Colored Tverberg problem, extensions and new results
aa r X i v : . [ m a t h . M G ] F e b Colored Tverberg problem,extensions and new results
Duško Jojić
Faculty of ScienceUniversity of Banja Luka
Gaiane Panina
St. Petersburg State UniversitySt. Petersburg Department ofSteklov Mathematical Institute
Rade T. Živaljević
Mathematical InstituteSASA, Belgrade
February 24, 2020
Abstract
We prove a “multiple colored Tverberg theorem” and a “balanced colored Tverbergtheorem”, by applying different methods, tools and ideas. The proof of the firsttheorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proofof the second result relies on high connectivity of the configuration space, establishedby discrete Morse theory.
Tverberg-Van Kampen-Flores type results have been for decades one of the central researchthemes in topological combinatorics. The last decade has been particularly fruitful, bring-ing the resolution (in the negative) of the general “Topological Tverberg Problem” [MW14,F, BFZ2, MW15, MW16], as summarized by several review papers [BBZ, BS, Sk18, Ž17].New positive results include the proof [JVZ-2, Theorem 1.2] of the “Balanced VanKampen-Flores theorem” and the development of “collectively unavoidable simplicial com-plexes”, leading to very general theorems of Van Kampen-Flores type [JPZ-1].Somewhat surprisingly the colored Tverberg problem , which in the past also occupiedone of the central places [M03, Ž17], doesn’t seem to have been directly affected by thesedevelopments.For example the
Topological type A colored Tverberg theorem (Theorem 2.2 in [BMZ]) isknown to hold if r is a prime number (see Section 1.1 for the definition of the parameter r )and at present we don’t know what happens in other cases. For comparison, the topologicalTverberg theorem is known to be true if r = p ν is a prime power, however we know todaythat this condition is essential (not an artefact of the topological methods used in theproof). 1oreover, while there has been a notable, more recent activity [JVZ-2, JPZ-1] in thearea of “monochromatic” Tverberg-Van Kampen-Flores theorems, the corresponding newpositive colored-type results seem to be virtually non-existent after the arXiv announce-ment of [BMZ](more than ten years ago), where the type A colored Tverberg theorem wasestablished.In this paper we open two new directions of studying colored Tverberg problem by prov-ing a “multiple Colored Tverberg theorem” (Theorem 1.3) and “balanced Colored Tverbergtheorem” (Theorem 1.7).The “multiple Colored Tverberg theorem” has evolved from the simplified proof of thetype A colored Tverberg theorem given in [VŽ11] and relies on the Eilenberg-Krasnoselskiitheory of degrees of equivariant maps for non-free action.The “balanced colored Tverberg theorem” is a relative of the type B colored Tverbergtheorem [ŽV92, VŽ94] and the “balanced Van Kampen-Flores theorem” [JVZ-2]. It usesthe methods of Forman’s discrete Morse theory, and builds on the methods and ideasdeveloped in our earlier papers [JNPZ, JPVZ, JPZ-1]. Acknowledgements.
Proposition 3.2 is supported by the Russian Science Foundationunder grant 16-11-10039. R. Živaljević was supported by the Ministry of Education, Scienceand Technological Development of Serbia (via grants to Mathematical Institute SASA). “Tverberg type theorems” is a common name for a growing family of theorems, conjecturesand problems about special partitions (patterns) of finite sets of points (point clouds) inthe affine euclidean space R d .The original Tverberg’s theorem claims that every set S ⊂ R d with ( r − d + 1) + 1 elements can be partitioned S = S ∪ . . . ∪ S r into r nonempty, pairwise disjoint subsets S , . . . , S r such that the corresponding convex hulls have a nonempty intersection: r \ i =1 conv( S i ) = ∅ . (1.1)Among the numerous relatives (refinements, predecessors, generalizations) of Tverberg’stheorem are the classical Radon’s lemma, Topological Tverberg theorem, Colored Tverbergtheorems, as well as the classical Van Kampen-Flores theorem and its generalizations.The original Tverberg’s theorem can be reformulated as the statement that for eachlinear (affine) map f : ∆ N a −→ R d from a N -dimensional simplex ( N = ( r − d + 1) ) thereexist r nonempty disjoint faces ∆ , . . . , ∆ r of the simplex ∆ N such that f (∆ ) ∩ . . . ∩ f (∆ r ) = ∅ . This form of Tverberg’s result can be abbreviated as follows, (∆ ( r − d +1) a −→ R d ) ⇒ ( r − intersection) (1.2)2here the input on the left is (as indicated) an affine map and as the output on the rightwe obtain the existence of a “Tverberg r -intersection”, that is a collection of r disjoint facesof the simplex ∆ ( r − d +1) which overlap in the image.The reformulation (1.2) is very useful since it places the theorem in a broader contextand motivates (potential) extensions and generalizations of Tverberg’s (affine) theorem.For example the affine input map can be replaced by an arbitrary continuous map f :∆ N → R d . The corresponding