Generalized chessboard complexes and discrete Morse theory
Duško Jojić, Gaiane Panina, Siniša T. Vrećica, Rade T. Živaljević
GGeneralized chessboard complexesand discrete Morse theory ∗ Duško Jojić
Faculty of ScienceUniversity of Banja Luka
Gaiane Panina
St. Petersburg State UniversitySt. Petersburg Department ofSteklov Mathematical Institute
Siniša T. Vrećica
Faculty of MathematicsUniversity of Belgrade
Rade T. Živaljević
Mathematical InstituteSASA, Belgrade
January 25, 2020
Abstract
Chessboard complexes and their generalizations, as objects, and Discrete Morse the-ory, 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 discreteMorse function on the Bier sphere
Bier ( K ) of an associated simplicial complex K .We illustrate the use of “standard discrete Morse functions” on generalized chessboardcomplexes by proving a connectivity result for chessboard complexes with multiplic-ities. Applications include new Tverberg-Van Kampen-Flores type results for j -wisedisjoint partitions of a simplex. Chessboard complexes and their relatives have been for decades an important theme oftopological combinatorics. They have found numerous and often unexpected applicationsin group theory, representation theory, commutative algebra, Lie theory, discrete and com-putational geometry, algebraic topology, and geometric and topological combinatorics, see[A04], [A-F], [Au10] [BLVŽ], [FH98], [G79], [J08], [KRW], [S-W], [VŽ94], [VŽ09], [W03],[Z], [ŽV92].The books [J] and [M03], as well as the review papers [W03] and [Ž17], cover selectedtopics of the theory of chessboard complexes and contain a more complete list of relatedpublications. ∗ This research was supported by the Grants 174020 and 174034 of the Ministry of Education, Scienceand Technological Development of the Republic of Serbia. The authors acknowledge the hospitality ofMathematical Institute in Oberwolfach, where this paper was completed. a r X i v : . [ m a t h . M G ] M a r hessboard complexes and their generalizations are some of the most studied graphcomplexes [J]. From this point of view chessboard complexes can be interpreted as relativesof L. Lovász Hom -complexes [Ko], matching complexes, clique complexes, and many otherimportant classes of simplicial complexes.More recently new classes of generalized chessboard complexes have emerged and newmethods, based on novel shelling techniques and ideas from Forman’s discrete Morse theory ,were introduced. Examples include multiple and symmetric multiple chessboard complexes[JVZ-1, JVZ-2], Bier complexes [JNPZ], and deleted joins of collectively unavoidable com-plexes, see [JNPZ] and [JPZ-1, JPZ-2]. Among applications are the resolution of thebalanced case of the “admissible/prescribable partitions conjecture” [JVZ-2], general VanKampen-Flores type theorem for balanced, collectively unavoidable complexes [JPZ-1], and“balanced splitting necklace theorem” [JPZ-2].This paper is both a leisurely introduction and an invitation to this part of topologicalcombinatorics, 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 Ed-monds and Fulkerson bottleneck (minmax) theorem (Section 5) and the construction of“standard discrete Morse functions” on generalized chessboard complexes with multiplic-ities (Section 6). This leads to a frequently optimal connectivity result for generalizedchessboard complexes with multiplicities (Theorem 6.1 in Section 6), which is used in Sec-tion 7 for the proof of new Tverberg-Van Kampen-Flores type results for j -wise disjointpartitions of a simplex. Chessboard complexes naturally arise in the study of the geometry of admissible rookconfigurations on a general ( m × 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 bein the same row or in the same column. The collection of all these placements forms asimplicial complex which is called the chessboard complex and denoted by ∆ m,n .More formally, the set of vertices of ∆ m,n is Vert(∆ m,n ) = [ m ] × [ n ] and S ⊆ [ m ] × [ n ] is a simplex of ∆ m,n if and only if for each two distinct elements ( i , j ) , ( i , j ) ∈ S neither i = i nor j = j . Let us take a closer look at one of the simplest chessboard complexes, the complex ∆ , ,based on the × chessboard (see Figure 1).The f -vector of ∆ , is f (∆ , ) = (12 , , so its Euler characteristics is χ (∆ , ) = 0 .Moreover, the geometric realization of ∆ , is an orientable -dimensional manifold.Indeed, the link of each vertex is isomorphic to ∆ , (= hexagonal triangulation ofthe circle S ) while the link of each edge is the circle S . Each -dimensional simplex2 B A BC B A A A B B B B B C C C C C C B C C Figure 1: Chessboard complex ∆ , σ = { A i , B j , C k } is uniquely completed to a permutation π = ( i, j, k, l ) of the set [4] = { , , , } and Sign( σ ) := Sign( π ) defines an orientation on ∆ , .From here we immediately conclude that ∆ , is a triangulation of the -dimensionaltorus T . The universal covering of ∆ , is identified as the honeycomb tiling of the planeand the corresponding fundamental domain is exhibited in Figure 1. From here we caneasily read off the generators of the group H (∆ , ; Z ) ∼ = Z as the geodesic edge-pathsconnecting the three copies of vertex C , shown in Figure 1. Let G be a finite graph with