points of different color, such that Ai n A2 n A3 = 0.
A general form of this result was deduced in [35] from a Borsuk-Ulam type result claiming that each Zr-equivariant map
(Ar>-ir(fc+1) —^ W®(d+1) (1)
must have a zero if r < d/(d — k) (this is a necessary condition), r is a prime power, Ar,2r-1 is a chessboard complex, and Wr = {x G Rr | x1 + ■ ■ ■ + xr = 0}.
The reader is referred to [41] for an overview of these and more recent results, as well as for a more complete list of references.
3. Generalized chessboard complexes
Motivated primarily by applications to problems in discrete geometry, especially the problems of Tverberg and Van Kampen-Flores type, more general chessboard complexes were introduced and studied. Closely related complexes previously emerged in algebraic combinatorics [28, 38].
These complexes are also referred to as generalized chessboard complexes, since the set of vertices remains the (m x n)-chessboard [m] x [n], but the criterion for S C [m] x [n] to be a simplex ("admissible rook placement") may be quite different and vary from problem to problem.
The following definition includes most if not all of the currently studied examples and provides a natural ecological niche for all these complexes and their relatives.
De pinition 1. Suppose that K. = {Ki}™=i and L = {Lj}rp=i are two collections of simplicial complexes where Vert(K^) = [m] for each i G [n] and Vert(Lj) = [n] for each j G [m\. Define,
Arn% = Am,n( fc, L) (2)
as the complex of all subsets (rook-placem,ents) A C [m] x [n] such that {i G [m] | (i,j) G A} G Kj for each j G [n] an d {j G [n] | (i,j) G A} G Li for ea ch i G [m\.
Definition 1 can be specialized in many ways. Again, we focus on the special cases motivated by intended applications to the generalized Tverberg problem.
De pinition 2. Suppose that k = (ki)™=i and l = (lj)7=i are two sequences of non-negative integers. Then the complex,
ak,l = \ki,...,k„;li,...,lm /on
Am,n Am,n \'J)
arises as the complex of all rook-placements A C [m] x [n] such that at most ki rooks are allowed to be in the i-th row (for i = 1,... ,n), and at most I j rooks are allowed to be in the j-th column (for j = 1,... ,m).
Remark 1. The complexes Aj^ = Am}n,kn,ll,...,lm are sometimes referred to as the chessboard complexes with multiplicities or multiple chessboard complexes. Closely related are "bounded degree graph complexes", studied in [28] and [38].
When k1 = ■ ■ ■ = kn = p and l1 = ■ ■ ■ = lm = q, we obtain the complex A™n. For the reasons which will become clear in the following section of the paper, in our earlier papers [22, 23] we focused to the case l1 = ■ ■ ■ = lm = 1, i.e. to the complexes,
a ki,...,k„;1 := a fci,...,fcn;1,...,1 /a\
Am,n : Am,n . K^J
In Section 6 of this paper we fill this "gap" and return to the case of general chessboard complexes with multiplicities.
3.1. n-fold j-wise deleted join
Joins and deleted joins of simplicial complexes, as well as their generalizations, have found numerous applications in topological combinatorics, see [31, Section 6.3] for motivation and an introduction.
For a simplicial complex K, the n-fold j-wise deleted join of K is
KA(j) ■= Mi W A2 W ••• W An £ K * | (A1,A2,..., An) is j-wise disjoint} (5)
where an n-tuple (A1, A2,..., An) is j-wise disjoint if every sub-collection {A^}l=1, where k1 <k2 < ■ ■ ■ <kj, has an empty intersection.
It immediately follows that K*™{j) C K™j+1) and that K™n+1) = K*"Md K*£{2) = K**? are respectively the n-fold join and the n-fold deleted join of the complex K.
A simple but very useful property of these operations is that they commute
(AA(J))A(k) - (AA(k))A(j) . For example UK = pt is a one-point simplicial complex we obtain the isomorphsim
Am,n = mT)A - mT)T = An,m.