more general statement (known as the Topological Tverbergtheorem) (∆ ( r − d +1) −→ R d ) ⇒ ( r − intersection) (1.3)is also true provided r = p ν is a prime power.Another possibility is to prescribe in advance which pairwise disjoint faces ∆ , . . . , ∆ r of ∆ N are acceptable, preferred or “admissible”, say by demanding that ∆ i ∈ K for a chosensimplicial subcomplex K ⊆ ∆ N . The input map is now a continuous (affine, simplicial)map f : K → R d and the conclusion is the same as in the Topological Tverberg theorem.The following four statements are illustrative for results of ‘colored Tverberg type’ (see[Ž17] for more detailed presentation). ( K , −→ R ) ⇒ (2 − intersection) (1.4) ( K , , a −→ R ) ⇒ (3 − intersection) (1.5) ( K , , −→ R ) ⇒ (3 − intersection) (1.6) ( K , , , −→ R ) ⇒ (4 − intersection) (1.7)The complex K t ,t ,...,t k = [ t ] ∗ [ t ] ∗ . . . ∗ [ t k ] is by definition the complete multipartitesimplicial complex obtained as a join of -dimensional complexes (finite sets). For example K p,q = [ p ] ∗ [ q ] is the complete bipartite graph obtained by connecting each of p ‘red vertices’with each of q ‘blue vertices’.More generally a coloring of vertices of a simplex by k + 1 colors is a partition V = V ert (∆ N ) = C ⊎ C ⊎ · · · ⊎ C k into “monochromatic” subsets C i . A subset ∆ ⊆ V is calleda multicolored set or a rainbow simplex if and only if | ∆ ∩ C i | ≤ for each i = 0 , . . . , k .If the cardinality of C i is t i we observe that K t ,t ,...,t k is precisely the subcomplex of allrainbow simplices in ∆ N .We refer the reader to [ŽV92, VŽ94, BMZ], [M03, Živ98] and [BBZ, BS, Sk18, Ž17] formore general statements, proofs, history and applications of colored Tverberg theorems.Following the classification proposed in [Ž17] we say that a colored Tverberg theoremis of type A if k ≥ d (where k + 1 is the number of colors and d is the dimension of theambient euclidean space). In the case of the opposite inequality k < d we say that it is of type B . The main difference between these two types is that in the type B case the number r of intersecting rainbow simplices must satisfy the inequality r ≤ d/ ( d − k ) , while in thetype A case there are no a priori constraints on these numbers.3n agreement with this classification (1.5) and (1.7) are classified as topological type Acolored Tverberg theorems while (1.4) and (1.6) are instances of topological type B coloredTverberg theorem .The following general results (Theorems 1.1 and 1.2) are the main representatives ofthese two classes of colored Tverberg theorems. In particular (1.4) (1.6) and (1.7) are theireasy consequences. Caveat:
Here and elsewhere in the paper we do not distinguish the N -dimensional (geo-metric) simplex ∆ N from a (combinatorial) simplex ∆ [ m ] = 2 [ m ] (abstract simplicial com-plex) spanned by m vertices (if m = N + 1 ). In agreement with this convention, subsets S ⊂ [ m ] are interpreted as simplices, faces of ∆ [ m ] . For S ⊂ [ m ] we have dim( S ) = | S | − where | S | is the cardinality of S . Theorem 1.1. (Type A) [BMZ]
Let r ≥ be a prime, d ≥ , and N := ( r − d + 1) .Let ∆ N be an N -dimensional simplex with a partition (coloring) of its vertex set into d + 2 parts, V = [ N + 1] = C ⊎ · · · ⊎ C d ⊎ C d +1 , with | C i | = r − for i ≤ d and | C d +1 | = 1 . Thenfor any continuous map f : ∆ N → R d , there are r disjoint rainbow simplices ∆ , . . . , ∆ r of ∆ N satisfying f (∆ ) ∩ · · · ∩ f (∆ r ) = ∅ . Theorem 1.2. (Type B) [VŽ94, Živ98]
Assume that r = p ν is a prime power, d ≥ , and k < d . Let [ m ] = C ⊎ · · · ⊎ C k be a coloring partition of vertices of the simplex ∆ [ m ] where m = (2 r − k + 1) such that r ≤ d/ ( d − k ) and | C i | = 2 r − for each i . Then for anycontinuous map f : ∆ [ m ] → R d , there are r disjoint rainbow simplices ∆ , . . . , ∆ r of ∆ [ m ] satisfying f (∆ ) ∩ · · · ∩ f (∆ r ) = ∅ . The implication (1.5) (Section 1.1) is an instance of a result of Bárány and Larman [BL92].It says that each collection of nine points in the plane, evenly colored by three colors, canbe partitioned into three ‘rainbow triangles’ which have a common point.At present it is not known if the following non-linear (topological) version of (1.5) istrue or not: ( K , , −→ R ) ⇒ (3 − intersection) . (1.8)In other words we don’t know if for each continuous map f : K , , → R there exist threepairwise vertex disjoint simplices ∆ , ∆ , ∆ in K , , such that f (∆ ) ∩ f (∆ ) ∩ f (∆ ) = ∅ .The implication (1.8) clearly follows from the following stronger statement: ( K , , , −→ R ) ⇒ (4 − intersection) . (1.9)However the implication (1.9) is also not known to hold in full generality (and we stronglysuspect that it is not the case). 4he following “multiple Colored Tverberg theorem” claims that the implication (1.9) istrue for continuous maps f : K , , , → R which satisfy an additional (3-to-2) constraint.(The reader may find it instructive to read first its affine version, Corollary 1.4.) Theorem 1.3.