vertex set V = V G and edge set E = E G . A graph complexon G is an abstract simplicial complex consisting of subsets of E . We usually interpretsuch a complex as a family of subgraphs of G . The study of graph complexes, with theemphasis on their homology, homotopy type, connectivity degree, Cohen-Macaulayness,etc., has been an active area of study in topological combinatorics, see [J].The chessboard complex ∆ m,n can be interpreted as a graph complex of the completebipartite graph K m,n , where the simplices S ⊂ [ m ] × [ n ] are interpreted as “matchings” in K m,n . Recall that Γ ⊆ E G is a matching on the graph G if each v ∈ V G is incident to atmost one edge in Γ .All “generalized chessboard complexes”, introduced in Section 3, can be also describedas graph complexes of the graph K m,n . Perhaps the first appearance of chessboard complexes was in the thesis of Garst [G79], as
Tits coset complexes . Recall that a Tits coset complex ∆( G ; G , . . . , G n ) , associated to a3roup G and a family { G , . . . , G n } of its subgroups is the nerve N erve ( F ) of the associatedfamily of cosets F = { gG i | g ∈ G, i ∈ [ n ] } . More explicitly vertices of ∆( G ; G , . . . , G n ) are cosets gG i and a colection S = { g j G i } ( i,j ) ∈ I , for some I ⊆ [ n ] × G , is a simplex of ∆( G ; G , . . . , G m ) if and only if (cid:92) ( i,j ) ∈ I g j G i (cid:54) = ∅ . If G = S m is the symmetric group and G i := { π ∈ S m | π ( i ) = i } for i = 1 , . . . , n , theassociated Tits coset complex is the chessboard complex ∆ m,n . Chessboard complexes made their first appearance in discrete geometry in [ŽV92], in thecontext of the so called colored Tverberg problem .For illustration, an instance of the type B colored Tverberg theorem [VŽ94, Ž17] claimsthat for each collection C ⊂ R of fifteen points in the -space, evenly colored by three col-ors, there exist three vertex disjoint triangles ∆ , ∆ , ∆ , formed by the points of differentcolor, such that ∆ ∩ ∆ ∩ ∆ (cid:54) = ∅ .A general form of this result was deduced in [VŽ94] from a Borsuk-Ulam type resultclaiming that each Z r -equivariant map (∆ r, r − ) ∗ ( k +1) Z r −→ W ⊕ ( d +1) r (2.1)must have a zero if r ≤ d/ ( d − k ) (this is a necessary condition), r is a prime power, ∆ r, r − is a chessboard complex, and W r = { x ∈ R r | x + · · · + x r = 0 } .The reader is referred to [Ž17] for an overview of these and more recent results, as wellas for a more complete list of references. Motivated primarily by applications to problems in discrete geometry, especially the prob-lems of Tverberg and Van Kampen-Flores type, more general chessboard complexes wereintroduced and studied. Closely related complexes previously emerged in algebraic combi-natorics [KRW, W03].These complexes are also referred to as generalized chessboard complexes , since the setof vertices remains the ( m × n ) -chessboard [ m ] × [ n ] , but the criterion for S ⊆ [ m ] × [ n ] tobe a simplex (“admissible rook placement”) may be quite different and vary from problemto problem.The following definition includes most if not all of the currently studied examples andprovides a natural ecological niche for all these complexes and their relatives. Definition 3.1.
Suppose that K = { K i } ni =1 and L = { L j } mj =1 are two collections of simpli-cial complexes where Vert( K i ) = [ m ] for each i ∈ [ n ] and Vert( L j ) = [ n ] for each j ∈ [ m ] .Define, ∆ K , L m,n = ∆ m,n ( K , L ) (3.1)4 s the complex of all subsets (rook-placements) A ⊂ [ m ] × [ n ] such that { i ∈ [ m ] | ( i, j ) ∈ A } ∈ K j for each j ∈ [ n ] and { j ∈ [ n ] | ( i, j ) ∈ A } ∈ L i for each i ∈ [ m ] . Definition 3.1 can be specialized in many ways. Again, we focus on the special casesmotivated by intended applications to the generalized Tverberg problem.
Definition 3.2.
Suppose that k = ( k i ) ni =1 and l = ( l j ) mj =1 are two sequences of non-negativeintegers. Then the complex, ∆ k , l m,n = ∆ k ,...,k n ; l ,...,l m m,n (3.2) arises as the complex of all rook-placements A ⊂ [ m ] × [ n ] such that at most k i rooks areallowed to be in the i -th row (for i = 1 , . . . , n ), and at most l j rooks are allowed to be inthe j -th column (for j = 1 , . . . , m ). Remark 3.3.
The complexes ∆ k , l m,n = ∆ k ,...,k n ; l ,...,l m m,n are sometimes referred to as the chessboard complexes with multiplicities or multiple chessboard complexes . Closely relatedare “bounded degree graph complexes”, studied in [KRW] and [W03].When k = · · · = k n = p and l = · · · = l m = q , we obtain the complex ∆ p,qm,n . For thereasons which will become clear in the following section of the paper, in our earlier papers[JVZ-1, JVZ-2] we focused to the case l = · · · = l m = 1 , i.e. to the complexes, ∆ k ,...,k n ; m,n := ∆ k ,...,k n ;1 ,..., m,n . (3.3)In Section 6 of this paper we fill this “gap” and return to the case of general chessboardcomplexes with multiplicities. n -fold j -wise deleted join Joins and deleted joins of simplicial complexes, as well as their generalizations, have foundnumerous applications in topological combinatorics, see [M03, Section 6.3] for motivationand an introduction.For a simplicial complex K , the n -fold j -wise deleted join of K is K ∗ n ∆( j ) := { A (cid:93) A (cid:93) · · · (cid:93) A n ∈ K ∗ | ( A , A , . . . , A n ) is j -wise disjoint } (3.4)where an n -tuple ( A , A , . . . , A n ) is j -wise disjoint if every sub-collection { A k i } ji =1 , where k < k < · · · < k j , has an empty intersection.It immediately follows that K ∗ n ∆( j ) ⊆ K ∗ n ∆( j +1) and that K ∗ n ∆( n +1) = K ∗ n and K ∗ n ∆(2) = K ∗ n ∆ 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 ( K ∗ n ∆( j ) ) ∗ m ∆( k ) ∼ = ( K ∗ m ∆( k ) ) ∗ n ∆( j ) . For example if K = pt is a one-point simplicial complex we obtain the isomorphsim ∆ m,n = (( pt ) ∗ m ∆ ) ∗ n ∆ ∼ = (( pt ) ∗ n ∆ ) ∗ m ∆ = ∆ n,m . K in equation (3.4) can be replaced by a collection K = { K j } nj =1 ofcomplexes K j ⊆ [ m ] which leads to the definition of the j -wise deleted join of K , K ∆( j ) := { A (cid:93) · · · (cid:93) A n ∈ K ∗ · · · ∗ K n | ( A , . . . , A n ) is j -wise disjoint } . All simplicial complexes described in this section are generalized chessboard complexesin the sense of Definition 3.1. For example if K ⊆ [ m ] then its n -fold j -wise deleted joinis the complex K n ∆( j ) ∼ = ∆ K , L m,n where K = · · · = K n and L = · · · = L m = (cid:0) [ m ] ≤ j − (cid:1) is the collection of all subsets of [ m ] ofcardinality strictly less than j . Let K (cid:32) [ 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 ofthe complements of all nonsimplices of KK ◦ := { A c | A / ∈ K } . By definition the “Bier sphere” is the deleted join
Bier ( K ) := K ∗ ∆ K ◦ . (A face A (cid:93) A ∈ Bier ( K ) is often denoted as a triple ( A , A ; B ) where B := [ m ] \ ( A ∪ A ) .)It turns out that Bier ( K ) is indeed a triangulation of an ( m − -dimensional sphere[Bi92], see [M03] and [Lo04] for different, very short and elegant proofs.The Bier sphere Bier ( K ) is also a generalized chessboard complex where K = K, K = K ◦ and L = · · · = L m = {∅ , { } , { }} ⊂ [2] . Alexander r -tuples K = { K i } ri =1 of simplicial complexes were introduced in [JNPZ] asa generalization of pairs ( K, K ◦ ) of Alexander dual complexes. The associated general-ized Bier complexes , defined as the r -fold deleted joins K ∗ r ∆ of Alexander r -tuples are alsogeneralized chessboard complexes in the sense of Definition 3.1. A discrete Morse function on a simplicial complex K ⊆ V is, by definition, an acyclicmatching on the Hasse diagram of the partially ordered set ( K, ⊆ ) . Here is a brief reminderof the basic facts and definitions of discrete Morse theory.Let K be a simplicial complex. Its p -dimensional simplices ( p -simplices for short) are de-noted by α p , α pi , β p , σ p , etc. A discrete vector field is a set of pairs D = { . . . , ( α p , β p +1 ) , . . . } (called a matching) such that:(a) each simplex of the complex participates in at most one pair;(b) in each pair ( α p , β p +1 ) ∈ D , the simplex α p is a facet of β p +1 ;(c) the empty set ∅ ∈ K is not matched, i.e. if ( α p , β p +1 ) ∈ D then p ≥ .6he pair ( α p , β p +1 ) can be informally thought of as a vector in the vector field D . For thisreason it is occasionally denoted by α p → β p +1 or α p (cid:37) β p +1 (and in this case α p and β p +1 are informally referred to as the beginning and the end of the arrow α p → β p +1 ).Given a discrete vector field D , a gradient path in D is a sequence of simplices (a zig-zagpath) α p (cid:37) β p +10 (cid:38) α p (cid:37) β p +11 (cid:38) α p (cid:37) β p +12 (cid:38) · · · (cid:38) α pm (cid:37) β p +1 m (cid:38) α pm +1 satisfying the following conditions:1. (cid:0) α pi , β p +1 i (cid:1) is a pair in D for each i ;2. for each i = 0 , . . . , m the simplex α pi +1 is a facet of β p +1 i ;3. for each i = 0 , . . . , m − , α i (cid:54) = α i +1 .A path is closed if α pm +1 = α p . A discrete Morse function (DMF for short) is a discretevector field without closed paths.Assuming that a discrete Morse function is fixed, the critical simplices are those sim-plices of the complex that are not matched. The Morse inequality [Fo02] implies thatcritical simplices cannot be completely avoided.A discrete Morse function D is perfect if the number of critical k -simplices equals the k -th Betty number of the complex. It follows that D is a perfect Morse function if andonly 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 ofR. Forman, is to contract all matched pairs of simplices and to reduce the simplicial complex K to a cell complex (where critical simplices correspond to the cells). Theorem 4.1. [Fo02] Assume that a discrete Morse function on a simplicial complex K has a single zero-dimensional critical simplex σ and that all other critical simplices havethe same dimension N > . Then K is homotopy equivalent to a wedge of N -dimensionalspheres.More generally, if all critical simplices, aside from σ , have dimension ≥ N , then thecomplex K is ( N − -connected. It is known that all Bier spheres are shellable, see [BPSZ] and [Č-D]. A method of Chari[Cha] can be used to turn this shelling into a perfect discrete Morse function (DMF). Theconstruction of our perfect DMF on a Bier sphere essentially follows this path, see [JNPZ]for more details. For the reader’s convenience here we reproduce this construction since itwill be needed in Section 5.