A single complex K in equation (5) can be replaced bv a collection K = {Kj}™=1 of complexes Kj C 2[ml which leads to the definitio n of the j-wise deleted jo in of K.,
^A(j) ■= {A1 W ■ ■ ■ W An £ K1 * ■ ■ ■ * Kn | (A1,..., A^) is j-wise disjoint} .
All simplicial complexes described in this section are generalized chessboard complexes in the sense of Definition 1. For example UK C 2^ then i ts n-fold j-wise deleted join is the complex
jy-n c^ \IC,C ^A (j) = Am,n
where K1 = ••• = Kn and L1 = ••• = Lm = (<—J is the collection of all subsets of [m] of
j
3.2. Bier spheres as generalized chessboard complexes
Let K £ 2[m^ be a simplicial complex on the ground set [m] (meaning that we allow {j} £ K for some j £ [m\). The Alexander dual of K is the simplicial complex K° that consists of the complements of all nonsimplices of K
K ° := {Ac | A/K} .
By definition the "Bier sphere" is the deleted join Bier(K) ■= K *a K(A face A1W A2 £ Bier(K) is often denoted as a triple (A1,A2; B) where B ■= [m\ \ (A1 U A2).)
It turns out that Bier(K) is indeed a triangulation of an (m — 2)-dimensional sphere [4], see [31] and [30] for different, very short and elegant proofs.
The Bier sphere Bier(K) is also a generalized chessboard complex where K1 = K,K2 = K° and Li = ••• = Lm = {0, {1}, {2}} c 2[2].
Alexander r-tuples K = {Ki}ri=1 of simplicial complexes were introduced in [18] as a generalization of pairs (K, K°) of Alexander dual complexes. The associated generalized Bier complexes, defined as the r-fold deleted joins K,*A of Alexander r-tuples are also generalized chessboard complexes in the sense of Definition 1.
4. Discrete Morse theory
A discrete Morse function on a simplicial complex K C 2y is, by definition, an acyclic matching on the Hasse diagram of the partially ordered set (K, C). Here is a brief reminder of the basic facts and definitions of discrete Morse theory.
Let K be a simplicial complex. Its ^dimensional simplices (p-simplices for short) are denoted by ap, a^, , ap, etc. A discrete vector field is a set of pairs D = {..., (ap, fip+1),... } (called a matching) such that:
(a) each simplex of the complex participates in at most one pair;
(b) in each pair (ap,fip+1) G the simplex ap is a facet of fip+1;
(c) the empty set 0 G K is not matched, i.e. if (ap,ftp+1) G D then p > 0.
The pair (ap, fip+1) can be informally thought of as a vector in the vector field D. For this reason it is occasionally denoted by ap ^ ,0p+1 or ap ^ fip+1 (and in this case ap and fip+1 are informally referred to as the beginning and the end of the arrow ap ^ fip+1).
Given a discrete vector field D, & gradient path in D is a sequence of simplices (a zig-zag path)
a0 ^ fii ^ $2 ' ' ' firn ^ °yn+1
satisfying the following conditions:
1. (/jp, iyp+l) is a pair in D for each i;
2. for each i = 0,..., m the simplex ap+1 is a facet of
3. for each i = 0,... ,m — 1, a.% = ai+1.
A path is closed if apm+1 = a0- A discrete Morse function (DMF for short) is a discrete vector field without closed paths.
Assuming that a discrete Morse function is fixed, the critical simplices are those simplices of the complex that are not matched. The Morse inequality [13] implies that critical simplices cannot be completely avoided.
A discrete Morse function D is perfect if the number of critical fc-simplices equals the fc-th Betty number of the complex. It follows that D is a perfect Morse function if and only if the number of all critical simplices equals the sum of all Betty numbers of K.
A central idea of discrete Morse theory, as summarized in the following theorem of R. Forman, is to contract all matched pairs of simplices and to reduce the simplicial complex Ktoa cell complex (where critical simplices correspond to the cells).
Theorem 1. [13] Assume that a discrete Morse function on a simplicial complex K has a single zero-dimensional critical simplex a0 and that all other critical simplices have the same dimension N > 1. Then K is homotopy equivalent to a wedge of N-dimensional spheres.