Suppose that K = K , , , ∼ = [3] ∗ [3] ∗ [3] ∗ [1] is a -dimensional simplicialcomplex on a ten-element vertex set, divided into four color classes. Assume that f is a(3-to-2) map, meaning that f = b f ◦ α for some map b f : K , , , −→ R where α : K , , , −→ K , , , is the simplicial map arising from a choice of epimorphisms [3] → [2] . Then there existfour pairwise vertex disjoint simplices ( vertex-disjoint rainbow simplices) ∆ , ∆ , ∆ , ∆ in K such that f (∆ ) ∩ f (∆ ) ∩ f (∆ ) ∩ f (∆ ) = ∅ . (1.10) Corollary 1.4.
Suppose that X is a collection of points in the plane R . Moreover,assume that these points are colored by colors, meaning that there is a partition X = A ⊔ B ⊔ C ⊔ D into monochromatic sets, where A = { a , a } , B = { b , b } , C = { c , c } are -element sets and D = { d } is a singleton.Then there exist four rainbow sets ∆ i ⊂ A ∪ B ∪ C ∪ D ( i = 1 , , , such that (1) Conv (∆ ) ∩ Conv (∆ ) ∩ Conv (∆ ) ∩ Conv (∆ ) = ∅ (2) each a , b , c , d appears as a vertex in exactly one of the sets ∆ i and each a , b , c appears as a vertex in exactly two of the sets ∆ i . The following corollary says that the implication (1.8) is true for a special class ofnon-linear maps.
Corollary 1.5.
Assume that f : K , , −→ R is a continuous map which admits a factor-ization K , , α −→ K , , b f −→ R (1.11) for some b f , where α is a (3-to-2) map. Then there exist three disjoint triangles ∆ , ∆ , ∆ in K , , such that f (∆ ) ∩ f (∆ ) ∩ f (∆ ) = ∅ . The proof of both Theorem 1.3 and its corollaries is postponed for Section 2. Theproofs rely on Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-freeactions, see the monograph [KB] for a detailed presentation of the theory.5 .3 Balanced colored Tverberg-type theorem
Our “balanced colored Tverberg theorem” (Theorem 1.7) was originally envisaged as anextension of the type B colored Tverberg theorem (Theorem 1.2) in the direction of thefollowing theorem which is often referred to as the balanced extension of the generalizedVan Kampen-Flores theorem . Theorem 1.6. ([JVZ-2, Theorem 1.2])
Let r ≥ be a prime power, d ≥ , N ≥ ( r − d + 2) , and rk + s ≥ ( r − d for integers k ≥ and s < r . Then for everycontinuous map f : ∆ N → R d , there are r pairwise disjoint faces ∆ , . . . , ∆ r of ∆ N suchthat f (∆ ) ∩ · · · ∩ f (∆ r ) = ∅ , with dim ∆ i k + 1 for i s and dim ∆ i k for s < i r . If one assumes that ( r − d is divisible by r , in which case s = 0 and dim ∆ i k for each i , then Theorem 1.6 reduces to the ‘equicardinal’ generalized Van Kampen-Florestheorem of Sarkaria [Sar], Volovikov [V96] and Blagojević, Frick and Ziegler [BFZ1].The following “balanced colored Tverberg theorem” can be described as a relative ofTheorem 1.6 and “balanced” extension of Theorem 1.2. Theorem 1.7.