A perfect DMF on
Bier ( K ) We construct a discrete vector field D on the Bier sphere Bier ( K ) in two steps:71) We match the simplices α = ( A , A ; B ∪ i ) and β = ( A , A ∪ i ; B ) iff the following holds:(i) i < B, i < A (that is, i is smaller than all the entries of B and A ).(ii) A ∪ i ∈ K ◦ .Before we pass to step 2, let us observe that the non-matched simplices are labelled by ( A , A ; B ∪ i ) such that A ∈ K ◦ , but A ∪ i / ∈ K ◦ . As a consequence, for non-matchedsimplices A ∪ B ∈ K .(2) In the second step we match together the simplices α = ( A , A ; B ∪ j ) and β = ( A ∪ j, A ; B ) iff the following holds:(a) None of the simplices α and β is matched in the first step.(b) j > B, j > A .(c) A ∪ j ∈ K .Observe that the condition (c) always holds (provided that the condition (a) is satisfied),except for the case B = ∅ . Lemma 4.2. (see [JNPZ, Lemma 6.1])
The discrete vector field D is a discrete Morsefunction on the Bier sphere Bier ( K ) .Proof. Since D is (by construction) a discrete vector field, it remains to check thatthere are no closed gradient paths. Observe that in each pair of simplices in the discretevector field D there is exactly one migrating element . More precisely, in the case (1) theelement i migrates to A , and in the case (2) the element j migrates to A .The lemma follows from the observation that (along a gradient path) the values of themigrating element that move to A strictly decreases. Similarly, the values of migratingelements that move to A can only increase. This is certified through the following simplecase analysis: (1) After a first step pairing comes a splitting of A . Then the gradient pathterminates. (2) After a first step pairing (with migrating element i ) comes a splitting of A . The gradient path proceeds only if the splitted element is smaller than i . (2) After asecond step pairing comes a splitting of A . Then the gradient path terminates. (2) Aftera second step pairing (with migrating element i ) comes a splitting of A . The gradientpath 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. Wehave: 8 A ∪ k, A ; B ∪ i ) → ( A ∪ k, A ∪ i ; B ) → ( A , A ∪ i ; B ∪ k ) → ( A , A ∪ k ∪ i ; B ) The migrating elements here are i and k . The definition of the matching D implies k < i .Otherwise ( A , A ∪ i ; B ∪ k ) is matched with ( A , A ; B ∪ k ∪ i ) , and the path wouldterminate after its second term.It is not difficult to see that there are precisely two critical simplices in D :1. An ( n − -dimensional simplex, ( A , A ; i ) where A < i < A , (this condition describes this simplex uniquely, in light of thefact that A ∈ K and A ∈ K ◦ ),2. and the -dimensional simplex, ( ∅ , { } ; { , , , ..., n } ) . (Here we make a simplifying assumption that { } ∈ K ◦ , which can be always achieved bya re-enumeration, except in the trivial case K ◦ = {∅} .) The construction of the discrete Morse function on the Bier sphere
Bier ( K ) illustratesthe fruitful idea which can be extended and further developed to cover the case of othergeneralized chessboard complexes.Examples of this construction can be found in [JNPZ] and [JPZ-1], see also Section 6for a construction of such a discrete Morse function on the multiple chessboard complex ∆ k ,...,k n ; l ,...,l m m,n .All these constructions of DMF share the same basic idea, for this reason we sometimesrefer to them as standard DMF on generalized chessboard complexes . Note that the proofsthat they indeed form an acyclic matching may vary from example to example and usesome special properties of the class under investigation. In this section we connect, via discrete Morse theory, the combinatorial topology of Bierspheres 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 treatmentof this classical result of combinatorial optimization.Figure 2 shows the abstract of the published version of [E-F], which originally appearedas a RAND-corporation preprint AD 664879 in January of 1966.9igure 2: Edmonds-Fulkerson bottleneck theoremThis is a purely combinatorial result which is often referred to as the Edmonds-Fulkresonbottleneck lemma (theorem). Minmax theorems are ubiquitous in mathematics, notablyin 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 andpublished in 1928) which gives conditions on a function f : C × D → R , defined on theproduct of two closed, convex sets in R n , to satisfy the minmax equality, min y ∈ D max x ∈ C f ( x, y ) = max x ∈ C min y ∈ D f ( x, y ) . (5.1)It is interesting to compare the Edmonds-Fulkerson minmax theorem with their geometriccounterparts. For example in a vicinity of a non-degenerate critical point a Morse functionhas the form f ( x, y ) = −| x | + | y | = − x − · · · − x p + y + · · · + y q . Moreover, this functionsatisfies the concave/convex condition of von Neumann’s minmax theorem and the relation(5.1) is valid.There is a formal resemblance of these results, for example the x -sections (respectively y -sections) of the convex sets C × D in (5.1) formally play the role of complementaryclutters R and S from the result of Edmonds and Fulkerson. At first sight it appears tobe naive and hard to expect a deeper connection between these results. Indeed, the clutter { C × { y }} y ∈ D of y -sections is nowhere near to be the complementary clutter of the set {{ x } × D } x ∈ C of all x -sections, which is a consequence of the following lemma (see theproperty (3) on page 301 in [E-F]). Lemma 5.1.
The clutter
S ⊂ E is the complementary clutter of the clutter R ⊂ E , ifand only if for each partition E = E (cid:93) E of E either an element of R is contained in E 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 theEdmonds-Fulkerson bottleneck minmax equality, provided we are willing to replace thesmooth by discrete Morse theory. 10 .1 Edmonds-Fulkerson minmax lemma revisited
Here we use the results from Section 4.1 to give a new proof and a new interpretation ofEdmonds-Fulkerson minmax lemma. As before (Figure 2) the clutters R and S are bothsubfamilies of E .Let (cid:98) R := { A ⊆ E | ( ∃ X ∈ R ) X ⊆ A } be the upper closure of the clutter R and let K := 2 E \ (cid:98) R be the complementary simplicial complex. Lemma 5.2.
Let K ◦ be the Alexander dual of the simplicial complex K := 2 E \ (cid:98) R . Then K ◦ = 2 E \ (cid:98) S is the complementary simplicial complex of the upper closure (cid:98) S of the clutter S . Proof:
This is an immediate consequence of Lemma 5.1 since the pair of complexes ( K, K ◦ ) is also characterized by the property that for each partition E = E (cid:93) E precisely one ofthe following two relations E ∈ K, E ∈ K ◦ is satisfied. (cid:3) Let f : E → R be a real function. We may assume that f is - . Moreover, we mayreplace 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 ∗ ∆ K ◦ , constructed in Section 4.1, there is a unique ( n − -dimensional critical simplex ( A , A ; i ) , characterized by the conditions A < i < A , A ∈ K, A ∈ K ◦ . Let us showthat a := min I ∈R max x ∈ I f ( x ) = f ( i ) = max J ∈S min x ∈ J f ( x ) =: b . (5.2)Indeed, A ∪ { i } / ∈ K implies A ∪ { i } ∈ R and from max x ∈ A ∪{ i } f ( x ) = f ( i ) we deducethe relation a ≤ f ( i ) .For the opposite inequality observe that if I ∈ R then I ∩ ( A ∪{ i } ) (cid:54) = ∅ (otherwise, since A ∪ { i } ∈ S , Lemma 5.1 would be violated). Hence, max x ∈ I f ( x ) ≥ f ( i ) and a ≥ f ( i ) .The proof of the equality b = f ( i ) is similar. (cid:3) Remark 5.3.