More generally, if all critical simplices, aside from a0, have dimension > N, then the complex K is (N — 1)-connected. □
4.1. Discrete vector fields on Bier spheres
It is known that all Bier spheres are shellable, see [6] and [10]. A method of Chari [9] can be used to turn this shelling into a perfect discrete Morse function (DMF). The construction of our perfect DMF on a Bier sphere essentially follows this path, see [18] for more details. For the reader's convenience here we reproduce this construction since it will be needed in Section 5.
A perfect DMF on Bier(K)
We construct a discrete vector field D1 on the Bier sphere Bier(K) in two steps:
(1) We match the simplices
a = (A1,A2; B U f) and ft = (A1,A2 U i; B)
iff the following holds:
(i) i<B, i < A2
(that is, i is smaller than all the entries of B and A2).
(ii) A2 U i £ K°.
Before we pass to step 2, let us observe that the non-matched simplices are labelled bv (A1,A2; BUi) such that A2 £ Kbut A2 U i £ KAs a consequence, for non-matched simplices A1 U B £ K.
(2) In the second step we match together the simplices
a = (A1, A2; B U j) and ft = (A1 U j, A2; B)
iff the following holds:
(a) None of the simplices a and ft is matched in the first step.
(b) j >B, j> Ai.
(c) Ai U j £ K.
Observe that the condition (c) always holds (provided that the condition (a) is satisfied), except for the case B = 0.
Lemma 1. (see [18, Lemma 6.1]) The discrete vector field D1 is a discrete Morse function on the Bier sphere Bier(K).
Proof. Since D1 is (by construction) a discrete vector field, it remains to check that there are no closed gradient paths. Observe that in each pair of simplices in the discrete vector field D1 there is exactly one migrating element. More precisely, in the case (1) the element ¿migrates to A2, and in the case (2) the element j migrates to A1.
The lemma follows from the observation that (along a gradient path) the values of the migrating element that move to A2 strictly decreases. Similarly, the values of migrating elements that move to A1 can only increase. This is certified through the following simple case analysis: (1) After a first step pairing comes a splitting of A2. Then the gradient path terminates. (2) After a first step pairing (with migrating element i) comes a splitting of A1. The gradient path proceeds only if the splitted element is smaller than i. (2) After a second step pairing comes a splitting of A1. Then the gradient path terminates. (2) After a second step pairing (with migrating element i) comes a splitting of A2. The gradient path proceeds only if the splitted element is bigger than i.
Let us illustrate this observation by an example which captures the above case analysis. Assume we have a fragment of a gradient path that contains two matchings of type 1. We have:
(Ai U k, A2; B U i) ^ (Ai U k, A2 U i; B) ^
(A-i_,A2 U i; B U k) ^ (Ai,A2 U k U i; B)
The migrating elements here are i and k. The definition of the matching Di implies k < i. Otherwise (Ai,A2 U i; B U k) is matched with (Ai,A2; B U k U i), and the path would terminate after its second term. □
It is not difficult to see that there are precisely two critical simplices in Di:
1. An (n — 2)-dimensional simplex,
(¿1 2; i)
where A1 < i < A2, (this condition describes this simplex uniquely, in light of the fact that Ai G K and A2 G K°), 0
(0, {1}; {2, 3, 4, ...,n}).
(Here we make a simplifying assumption that {1} G Kwhich can be always achieved by a re-enumeration, except in the trivial case K° =
4.2. Discrete vector fields on generalized chessboard complexes
The construction of the discrete Morse function on the Bier sphere Bier(K) illustrates the fruitful idea which can be extended and further developed to cover the case of other generalized chessboard complexes.
Examples of this construction can be found in [18] and [20], see also Section 6 for a construction of such a discrete Morse function on the multiple chessboard complex A^.,fcn;il,.",im.
All these constructions of DMF share the same basic idea, for this reason we sometimes refer to them as standard DMF on generalized chessboard complexes. Note that the proofs that they indeed form an acyclic matching may vary from example to example and use some special properties of the class under investigation.
5. Edmonds-Fulkerson bottleneck extrema
In this section we connect, via discrete Morse theory, the combinatorial topology of Bier spheres with Edmonds-Fulkreson theorem on bottleneck extrema of pairs of dual clutters. We will show that there is much more than meets the eye in the standard concise treatment of this classical result of combinatorial optimization.