Assume that r = p ν is a prime power and let d ≥ . Let integers k ≥ and < s r be such that r ( k −
1) + s = ( r − d, or more explicitly, (1.12) k := ⌈ ( r − d/r ⌉ and s := ( r − d − r ( k − . (1.13) Let [ m ] = C ⊎· · ·⊎ C k +1 be a coloring partition of vertices of ∆ [ m ] , where m = (2 r − k +1) and | C i | = 2 r − for each i . Then for any continuous map f : ∆ [ m ] → R d there are r disjoint rainbow simplices ∆ , . . . , ∆ r of ∆ [ m ] satisfying f (∆ ) ∩ · · · ∩ f (∆ r ) = ∅ such thatdim (∆ i ) = | ∆ i | − k for i s and dim (∆ i ) k − for s < i r . (1.14) Remark 1.8.
Theorem 1.2 is a special case of Theorem 1.7 for s = r . Indeed, thecondition r ≤ d/ ( d − k ) (in Theorem 1.2) is easily checked to be equivalent to the condition rk ≥ ( r − d .The proof of Theorem 1.7 is based on high connectivity of the appropriate configurationspace (Proposition 3.2), which is proved by the methods of discrete Morse theory.For the reader’s convenience we briefly outline basic facts and ideas from discrete Morsetheory in Appendix 1. For a more complete presentation the reader is referred to [Fo02]. The first step in the proof of the multiple Colored Tverberg theorem (Theorem 1.3) is astandard reduction, via the
Configuration Space/Test Map scheme [Ž17, M03, Živ98], to aproblem of equivariant topology. 6tarting with a continuous map f : K , , , → R we build the associated configurationspace as the deleted join ( K , , , ) ∗ = ([3] ∗ [3] ∗ [3] ∗ [1]) ∗ ∼ = (∆ , ) ∗ ∗ [4] where ∆ , is the standard chessboard complex of all non-taking rook placements on a (3 × -chessboard.The associated test map, designed to test if a simplex τ = (∆ , ∆ , ∆ , ∆ ) ∈ ( K , , , ) ∗ satisfies the condition (1.10), is defined as a Σ -equivariant map Φ : ( K , , , ) ∗ −→ ( R ) ∗ /D ֒ → ( W ) ⊕ (2.1)where D ⊂ ( R ) ∗ is the diagonal ( -dimensional) subspace and W is the standard, -dimensional real permutation representation of Σ .Summarizing, in order to show that there exists a -tuple (∆ , ∆ , ∆ , ∆ ) satisfying(1.10) it is sufficient to prove that the Σ -equivariant map (2.1) must have a zero.For the next step we need to use, side by side with the standard chessboard complex ∆ , (on a (3 × -chessboard), a ‘multiple chessboard complex’ ∆ ; L , ) , defined as the complex ofall rook placements on a (2 × -chessboard such that in the first column up to two rooksare permitted, while in all rows and in the second column at most one rook is allowed.More general ‘multiple chessboard complexes’ are studied in [JVZ-1], and the notationfollows this paper. In particular the vectors = (1 , , , (respectively L = (1 , )describe the restriction on the rook placements in the rows (respectively the columns) ofthe (2 × -chessboard.Now we use the fact that f satisfies the (3-to-2) condition, which allows us to prove thefollowing lemma. Lemma 2.1.
Assume that f is a (3-to-2) map, meaning that f = b f ◦ α for some map b f : K , , , −→ R where α : K , , , −→ K , , , is the simplicial map arising from a choice of epimorphisms [3] → [2] . Under this conditionthe equivariant map (2.1) admits a factorization Φ = b Φ ◦ π into Σ -equivariant maps, asdisplayed in the following commutative diagram (∆ ; L , ) ∗ (3) ∗ [4] b Φ −−−→ ( W ) ∗ (3) π x ∼ = x (∆ , ) ∗ ∗ [4] Φ −−−→ ( W ) ∗ (3) (2.2) where ∆ ; L , is the multiple chessboard complex defined above and π is an epimorphism. Proof:
The proof is by elementary inspection. Note that the map b π : ∆ , → ∆ ; L , , whichinduces the map π in the diagram (2.2), is informally described as the map which unifiestwo columns of the (3 × -chessboard into one column of the (2 × -chessboard. (cid:3) Σ -equivariant map b Φ always has a zero. (Here we tacitlyuse the fact that π is an epimorphism.) (∆ ; L , ) ∗ (3) The Σ -representation W can be described as R with the action coming from the sym-metries of regular tetrahedron ∆ [4] , centered at the origin. If the map b Φ has no zeros thanthere exists a Σ -equivariant map g : (∆ ; L , ) ∗ (3) ∗ [4] −→ ( ∂ ∆ [4] ) ∗ (3) where ∂ ∆ [4] is the boundary sphere of the simplex ∆ [4] . This is ruled out by the followingtheorem. Theorem 2.2.