One of the consequences is that the (algorithmic) complexity of determiningthe critical cell ( A , A ; i ) in the Bier sphere Bier ( K ) is at least as big as the complexityof evaluating the maxmin (minmax) of a function on a family of sets (clutter). Suppose that k , . . . , k n and l , . . . , l m are two sequences of non-negative integers. Thegeneralized chessboard complex ∆ k ,...,k n ; l ,...,l m m,n contains all rooks placements on [ n ] × [ m ] table such that at most k i rooks are in the i -th row and at most l j rooks are in the j -thcolumn. We use Forman’s discrete Morse theory to obtain a generalization of Theorem 3.2from [JVZ-1]. 11 heorem 6.1. If l + l + · · · + l m (cid:62) k + k + · · · + k n + n − ∗ ) then ∆ k ,...,k n ; l ,...,l m m,n is ( k + k + · · · + k n − -connected. Proof:
A column (or a row) is called full if it contains the maximal allowed number ofrooks. Otherwise, it is called free .We now define a Morse matching for ∆ = ∆ k ,...,k n ; l ,...,l m m,n . For a given face R we describea face R (cid:48) that is paired with R , or we recognize that R is a critical face. Let us do it stepwise. Step 1.
Take the minimal a such that either (1) there is a rook positioned at (1 , a ) , or (2) the a column is free.In the first case (there is a rook at (1 , a ) ), we match R and R (cid:48) = R \ { (1 , a ) } .This is always possible except for the unique exception, when R contains exactly onerook at (1 , .In the second case we match R and R (cid:48) = R ∪ { (1 , a ) } provided that R (cid:48) belongs to ∆ .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 , a ) ,and a free column a . 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 a := a and match R and R (cid:48) = R \ { (2 , a ) } .2. If(a) there is no rook at (2 , a ) , and(b) the number of rooks in column a is smaller than l a − ,set a := a and match R and R (cid:48) = R ∪ { (2 , a ) } provided that R (cid:48) belongs to ∆ . Thelatter condition means that the second row in R is not full.Introduce also T ( R ) := 2 . Its meaning is "the column a = a has been used twice".3. If none of the above cases holds, set a > a to be the minimal number such thateither (1) there is a rook positioned at (2 , a ) , or (2) the a column is free.The condition ( ∗ ) guarantees that a is well-defined.If there is a rook at (2 , a ) , we match R and R (cid:48) = R \ { (2 , a ) } .Otherwise, we match R and R (cid:48) = R ∪ { (2 , a ) } provided that R (cid:48) belongs to ∆ . Thelatter condition means that the second row in R is free.In this case we set T ( R ) := 1 , since the column a has been used once.Clearly, after Step 2 the unmatched simplices are those with full first and second rows,empty (2 , a ) , and a free column a . 12e proceed in the same manner. During the first k − steps, some of the simplicesbecome matched. Unmatched simplices have first k − rows full. They also have no rookat ( k − , a k − ) . 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, a k − ) , then match R and R (cid:48) = R \ { ( k, a k ) } .2. If(a) there is no rook at ( k, a k − ) , and(b) the number of rooks in column a k − is smaller than l a k − − T ( R ) ,set a k := a k − and match R and R (cid:48) = R ∪ { ( k, a k ) } provided that R (cid:48) belongs to ∆ .The latter condition means that the k -th row in R is free.Set T ( R ) := T ( R ) + 1 ; this means that “now the column a k = a k − has been used T ( R ) times”.3. Otherwise, set a k > a k − to be the minimal number such that either (1) there is arook positioned at ( k, a k ) , or (2) the a k column is free.Next, we match R and R (cid:48) = R \ { (2 , a ) } or R (cid:48) = R ∪ { (2 , a ) } provided that R (cid:48) belongs to ∆ .If R is not matched, set T ( R ) := 1 . Remark. If k < n , then ( ∗ ) guarantees that a k is well-defined. For the last row a n isill-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 uniquezero-dimensional simplex).Now let us prove that the above defined matching is acyclic. Take a directed path R (cid:37) Q (cid:38) R (cid:37) Q (cid:38) · · · . Recall that R i (cid:37) Q i if and only if Q i = R i ∪ { ( s i , a s i ) } , the first s i − rows of R i are full,and a s i is the first free column after a s i − .Let us prove that ( s i , a s i ) strictly decreases along the path wrt lexicographic order.This will imply the acyclicity.For Q i (cid:38) R i +1 , we have R i +1 = Q i \ { ( p i , q i ) } for some ( p i , q i ) ∈ Q i (there are noconditions when we remove a rook from Q i ). It suffices to consider the first two steps inour directed path: R (cid:37) Q = R ∪ { ( s , a s ) } (cid:38) R = Q \ { ( p , q ) } . • If p > s or p = s and a s < q (the removed rook is below or right on ( s , a s ) , theadded rook at the first step) our path stop, because R is paired with R \ { ( s , a s ) } . • If p < s or p = s and a s > q (the removed rook is above or left ( s , a s ) ), thenwe have that s < s or s = s and a s < a s .Summarizing, all critical faces (except for the unique zero-dimensional one) have all therows full. Therefore ∆ k ,...,k n ; l ,...,l m m,n is ( k + k + · · · + k n − -connected. (cid:3) Tverberg-Van Kampen-Flores type results for j -wisedisjoint partitions of a simplex Recall that a coloring of a set S ⊂ R d is a partition S = S (cid:93) · · · (cid:93) S k , where S i are thecorresponding monochromatic sets. By definition a subset C ⊆ S is a rainbow set if itcontains at most point from each of the color classes S i . Theorem 7.1.