Let E be a finite set. Call a family of mutually noncomparable subsets of E a clutter on E. It is shown that for any cluttcr 3t on E, there exists a unique clutter y on E
Specifically, Sf consists of the minimal subsets of E that have non-empty intersection with every member of ¿it. The pair ■¥) is called a blocking system on E. An algorithm is described and several examples of blockings systems are discussed.
Figure 2: Edmonds-Fulkerson bottleneck theorem
Figure 2 shows the abstract of the published version of [11], which originally appeared as a RAND-corporation preprint AD 664879 in .January of 1966.
This is a purely combinatorial result which is often referred to as the Edmonds-Fulkreson bottleneck lemma (theorem). Minmax theorems are ubiquitous in mathematics, notably in geometry, polyhedral combinatorics, critical point theory, game theory and other areas. One of
early examples is the minimax theorem of John von Neumann (first proven and published in 1928) which gives conditions on a function f ■ C x D ^ R, defined on the product of two closed, convex sets in Rra, to satisfy the minmax equality,
Res max f (x, y) = max Res f (x, y). (6)
yeD xeC xeC yeD
It is interesting to compare the Edmonds-Fulkerson minmax theorem with their geometric counterparts. For example in a vicinity of a non-degenerate critical point a Morse function has the form f (x, y) = — |x|2 + |y|2 = — x2 — ■ ■ ■ — x2V + y2 + ■ ■ ■ + y\. Moreover, this function satisfies the concave/convex condition of von Neumann's minmax theorem and the relation (6) is valid.
There is a formal resemblance of these results, for example the ^-sections (respectively y-sections) of the convex sets C x D in (6) formally play the role of complementary clutters K and S from the result of Edmonds and Fulkerson. At first sight it appears to be naive and hard to expect a deeper connection between these results. Indeed, the clutter {C x {y}}yeD of y-sections is nowhere near to be the complementary clutter of the set {{^} x D}xec of all ^-sections, which is a consequence of the following lemma (see the property (3) on page 301 in [11]).
Lemma 2. The clutter S c 2E is the complementary clutter of the clutter K c 2E, if and only if for each partition E = E0 W E\ of E either an eleme nt ofK is containe d in E0 or an element of S is contained in E\, but not both.
In the next section we show that there does exist a geometric interpretation of the Edmonds-Fulkerson bottleneck minmax equality, provided we are willing to replace the smooth by discrete Morse theory.
5.1. Edmonds-Fulkerson minmax lemma revisited
Here we use the results from Section 4.1 to give a new proof and a new interpretation of Edmonds-Fulkerson minmax lemma. As before (Figure 2) the clutters K and S are both subfamilies of 2e.
Let 77 ■= {A C E | (3X £ K) X C A} be the upper closure of the clutter Urnd let K :=2E be the complementary simplicial complex.
Lemma 3. Let K° be the Alexander dual of the simplicial complex K ■= 2E \ K. Then
K° = 2e \S
is the complementary simplicial complex of the upper closure S of the clutter S.
proof. This is an immediate consequence of Lemma 2 since the pair of complexes (K, K°) is also characterized bv the property that for each partition E = E0 W E\ precisely one of the following two relations E0 £ K, E\ £ K° is satisfied. □
Let f ■ E ^ R be a real function. We may assume that / is 1-1. Moreover, we may replace E by the set [n] (where n is the cardinality of E) and assume that f = id ■ [n] ^ [n] is the identity function.
By construction and properties of the perfect DMF on the Bier sphere Bier(K) = K *a K constructed in Section 4.1, there is a unique (n — 2)-dimensional critical simplex (Ai,A2; i), characterized by the conditions A\ < i < A2, A\ £ K, A2 £ KLet us show that
a ■= Res max f (x) = f (i) = max Res f (x) =■ b. ien xei ' ' Jes xeJ '
Indeed, A1 U{f} G K implies A1 U{f} G ^ and from maxx£^lU{i} f (x) = f (i) we deduce the relation a < f (i).