Let G = ( Z ) = { , α, β, γ } be the Klein four-group . Let ∆ ; L , be themultiple chessboard complex (based on a × chessboard), where = (1 , , , and L =(2 , , and let ∂ ∆ [4] ∼ = S be the boundary of a simplex spanned by vertices in [4] . Both ∆ ; L , and ∂ ∆ [4] ∼ = S are G -spaces, where the first action permutes the rows of the chessboard [2] × [4] , while the second permutes the vertices of the -simplex ∆ [4] . Under these conditionsthere does not exist a G -equivariant map f : (∆ ; L , ) ∗ (3) ∗ [4] −→ ( ∂ ∆ [4] ) ∗ (3) ∼ = ( S ) ∗ (3) ∼ = S where the joins have the diagonal G -action. Theorem 2.2 is proved by an argument involving the degree of equivariant maps whichcan be traced back to Eilenberg and Krasnoselskii, see [KB] for a thorough treatment andAppendix 1 for the statement of one of the main theorems.Before we commence the proof of Theorem 2.2 let us describe a convenient geometricmodel for the complex ∆ ; L , . Recall that the Bier sphere Bier ( K ) of a simplicial complex K ⊂ [ m ] is the deleted join K ∗ ∆ K ◦ of K and its Alexander dual K ◦ , see [M03] for moredetails on this subject. Lemma 2.3.
The multiple chessboard complex ∆ ; L , is a triangulation of a -sphere. Moreexplicitly, there is an isomorphism ∆ ; L , ∼ = Bier (∆ (1)[4] ) , where ∆ (1)[4] is the -skeleton of thetetrahedron ∆ [4] and Bier ( K ) = K ∗ ∆ K ◦ is the Bier sphere associated to a simplicialcomplex K (and its Alexander dual K ◦ ). The proof of Lemma 2.3 is straightforward and relies on the observation that the sub-complexes of ∆ ; L , , generated by the vertices in the first (second) column of the chessboard [2] × [4] , are respectively K = ∆ (1)[4] and K ◦ = (∆ (1)[4] ) ◦ = ∆ (0)[4] . The following lemma clarifiesthe structure of the sphere ∆ ; L , as a G -space where G = ( Z ) = { , α, β, γ } is the Kleinfour-group. 8 emma 2.4. As a G -space the sphere ∆ ; L , is homeomorphic to the regular octahedralsphere (positioned at the origin), where the generators α, β, γ are interpreted as the ◦ -rotations around the axes connecting pairs of opposite vertices of the octahedron.More explicitly, let R α be the -dimensional G -representation characterized by the con-ditions αx = x, βx = γx = − x ( R β and R γ are defined similarly) and let S α , S β , S γ bethe corresponding -dimensional G -spheres. Then ∆ ; L , is G -isomorphic to the -sphere S ( R α ⊕ R β ⊕ R γ ) ∼ = S α ∗ S β ∗ S γ with the induced G -action. Remark 2.5.
Here is a geometric interpretation (visualization) of the G -isomorphism ∆ ; L , ∼ = Bier (∆ (1)[4] ) . The geometric realizations of K = ∆ (1)[4] and its Alexander dual K ◦ =∆ (0)[4] are respectively constructed in the tetrahedron ∆ [4] and its polar body ∆ ◦ [4] . If bothtetrahedra are inscribed in the cube I , the geometric realization of Bier ( K ) is naturallyinterpreted as a triangulation of the boundary ∂ ( I ) of the cube I . Lemma 2.6.
As a G -space the boundary sphere of the tetrahedron ∂ ∆ [4] is also home-omorphic to the octahedral sphere described in Lemma 2.4. Moreover, there is a radial G -isomorphism ρ : ∂ ( I ) → ∂ ∆ [4] . Summarizing we conclude that the G -sphere we are studying in this section has twocombinatorial ∆ ; L , , ∂ ∆ [4] = 2 [4] \ { [4] } and three equivalent geometric incarnations, theboundary of the cube ∂ ( I ) , the boundary of the tetrahedron ∂ ∆ [4] , and the boundary ofthe octahedron S α ∗ S β ∗ S γ . Proposition 2.7.