Let r be a prime power and j ≥ . Suppose that { S i } ki =1 is a collection of k finite sets of points in R d (called colors). Assume that the cardinalities m i = | S i | satisfythe inequality jm i − ≤ r for each i = 1 , ..., k . If ( r − d + 1) ≤ ( j − m − , where m := m · · · + m k , then it is possible to partition the set S = S (cid:93) · · · (cid:93) S k into r rainbow, j -wise disjoint sets S = C (cid:93) · · · (cid:93) C r , so that their convex hulls intersect, conv( C ) ∩ · · · ∩ conv( C r ) (cid:54) = ∅ . Proof:
The rainbow sets span the multicolored simplices which are encoded as the sim-plices of the simplicial complex ([ pt ] ∗ ( m )∆(2) ) ∗ · · · ∗ ([ pt ] ∗ ( m k )∆(2) ) . Indeed these are precisely thesimplices which are allowed to have at most vertex in each of k different colors. Theconfiguration space of all r -tuples of j -wise disjoint multicolored simplices is the simplicialcomplex, K = (([ pt ] ∗ ( m )∆(2) ) ∗ · · · ∗ ([ pt ] ∗ ( m k )∆(2) )) ∗ r ∆( j ) Since the join and deleted join commute, this complex is isomorphic to, K = ([ pt ] ∗ ( m )∆(2) ) ∗ r ∆( j ) ∗ · · · ∗ ([ pt ] ∗ ( m k )∆(2) ) ∗ r ∆( j ) where pt is a one-point simplicial complex.If we suppose, contrary to the statement of the theorem, that the intersection of imagesof any r , j -wise disjoint multicolored simplices is empty, the associated mapping F : K → ( R d ) ∗ r would miss the diagonal D ⊂ ( R d ) ∗ r . By composing this map with the orthogonalprojection to D ⊥ , and after the radial projection to the unit sphere in D ⊥ , we obtain a ( Z /p ) α -equivariant mapping, ˜ F : K → S ( r − d +1) − . The complex ([ pt ] ∗ ( m i )∆(2) ) ∗ r ∆( j ) is a multiple chessboard complex ∆ ,j − m i ,r . Since by assumption jm i − ≤ r , this complex is ( m i ( j − − -connected by the main result from [JVZ-1].Hence the complex K is ( m ( j − − -connected. By our assumption m ( j − − ≥ ( r − d + 1) − , so in light of Volovikov’s theorem [V96] such a mapping ˜ F does notexist. (cid:3) The following obvious corollary of Theorem 6.1 is more suitable for applications in therest of the section.
Corollary 7.2.
By interchanging the rows and the columns of the multiple chessboardcomplex in Theorem 6.1, we obtain that the complex ∆ k ,...,k n ; l ,...,l m m,n is ( l + · · · + l m − -connected if l + · · · + l m ≤ k + · · · + k n − m + 1 . heorem 7.3. Let r be a prime power. Assume that positive integers k, r, N, j and d satisfy the inequalities ( k + 1) r + r − ≤ ( N + 1)( j − and ( r − d + 1) + 1 ≤ r ( k + 1) .Then for every continuous map f : ∆ N → R d there exist r , j -wise disjoint faces of thesimplex ∆ N of dimension at most k , whose images have a nonempty intersection. Proof:
The faces of dimension at most k form the k -skeleton (∆ N ) ( k ) = [ pt ] ∗ ( N +1)∆( k +2) . Theconfiguration space of all r -tuples of j -wise disjoint k -dimensional faces of this skeleton isthe simplicial complex, K = ([ pt ] ∗ ( N +1)∆( k +2) ) ∗ r ∆( j ) . This is a generalized chessboard complex K = ∆ k +1; j − N +1 ,r . Since by our assumption ( k + 1) r ≤ ( N + 1)( j − − r + 1 , this complex K is by Corollary 7.2 (( k + 1) r − -connected.If we suppose, contrary to the statement of the theorem, that the intersection of imagesof any r , j -wise disjoint k -dimensional faces is empty, the associated mapping F : K → ( R d ) ∗ r would miss the diagonal D .As in the proof of the previous theorem we obtain a ( Z /p ) α -equivariant mapping, ˜ F : K → S ( r − d +1) − . We have already observed that K is (( k + 1) r − -connected, and by our assumption r ( k + 1) − ≥ ( r − d + 1) − , so in light of Volovikov’s theorem [V96] such a mapping ˜ F does not exist. (cid:3) Theorem 7.4.
Let r be a prime power. Suppose that q, r, j and d are positive integers andlet { S i } ki =1 ⊆ R d is a collection of colored points where all color classes S i are of the samecardinality m . Then if qr ≤ m ( j − − r + 1 and ( r − d + 1) + 1 ≤ qrk , then it is alwayspossible to partition the set S := ∪ ki =1 S i into r j -wise disjoint sets containing at most q points of each color, so that their convex hulls conv( S i ) have a non-empty intersection. Proof:
The sets containing at most q points of each color span the multicolored simpliceswhich are encoded as the simplices of the simplicial complex ([ pt ] ∗ m ∆( q +1) ) ∗ k . Indeed, theseare precisely the simplices which are allowed to have at most q vertices in each of k differentcolors. The configuration space of all r -tuples of j -wise disjoint multicolored simplices isthe simplicial complex, K = (([ pt ] ∗ m ∆( q +1) ) ∗ k ) ∗ r ∆( j ) . Since the join and deleted join commute, this complex is isomorphic to, K = (([ pt ] ∗ m ∆( q +1) ) ∗ r ∆( j ) ) ∗ k . If we suppose, contrary to the statement of the theorem, that the intersection of imagesof any r , j -wise disjoint multicolored simplices is empty, the associated mapping F : K → ( R d ) ∗ r would miss the diagonal D . As before, by composing this map with the orthogonal15rojection to D ⊥ , and after the radial projection to the unit sphere in D ⊥ , we obtain a ( Z /p ) α -equivariant mapping, ˜ F : K → S ( r − d +1) − . The complex ([ pt ] ∗ m ∆( q +1) ) ∗ r ∆( j ) is a multiple chessboard complex ∆ q,j − m,r . Since we assumed qr ≤ ( j − m − r + 1 , this complex is ( qr − -connected by Corollary 7.2. Hence thecomplex K is ( qrk − -connected. By our assumption qrk ≥ ( r − d + 1) + 1 , so in lightof Volovikov’s theorem [V96] such a mapping ˜ F does not exist. (cid:3) Theorem 7.5.