For the opposite inequality observe that if I G ^ then I n (A2 U {&}) = 0 (otherwise, since A2 U {¿} G 5, Lemma 2 would be violated). Hence, maxxej f (x) > f (j^d a > f (i).
The proof of the equality b = f (i) is similar. □
Remark 2. One of the consequences is that the (algorithmic) complexity of determining the critical cell (A1, A2; i) in the Bier sphere Bier(K) is at least as big as the complexity of evaluating the maxmin (minmax) of a function on a family of sets (clutter).
6. Discrete Morse theory for chessboard complexes with multiplicities
Suppose that k1 ,...,knmd l1,...,lm are two sequences of non-negative integers. The generalized chessboard complex contains all rooks placements on [n] x [m] table
such that at most ki rooks are in the f-th row and at most lj rooks are in the j-th column. We use Forman's discrete Morse theory to obtain a generalization of Theorem 3.2 from [22].
Theorem 2. If
h + 12 + ' ' ' + lm ^ fc1 + ^2 + ' ' ' + kn + n — 1 (*) then Am}n,kn;ll,...,lrn is (k1 + k2 + ■ ■ ■ + kn — 2)-connected.
proof. A column (or a row) is called full if it contains the maximal allowed number of rooks. Otherwise, it is called free.
We now define a Morse matching for A = A^.,fcn;il,.",im. For a given face R we describe a face R' that is paired with E, or we recognize that R is a critical face. Let us do it stepwise.
Step 1.
Take the minimal a1 such that either (1) there is a rook positioned at (1,a1) or (2) the a1 column is free.
In the first case (there is a rook at (1, a^), we match R and R' = R \ {(1, a1)}. This is alwavs possible except for the unique exception, when R contains exactlv one rook at (1,1)-
In the second case we match R and R' = RU{(1, a1)} provided that R' ^^togs to A. The latter condition means that the first row in R is not full.
Clearly, after Step 1 the unmatched simplices are those with full first row, empty (1, a1), and a free column a1.
Step 2. We match some of the simplices that are unpaired on the first step.
1. If there is a rook at (2, a^, set a2 := a1 and match R and R' = R \ {(2, a2)}.
2. If
(a) there is no rook at (2, a1), and
(b) the number of rooks in column a1 is smaller than lal — 1,
set a2 := a1 and mat ch R and R' = R U {(2, a2)} provided t hat R' ^^togs to A. The latter condition means that the second row in R is not full.
Introduce also T(R) := 2. ^^s meaning is "the column a1 = a2 has been used twice".
3. If none of the above cases holds, set a2 > ai to be the minimal number such that either (1) there is a rook positioned at (2, a2), or (2) the a2 column is free.
The condition (*) guarantees that a2 is well-defined.
If there is a rook at (2, a2), we match R and R' = R \ {(2, a2)}.
Otherwise, we match E and R' = R U {(2, a2)} provided that R' belongs to A. The latter condition means that the second row in R is free.
In this case we set T(R) ■= 1, since the column a2 has been used once.
Clearly, after Step 2 the unmatched simplices are those with full first and second rows, empty (2, a2), and a fee column a2.
We proceed in the same manner. During the first k — 1 steps, some of the simplices become matched. Unmatched simplices have first k — 1 rows full. They also have no rook at (k — 1, ak-i). Each unmatched simplex R is associated a number T(R).
This is how a generic step looks like: Step k.
1. If there is a rook at (k,ak-i), then match R and R' = R \ {(k,ak)}•
-
(a) there is no rook at (k,ak-i), and
(b) the number of rooks in column au-i is smaller than lak-1 — T(R),
set ak ■= ak-i and match R and R' = RU {(k, ak)} provided that R' belongs to A. The latter condition means that the fc-th row in R is free.
Set T(R) ■= T(R) + 1; this means that "now the column ak = ak-i has been used T(R) times".
3. Otherwise, set ak > ak-i to be the minimal number such that either (1) there is a rook positioned at (k, ak), or (2) the ak column is free.
Next, we match R and R' = R \ {(2, a2)} or R' = R U {(2, a2)} provided that R' belongs to A.