Let φ : (∆ ; L , ) ∗ (3) → ( ∂ ∆ [4] ) ∗ (3) be an arbitrary G -equivariant map. Then deg( φ ) ≡ mod . Proof:
It follows from Theorem 3.10 that deg( φ ) ≡ deg( ψ ) ( mod for each two equivariantmaps of the indicated spaces. Here we use that fact that (∆ ; L , ) ∗ (3) ∼ = ( S ) ∗ (3) ∼ = S is atopological manifold.Hence, it is enough to exhibit a single map ψ with an odd degree. In light of the resultsfrom the previous section (∆ ; L , ) ∗ (3) and ( ∂ ∆ [4] ) ∗ (3) are G -homeomorphic, -dimensionalspheres. If we choose the G -isomorphism ψ : (∆ ; L , ) ∗ (3) → ( ∂ ∆ [4] ) ∗ (3) then deg( ψ ) = ± . (cid:3) Proof of Theorem 2.2: (∆ ; L , ) ∗ (3) ∗ [4] f −−−→ ( ∂ ∆ [4] ) ∗ (3) e x ∼ = x (∆ ; L , ) ∗ (3) φ −−−→ ( ∂ ∆ [4] ) ∗ (3) (2.3)Suppose that a G -equivariant map f exists. Let e be the inclusion map and let φ = f ◦ e be the composition.The map e is homotopically trivial, since Image( e ) ⊂ Cone( v ) for each v ∈ [4] . This is,however, a contradiction since in light of Proposition 2.7 the map φ has an odd degree. (cid:3) Proof of the balanced Color Tverberg theorem
Following the “join variant” of the configuration space/test map -scheme [M03] [Ž17], a configuration space C ⊆ ∆ ∗ ( r )[ m ] , appropriate for the proof of Theorem 1.7, collects togetherall joins A ∗ · · · ∗ A r := A ⊎ · · · ⊎ A r of disjoint rainbow simplices A i ⊂ [ m ] , satisfying(after a permutation of indices) the condition (1.14) from Theorem 1.7. For the futurereference we record a more detailed description of this configuration space. Definition 3.1.
The configuration space C of r -tuples of disjoint rainbow simplices satis-fying the restrictions listed in Theorem 1.7 is the simplicial complex whose simplices arelabeled by ( A , ..., A r ; B ) where • [ m ] = A ⊔ ... ⊔ A r ⊔ B is a partition such that B = [ m ] . • Each A i is a rainbow set (simplex), in particular | A i | ≤ k + 1 for each i ∈ [ r ] . • The number of simplices A i with | A i | = k + 1 does not exceed s . Note that the dimension of a simplex ( A , ..., A r ; B ) is | A | + ... + | A r | − . Moreover,a facet of a simplex ( A , ..., A r ; B ) is formally obtained by moving an element of some A i to B . Proposition 3.2.
The configuration space C is ( rk + s − -connected. Let us briefly explain how Theorem 1.7 can be deduced from Proposition 3.2. This isa standard argument used for example in the proof of topological Tverberg theorem, see[M03, Section 6] or [Ž17].Suppose Theorem 1.7 is not true, which means that f ( A ) ∩ · · · ∩ f ( A r ) = ∅ for allcollections A , . . . , A r of r disjoint rainbow simplices satisfying (1.14). From here we deducethat there exists a ( Z /p ) ν - equivariant mapping Ψ f : C → R ( d +1) r missing the diagonal D = { ( y, y, ..., y ) : y ∈ R d +1 } .However Image(Ψ f ) ⊂ R ( d +1) r \ D contradicts Volovikov’s theorem [V96], since R ( d +1) r \ D is ( Z /p ) ν -homotopy equivalent to a sphere of dimension ( r − d + 1) − rk + s − and the configuration space C is by Proposition 3.2 ( rk + s − -connected. Proof of Proposition 3.2:
Let us begin by introducing some useful abbreviations.A set A ⊂ [ m ] is called C i -full if it contains a vertex colored by C i .A simplex ( A , ..., A r ; B ) is called C i -full if each of the A i is full, or equivalently, if | S ri =1 A i T C i | = r . 10 simplex ( A , ..., A r ; B ) is ( k + 1) -full if it contains (the maximal allowed number) s of k + 1 -sets among A i .A simplex ( A , ..., A r ; B ) is saturated if it is ( k + 1) -full, and | A i | ≥ k ∀ i .Saturated simplices are maximal faces of the configuration space C . Their dimension is rk + s − .We now define a Morse matching for C . For a given simplex ( A , ..., A r ; B ) we eitherdescribe a simplex that is paired with it, or alternatively recognize ( A , ..., A r ; B ) as acritical simplex.This is done stepwise. We shall have r “big” steps, each of them further splitting intoconsecutive k + 1 small steps. Big steps treat the sets A i one by one, and small steps treatcolors one by one. Step 1.Step 1.1
Assume that the vertices of each color are enumerated by { , , ..., r − } . Set a = min h ( A ∪ B ) ∩ C i and match ( A ∪ a , A , ..., A r ; B ) with ( A , A , ..., A r ; B ∪ a ) whenever both thesesimplices are elements of C .A simplex of type ( A ∪ a , A , ..., A r ; B ) ∈ C is not matched iff it equals ( { a } , ∅ , ..., ∅ ; [ m ] \ { a } ) . It is -dimensional and it will stay unmatched until the end of the matching process.If a simplex of type ( A , A , ..., A r ; B ∪ a ) is unmatched then either A is C -full, or | A | = k , and ( A , A , ..., A r ; B ∪ a ) is ( k + 1) -full. Step 1.2
Set a = min h ( A ∪ B ) ∩ C i and match ( A ∪ a , A , ..., A r ; B ) with ( A , A , ..., A r ; B ∪ a ) whenever both theseare elements of C that have not been matched on the Step 1.1. • If a simplex of type ( A , A , ..., A r ; B ∪ a ) is unmatched, then either A is C -full,or | A | = k , and ( A , A , ..., A r ; B ∪ a ) is ( k + 1) -full.Such simplices are called " Step 1.2-Type 1 unmatched simplices". • If a simplex of type ( A ∪ a , A , ..., A r ; B ) is not matched, then | A ∪ a | = k + 1 ,and ( A ∪ a , A , ..., A r ; B ) is ( k + 1) -full (these are necessary but not sufficientconditions). The reason is that in this case ( A , A , ..., A r ; B ∪ a ) belongs to C but might be matched on the Step 1.1.Such simplices are called " Step 1.2-Type 2 unmatched simplices."In the sequel we use similar abbreviations. Step i.j – Type 1 means, that one cannotmove an element colored by j from B to A i . Step i.j – Type 2 means, that onecannot move an element colored by j from A i to B .11 tep 1.3 and subsequent steps (up to Step .k + 1 ) go analogously.Summarizing, we conclude: Lemma 3.3.