Let r be a prime power. Suppose that q, r, j and d are positive integers andlet { S i } ki =1 ⊆ R d is a collection of colored points where all color classes S i are of the samecardinality m . If jm − ≤ qr and ( r − d + 1) + 1 ≤ ( j − mk , then it is possible todivide all points in r , j -wise disjoint sets containing at most q points of each color, so thattheir convex hulls conv( S i ) have a non-empty intersection. Proof:
As before the sets containing at most q points of each color span the multicoloredsimplices which are encoded as the simplices of the simplicial complex ([ pt ] ∗ m ∆( q +1) ) ∗ k . Indeedthese are precisely the simplices which are allowed to have at most q vertices in each of k different colors. The configuration space of all r -tuples of j -wise disjoint multicoloredsimplices is the simplicial complex, K = (([ pt ] ∗ m ∆( q +1) ) ∗ k ) ∗ r ∆( j ) . Since the join and deleted join commute, this complex is isomorphic to, K = (([ pt ] ∗ m ∆( q +1) ) ∗ r ∆( j ) ) ∗ k . If we suppose, contrary to the statement of the theorem, that the intersection of imagesof any r , j -wise disjoint multicolored simplices is empty, the associated mapping F : K → ( R d ) ∗ r would miss the diagonal D . As before, from here by an equivariant deformation weobtain a ( Z /p ) α -equivariant mapping, ˜ F : K → S ( r − d +1) − . The complex ([ pt ] ∗ m ∆( q +1) ) ∗ r ∆( j ) is the multiple chessboard complex ∆ q,j − m,r . Since we as-sumed ( j − m ≤ qr − m + 1 , this complex is (( j − m − -connected by Corollary7.2. Hence the complex K is (( j − mk − -connected. By our assumption ( j − mk ≥ ( r − d + 1) + 1 , and again this is in contradiction with Volovikov’s theorem [V96]. (cid:3) For illustration let us consider a very special case of this theorem q = 1 and j = 2 . Theorem 7.6.
Let r be a prime power. Given k finite sets of points in R d (called colors),of m points each, so that m − ≤ r and ( r − d + 1) + 1 ≤ mk , it is possible to dividethe points in r pairwise disjoint sets containing at most point of each color, so that theirconvex hulls intersect. Remark 7.7.
It is easy to see that the assumptions on the total number of points is thebest possible, since the set of ( r − d + 1) points in the general position could not bedivided in r disjoint sets whose convex hulls intersect.16 .1 A comparison with known results It is interesting to compare results from the previous section with similar results from [BFZ](Section 9). Note that the proof methods are quite different. We use high connectivity ofthe multiple chessboard complex, established in Section 6, while the authors of [BFZ] usethe ‘constraint method’, relying on the ‘optimal colored Tverberg theorem’ from [BMZ], asa ‘black box’ result.For illustration, let us compare our Theorem 7.6 to Theorem 9.1 from [BFZ].Let us choose k ≥ d + 1) in Theorem 7.6 and select the smallest m satisfying theinequality ( r − d + 1) + 1 ≤ mk , meaning that we are allowed to assume ( m − k < ( r − d + 1) + 1 ≤ mk . From here we immediately deduce the inequality m − ≤ r and, as a consequence ofTheorem 7.6, we have the following result. Corollary 7.8.
Let r be a prime power. Assume k ≥ d + 1) and choose m satisfyingthe inequality ( r − d + 1) + 1 ≤ mk . Suppose that S ⊂ R d is a set of cardinality mk ,evenly colored by k colors (meaning that S = ∪ ki =1 S i where | S i | = m for each i ). Then itis possible to select r pairwise disjoint subsets C i ⊂ S , containing at most point of eachcolor, so that ∩ ri =1 conv( C i ) (cid:54) = ∅ . This result clearly follows from Theorem 9.1 if we assume that r is a prime. Corollary7.8 illustrates the phenomenon that there exist instances of the ‘optimal colored Tverbergtheorem’ (Theorem 9.1 in [BFZ]) which remain valid if the condition on r being a prime isrelaxed to r is a prime power. In this section we briefly discuss the problem whether each admissible r-tuple is Tverbergprescribable. This problem, as formulated in [BFZ], will be referred to as the TverbergA-P problem or the Tverberg A-P conjecture.
Definition 7.9.
For d ≥ and r ≥ , an r -tuple d = ( d , ..., d r ) of integers is admissible if, [ d ] ≤ d i ≤ d for all i , and (cid:80) ri =1 ( d − d i ) ≤ d . An admissible r-tuple is Tverberg prescribableif there is an N such that for every continuous map f : ∆ N → R d there is a Tverbergpartition { σ , ..., σ r } for f with dim( σ i ) = d i . Question: (Tverberg A-P problem; [BFZ] (Question 6.9.)) Is every admissible r -tupleTverberg prescribable?As shown in [F], (Theorem 2.8.), the answer to the above question is negative. It wasalso demonstrated that a more realistic conjecture arises if the condition [ d ] ≤ d i ≤ d , in thedefinition of admissible r -tuple, is replaced by a stronger requirement ( r − r ( d − ≤ d i ≤ d for all i . 17ere we remark that a positive answer to the modified question is quite straightforwardin the case r ≥ d . Indeed, in this case we have for all id i ≥ ( r − r ( d − ≥ d − − ( d − r > d − . So, in this case each d i is equal to either d − or d , and the A-P conjecture reduces to the‘balanced case’, established in [JVZ-2]. References [A04] C.A. Athanasiadis. Decompositions and connectivity of matching and chessboardcomplexes.
Discrete Comput. Geom.
31 (2004), 395–403.[A-F] S. Ault, Z. Fiedorowicz. Symmetric homology of algebras.