If R is not matched, set T(R) ■= 1.
Remark. If k < n, then (*) guarantees that ak is well-defined. For the last row an is ill-defined if and only if (*) is an equality and R has all the rows full.
Eventually we have all the rows full for non-matched simplices (except for the unique zero-dimensional simplex).
Now let us prove that the above defined matching is acyclic. Take a directed path
Ri ^ Qi \ R2 ^ Q2 \ ••• .
Recall that Ri ^ Qi if and only if Qi = Ri U {(s^, aSi)} , the first Si — 1 rows of Ri are full, and aSi is the first free column after aSi-i.
Let us prove that (si, aSi) strictly decreases along the path wrt lexicographic order. This will imply the acvclicity
For Qi \ Ri+i, we have Ri+i = Qi \ {(pi, qi)} for some (pi,Qi) £ Qi (there are no conditions when we remove a rook from Qi). It suffices to consider the first two steps in our directed path:
Ri ^ Qi = Ri U {(si,aSi)} \ R2 = Qi \ {(;P2, Q2)}.
• If p2 > s1 or p2 = s1 and aSl < q2 (the removed rook is below or right on (s1, aSl), the added rook at the first step) our path stop, because R2 is paired with R2 \ {(s1, aSl)}.
• If p2 < s1 or p2 = s^d aSl > q2 (the removed rook is above or left (s1 ,aSl)), then we have that s2 < s1 or s2 = s1 and aS2 < aSl.
Summarizing, all critical faces (except for the unique zero-dimensional one) have all the rows full. Therefore A^yr^1 is (A* + ^ + • • • + kn — 2)-connected. □
7. Tverberg-Van Kampen-Flores type results for j-wise disjoint partitions of a simplex
Recall that a coloring of a set S C Rd is a partition S = S1 l±l ■ ■ ■ l±l Sk, where Si are the corresponding monochromatic sets. By definition a subset C C S is a rainbow set if it contains at most 1 point from each of the color classes Si.
Theorem 3. Let r be a prime power and j > 1. Suppose that {S}^ is a collection of k finite sets of points in Rd (called colors). Assume that the cardinalities mi = I Si | satisfy the inequality
jmi — 1 < r for each i = 1,...,k. If (r — 1)(d + 1) < (j — 1)m — 1, where m := m1----+ mk, then it is
possible to partition the set S = S11 ■ ■ ■ l Sk into r rainbow, j-wise disjoint sets S = C11 ■ ■ ■ l Cr, so that their convex hulls intersect,
conv(C1) n ■ ■ ■ n conv(CV) = 0 .
proof. The rainbow sets span the multicolored simplices which are encoded as the simplices of the simplicial complex ([pi]^)^) * ■ ■ ■ * (Ma^)- Indeed these are precisely the simplices which are allowed to have at most 1 vertex in each of k different colors. The configuration space of all r-tuples of j-wise disjoint multicolored simplices is the simplicial complex,
K = (([^0 *•••* (N^) )))lr(,)
Since the join and deleted join commute, this complex is isomorphic to,
K = (MÎ3x-) *•••* a^oAœ
where pt is a one-point simplicial complex.
If we suppose, contrary to the statement of the theorem, that the intersection of images of any r, j-wise disjoint multicolored simplices is empty, the associated mapping F : K ^ (Rd)*r would miss the diagonal D C (Rd)*r. By composing this map with the orthogona 1 projection to D±, and after the radial projection to the unit sphere in D±, we obtain a (Z/p)a-equivariant mapping,
F: K ^ 5 (r-m+i)-i_
The complex ([pt]A^f)*^) '1S a multiple ^tessboard complex Ami—1. Since by assumption jmi — 1 < r, ^to ^^mplex is (mi(j — 1) — 2)-connected by the main result from [22]. Hence the complex K is (m(j — 1) — By our assumption m(j — 1) — 2 > (r — 1)(d + 1) — 1, so
in light of Volovikov's theorem [34] such a mapping F does not exist. □
The following obvious corollary of Theorem 2 is more suitable for applications in the rest of the section.