With the exception of the unique zero-dimensional unmatched simplex, if asimplex ( A , ..., A r ; B ) is unmatched after Step 1 then one of the following is valid:1. either | A | = k + 1 , or2. | A | = k , and ( A , ..., A r ; B ) is ( k + 1) -full. Proof follows directly from the analysis of matching algorithm on small steps.
Step 2.
Now we treat A for the simplices that remained unmatched after Step 1. Step 2.1
Set a = min h(cid:16) ( A ∪ B ) \ [1 , a ] (cid:17) ∩ C i and match ( A , A ∪ a , ..., A r ; B ) with ( A , A , ..., A r ; B ∪ a ) whenever both theseare elements of C that are not matched on Step 1. • If a simplex ( A , A , ..., A r ; B ∪ a ) is not matched now, then either | A | = k ,and ( A , A , ..., A r ; B ∪ a ) is ( k + 1) -full, or A is C -full.Such simplices will be called Step 2.1 — Type 1 simplices. • If a simplex of type ( A , A ∪ a , ..., A r ; B ) is not matched, then it is ( k + 1) -full,and | A | = k + 1 . Such simplices will be called
Step 2.1 — Type 2 simplices.
Step 2.2
Set a = min h(cid:16) ( A ∪ B ) \ [1 , a ] (cid:17) ∩ C i and match ( A , A ∪ a , ..., A r ; B ) with ( A , A , ..., A r ; B ∪ a ) whenever both theseare elements of C that are not matched before, that is, on Step 1, and on Step 2.1. Step 2.3 and subsequent steps (up to Step .k + 1 ) go analogously.Summarizing, we conclude: Lemma 3.4.
With the exception of the unique zero-dimensional unmatched simplex, if asimplex ( A , ..., A r ; B ) is unmatched after Step 2 then it is unmatched after Step 1 (for thiswe have Lemma 3.3), and also one of the following is valid:1. either | A | = k + 1 , or2. | A | = k , and ( A , ..., A r ; B ) is ( k + 1) -full. Steps 3,4,..., and r − go analogously. 12 emma 3.5. For all the steps j = 1 , , ..., r − , the numbers a ij are well-defined. Proof. Indeed, for ( A , ..., A r ; B ) ∈ C , the set B ∩ C i contains at least r − points.(Here we use that | C i | = 2 r − and | A j ∩ C i | ≤ for each j .) The entries a i , a i , ..., a ij − are either not in B ∩ C i , or (by construction) are the smallest consecutive entries of B ∩ C i .Altogether there are strictly less than r − of them.A special attention should be paid to the last Step r .First, let us observe that (by construction) we already have: Lemma 3.6.
With the exception of the unique zero-dimensional unmatched simplex, if asimplex ( A , ..., A r ; B ) is unmatched after Step r − then one of the following is valid:1. | A | = | A | = ... = | A r − | = k + 1 , or2. for some i , | A i | = k , and ( A , ..., A r ; B ) is ( k + 1) -full. Proof: This follows from Lemma 3.4 and its analogs for Steps , ..., r − . Step r.