Algebr. Geom. Topol. ,Vol. 10, Number 4 (2010), 2343–2408. arXiv:0708.1575v54 [math.AT] 5 Nov 2007.[Au10] S. Ault. Symmetric homology of algebras,
Algebr. Geom. Topol. , Vol. 10, No 4(2010), 2343–2408.[Bi92] T. Bier. A remark on Alexander duality and the disjunct join. Unpublishedpreprint, 1992.[BLVŽ] A. Björner, L. Lovász, S.T. Vrećica, and R.T. Živaljević. Chessboard complexesand matching complexes.
J. London Math. Soc. (2)
49 (1994), 25–39.[BPSZ] A. Bjorner, A. Paffenholz, J. Sjostrand, G.M. Ziegler, Bier spheres and posets,
Discrete Comput. Geom.
34 (2005), no. 1, 71–86.[BFZ] P.V.M. Blagojević, F. Frick, G.M. Ziegler. Tverberg plus constraints.
Bull. Lond.Math. Soc. , 46:953–967, 2014.[BMZ] P.V.M. Blagojević, B. Matschke, G.M. Ziegler. Optimal bounds for the coloredTverberg problem.
J. Eur. Math. Soc. 17 , 4 (2015), 739–754.[Cha] M.K. Chari. On discrete Morse functions and combinatorial decompositions.
Dis-crete Math. , 217(1-3):101–113, 2000.[Č-D] S.Lj. Čukić, E. Delucchi. Simplicial shellable spheres via combinatorial blowups,
Proc. Amer. Math. Soc.
135 (2007), no. 8, 2403–2414.[E-F] J. Edmonds, D.R. Fulkerson, Bottleneck extrema,
Journal of Combinatorial The-ory
Algebraic Topology Discus-sion List (maintained by Don Davis), .[Fo02] R. Forman, A user’s guide to discrete Morse theory,
Sém. Lothar. Combin.
J.Algebraic Combin.
Cohen-Macaulay complexes and group actions . Ph.D. Thesis, Univ. ofWisconsin-Madison, 1979.[H] S. Hell. Tverberg’s theorem with constraints.
J. Combinatorial Theory , Ser. A115:1402–1406, 2008.[JNPZ] D. Jojić, I. Nekrasov, G. Panina, R. Živaljević. Alexander r-tuples and Bier com-plexes,
Publ. Inst. Math. (Beograd) (N.S.)
A Tverberg type theorem for collectively un-avoidable complexes , Israel J. Math. (accepted for publication), arXiv:1812.00366[math.CO].[JPZ-2] D. Jojić, G. Panina, R. Živaljević,
Splitting necklaces, with constraints ,arXiv:1907.09740 [math.CO].[JVZ-1] D. Jojić, S.T. Vrećica, R.T. Živaljević. Multiple chessboard complexes and thecolored Tverberg problem.
J. Combin. Theory Ser. A , 145:400–425, 2017.[JVZ-2] D. Jojić, S.T. Vrećica, R.T. Živaljević. Symmetric multiple chessboard complexesand a new theorem of Tverberg type.
J. Algebraic Combin. , 46:15–31, 2017.[JVZ] D. Jojić, S.T. Vrećica, R.T. Živaljević. Topology and combinatorics of unavoidablecomplexes, arXiv:1603.08472v1 [math.AT], (unpublished prepreint).[J] J. Jonsson.
Simplicial Complexes of Graphs . Lecture Notes in Mathematics, Vol.1928. Springer 2008.[J08] J. Jonsson. Exact sequences for the homology of the matching complex.
Journalof Combinatorial Theory , Series A 115 (2008), no. 8, 1504–1526.[J07] J. Jonsson. On the -torsion part of the homology of the chessboard complex. Annals of Combinatorics , 2010, (14) 4, 487–505.[KRW] D.B. Karaguezian, V. Reiner, M.L. Wachs. Matching Complexes, Bounded DegreeGraph Complexes, and Weight Spaces of GL -Complexes.
J. Algebra
Combinatorial Algebraic Topology , Algorithms and Computation inMathematics, Springer 2008. 19Lo04] M. de Longueville. Bier spheres and barycentric subdivision,
J. Comb. Theory Ser.A
105 (2004), 355–357.[M03] J. Matoušek.
Using the Borsuk-Ulam Theorem. Lectures on Topological Methodsin Combinatorics and Geometry . Universitext, Springer-Verlag, Heidelberg, 2003(Corrected 2nd printing 2008).[R-R] V. Reiner and J. Roberts. Minimal resolutions and homology of chessboard andmatching complexes.
J. Algebraic Combin.
11 (2000), 135–154.[S-W] J. Shareshian and M.L. Wachs. Torsion in the matching complex and chessboardcomplex.
Advances in Mathematics
212 (2007) 525–570.[V96] A.Y. Volovikov.
On the van Kampen-Flores theorem , Math. Notes, 59:477–481,1996.[VŽ94] S. Vrećica and R. Živaljević. New cases of the colored Tverberg theorem. InH. Barcelo and G. Kalai, editors,
Jerusalem Combinatorics ’93 , Contemp. Math.Vol. 178, pp. 325–334, A.M.S. 1994.[VŽ09] S. Vrećica and R. Živaljević. Cycle-free chessboard complexes and symmetric ho-mology of algebras.
European J. Combinatorics
30 (2009) 542–554.[VŽ11] S. Vrećica and R. Živaljević. Chessboard complexes indomitable.
J. Combin.Theory Ser. A , 118:2157–2166, 2011.[W03] M.L. Wachs. Topology of matching, chessboard, and general bounded degree graphcomplexes, Dedicated to the memory of Gian-Carlo Rota.
Algebra Universalis
Israel J. Math.
87 (1994), 97–110.[ŽV92] R.T. Živaljević and S.T. Vrećica. The colored Tverberg’s problem and complexesof injective functions.
J. Combin. Theory Ser. A
61 (1992), 309–318.[Ž17] R.T. Živaljević. Topological methods in discrete geometry. Chapter 21 in
Handbookof Discrete and Computational Geometry , third ed., J.E. Goodman, J. O’Rourke,and C.D. Tóth, CRC Press LLC, Boca Raton, FL, 2017.[Živ98] R. Živaljević. User’s guide to equivariant methods in combinatorics, I and II.