Corollary 1. By interchanging the rows and the columns of the multiple chessboard complex in Theorem 2, we obtain that the complex Am',n,kn,ll,"',lm is (li + ••• + lm — 2)-connected if h +-----+ lm < ki + ■ ■ ■ + kn m + 1.
Theorem 4. Let r be a prime power. Assume that positive integers k, r, N, j and d satisfy the inequalities (k + 1)r + r — 1 < (N + 1)(j — 1) and (r — 1)(d + 1) + 1 < r(k + 1). Then for every continuous map f ■ An ^ Rd there exist r, j -wise disjoint faces of the simplex AN of dimension at most k, whose images have a nonempty intersection.
proof. The faces of dimension at most k form the fc-skeleton (AN)(kj = M^+ij- The configuration space of all r-tuples of j-wise disjoint fc-dimensional faces of this skeleton is the simplicial complex,
K = ([Pt]^,(k+^j">A(jj.
This is a generalized chessboard complex K = A^^-1. Since by our assumption (k + 1)r < < (N + 1)(j — 1) — r + 1, ^to complex K is by Corollary 1 ((fc + 1)r — 2)-connected.
If we suppose, contrary to the statement of the theorem, that the intersection of images of any r, j-wise disjoint fc-dimensional faces is empty, the associated mapping F ■ K ^ (Rd)*r would miss the diagonal D.
As in the proof of the previous theorem we obtain a (Z/p)a-equivariant mapping,
F ■ K ^ 5(r-w+ij-\
We have already observed that K is ((fc + 1)r — 2)-connected, and by our assumption
r(k + 1) — 2 > (r — 1)(d + 1) — 1, so in light of Volovikov's theorem [34] such a mapping F
□
Theorem 5. Let, r be a prime power. Suppose that q,r,j and d are positive integers and let {Si}i=i c Rd is a collection of colored points where all color classes Si are of the same cardinality m. Then if qr < m(j — 1) — r + 1and(r — 1)(d + 1) + 1 < qrk, then it is always possible to partition the set S ■= U=iSi into r j-wise disjoint sets containing at most q points of each color, so that their convex hulls conv(Si) have a non-empty intersection.
proof. The sets containing at most q points of each color span the multicolored simplices which are encoded as the simplices of the simplicial complex ([pt]^q+ij)*k- Indeed, these are precisely the simplices which are allowed to have at most q vertices in each of fc different colors. The configuration space of all r-tuples of j-wise disjoint multicolored simplices is the simplicial complex,
k = (([Pt]r{q+ij)*k )lUj.
Since the join and deleted join commute, this complex is isomorphic to,
K = (([pt]r{q+ij)lUj)*k.
If we suppose, contrary to the statement of the theorem, that the intersection of images of any r, j-wise disjoint multicolored simplices is empty, the associated mapping F ■ K ^ (Rd)*r would miss the diagonal D. As before, by composing this map with the orthogonal projection to D±, and after the radial projection to the unit sphere in D±, we obtain a (Z/p)"-equivariant mapping,
F■ K5 (r-w+ij-\
The complex ([pt]is a multiple chessboard complex A^r 1. Since we assumed qr < (j — 1)^ — r + 1, this complex is (qr — 2)-connected by Corollary 1. Hence the complex K is (grfc — 2)-connected. By our assumption qrk > (r — 1)(d + 1) + 1, so in light of Volovikov's theorem [34] such a mapping F does not exist. □
Theorem 6. Let r he a prime power. Suppose that q,r,j and d are positive integers and let {Si},l=1 c Rd is a collection of colored points where all color classes Si are of the same cardinality m. If jm — 1 < qr and (r — 1)(d + 1) + 1 < (j — 1)mk, then it is possible to divide all points in r, j-wise disjoint sets containing at most q points of each color, so that their convex hulls conv(Si) have a non-empty intersection.
proof. As before the sets containing at most q points of each color span the multicolored simplices which are encoded as the simplices of the simplicial complex ([pt]*^q+1))*k- Indeed these are precisely the simplices which are allowed to have at most q vertices in each of fc different colors. The configuration space of all r-tuples of j-wise disjoint multicolored simplices is the simplicial complex,
K = (([Pt] ^+1))*fc
Since the join and deleted join commute, this complex is isomorphic to,
K = (([Pt] ^+1))^))*".
If we suppose, contrary to the statement of the theorem, that the intersection of images of any r, j-wise disjoint multicolored simplices is empty, the associated mapping F : K ^ (Rd)*r would miss the diagonal D. As before, from here by an equivariant deformation we obtain a (Z/p)a-equivariant mapping,
i^: K ^ s(r-1)(d+1)~1.
The complex ([pt]is the multiple chessboard complex Ai-1. Since we assumed (j — 1)m < qr — m + 1, this complex is ((j — 1)m — 2)-connected by Corollary 1. Hence the complex
K is ((j — 1)mk — By our assumption (j — 1)mk > (r — 1)(d + 1) + 1, and again this
□
For illustration let us consider a very special case of this theorem q = 1 and j = 2.
Theorem 7. Let r be a prime power. Given k finite sets of points in Rd (called colors), of m points each, so that 2m — 1 < r and (r — 1)(d + 1) + 1 < mk, it is possible to divide the points in r
1
Remark 3. It is easy to see that the assumptions on the total number of points is the best possible, since the set of (r — 1)(d + 1) points in the general position could not be divided in r disjoint sets whose convex hulls intersect.
7.1. A comparison with known results
It is interesting to compare results from the previous section with similar results from [7] (Section 9). Note that the proof methods are quite different. We use high connectivity of the multiple chessboard complex, established in Section 6, while the authors of [7] use the 'constraint method', relying on the 'optimal colored Tverberg theorem' from [8], as a 'black box' result.
For illustration, let us compare our Theorem 7 to Theorem 9.1 from [7].
Let us choose fc > 2(d + 1) in Theorem 7 and select the smallest m satisfying the inequality (r — 1)(d + 1) + 1 < mfc, meaning that we are allowed to assume
(m — 1)fc < (r — 1)(d + 1) + 1 < mfc.
From here we immediately deduce the inequality 2m — 1 < r and, as a consequence of Theorem 7, we have the following result.
Corollary 2. Let r be a prime power. Assume fc > 2(d + 1) and choose m satisfying the inequality (r — 1)(d + 1) + 1 < mfc. Suppose that S c Rd is a set of cardinality mfc, evenly colored by fc colors (meaning that S = U^=i Si where |5j| = m for each i). Then it is possible to select r pairwise disjoint subsets Ci c S, containing at most 1 point of each color, so that nl=iconv(Ci) = 0.
This result clearly follows from Theorem 9.1 if we assume that r is a prime. Corollary 2 illustrates the phenomenon that there exist instances of the 'optimal colored Tverberg theorem' (Theorem 9.1 in [7]) which remain valid if the condition on r being a prime is relaxed to r is a prime power.
7.2. A remark on Tverberg A-P conjecture
In this section we briefly discuss the problem whether each admissible r-tuple is Tverberg prescribable. This problem, as formulated in [7], will be referred to as the Tverberg A-P problem or the Tverberg A-P conjecture.
Definition 3. For d > 1 and r > 2, an r-tuple d = (d1,...,dr) of integers is admissible if, [f] < di < d for all i, and — di) < d. An admissible r-tu,ple is Tverberg prescribable if there
is an N such that for every continuous map f : AN ^ Rd there is a Tverberg partition {a1, ...,ar} for f with dim(aj) = di.
Question. (Tverberg A-P problem; [7] (Question 6.9.)) Is every admissible r-tuple Tverberg prescribable?
As shown in [14], (Theorem 2.8.), the answer to the above question is negative. It was also demonstrated that a more realistic conjecture arises if the condition [2] < di < d, in the definition of admissible r-tuple, is replied by a stronger requirement (r—^ (d — 1) < di < d to all i.
Here we remark that a positive answer to the modified question is quite straightforward in the case r > d. Indeed, in this case we have for all i
d% > ^(d — 1) > d — 1 — >d — 2.
So, in this case each di is equal to either d — 1 or d, and the A-P conjecture reduces to the 'balanced case', established in [23].
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