Now we turn our attention to A r . Step r.1
Set a r = min h(cid:16) ( A r ∪ B ) \ [1 , a r − ] (cid:17) ∩ C i . It might happen that the set h(cid:16) ( A r ∪ B ) \ [1 , a r − ] (cid:17) ∩ C i is empty for ( A , ..., A r ; B ) ,so a r is ill-defined.This means that ( A , ..., A r ; B ) is C -full. Such simplex is left unmatched and called Step r. — Type 3 simplex.If a r is well-defined, we proceed in our standard way: we match ( A , A , ..., A r ∪ a r ; B ) with ( A , A , ..., A r ; B ∪ a r ) whenever both these are elements of C that are notmatched before. Step r.2
Set a r = min h (( A r ∪ B ) \ [1 , a r − ]) ∩ C i . Again, if this number is ill-defined, this means that ( A , ..., A r ; B ) is C -full, and weleave the simplex Step r. — Type 3 unmatched.Otherwise we proceed standardly. Step r.3 and subsequent steps (up to Step r.k+1) go analogously.Summarizing, we conclude:
Lemma 3.7.
With the exception of the unique zero-dimensional unmatched simplex, if asimplex ( A , ..., A r ; B ) is unmatched after Step r , then it is saturated. | A i | ≥ k for i = 1 , ..., r − .If a simplex ( A , ..., A r ; B ) has | A i | < k for some i , then it has some missing color. Letthe smallest missing color be j . Then the simplex gets matched at Step i.j , since a ij is welldefined and can be added to A i .On each Step i.j , the simplex ( A , ..., A r ; B ) is either Type 1, or Type 2, or (this mighthappen for Step r.j ) Type 3. If it is at least once of Type 2 (does not matter on whichstep), then (by the same lemma) it is ( k + 1) -full, and therefore saturated.If it is always of Type 1 on steps , ..., r − and not saturated, then | A i | = k + 1 for all i = 1 , ..., r − .Since s < r , it is saturated.It remains to prove the acyclicity of the matching.Assume we have a gradient path α p ր β p +10 ց α p ր β p +11 ց α p ր β p +12 ց · · · ց α pm ր β p +1 m ց α pm +1 For each of the simplices α consider the sequence of numbers Π( α ) := ( a , a , ..., a k +11 , a , ..., a k +12 , ..., a r , ..., a k +1 r ) These are all the numbers a ij listed in the order that they appear in the matching algorithm;they are well-defined including the step where α gets matched. If a ir is ill-defined, set it tobe ∞ . Lemma 3.8.
During a path, Π( α ) strictly decreases w.r.t. lexicographic order. Therefore,the matching is acyclic. Firstly, it suffices to look at the two-step paths only: α p ր β p +10 ց α p ր β p +11 The proof is via an easy case analysis. Here are two examples of how it goes:(1) Assume that α p ր β p +10 means adding color i to A j , and β p +10 ց α p means removingcolor i ′ > i from A j . Then1. either α p is matched with some p − -dimensional simplex obtained by removing color i from A j , and the path terminates right here, or2. or α p is matched before Step j.i .(2) Assume that α p ր β p +10 means adding color i to A j , and β p +10 ց α p means removingcolor i ′ from A j ′ with j ′ < j . Then1. either α p is matched by adding color i ′ to A j ′ , or2. or α p is matched before Step j ′ .i ′ .This completes the proof of Theorem 1.7.14 ppendix 1. Discrete Morse theory A discrete Morse function (or a discrete vector field) on a simplicial complex K ⊆ V is, bydefinition, an acyclic matching on the Hasse diagram of the partially ordered set ( K, ⊆ ) .Here is a brief reminder of 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 ≥ .The 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 ր 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 ր β p +10 ց α p ր β p +11 ց α p ր β p +12 ց · · · ց α pm ր β p +1 m ց 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 = 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 turn to the cells). Theorem 3.9. [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. ppendix 2. Comparison principle for equivariant maps The following theorem is proved in [KB] (Theorem 2.1 in Section 2). Note that the con-dition that the H i -fixed point sets S H i are locally k -connected for k ≤ dim( M H i ) − isautomatically satisfied if S is a representation sphere. So in this case it is sufficient toshow that the sphere S H i is (globally) (dim( M H i ) − -connected which is equivalent tothe condition dim( M H i ) ≤ dim( S H i ) ( i = 1 , . . . , m ) . Theorem 3.10.
Let G be a finite group acting on a compact topological manifold M = M n and on a sphere S ∼ = S n of the same dimension. Let N ⊂ M be a closed invariant subsetand let ( H ) , ( H ) , . . . , ( H k ) be the orbit types in M \ N . Assume that the set S H i is bothglobally and locally k -connected for all k = 0 , , . . . , dim( M H i ) − , where i = 1 , . . . , k . Thenfor every pair of G -equivariant maps Φ , Ψ : M −→ S , which are equivariantly homotopicon N , there is the following relation deg(Ψ) ≡ deg(Φ) (mod GCD {| G/H | , . . . , | G/H k |} ) . (3.1) References [AMSW] S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, Eliminatinghigher-multiplicity intersections. III. Codimension 2, 2017 (v1 – 2015), arXiv:1511.03501.[BBZ] I. Bárány, P.V.M. Blagojević and G.M. Ziegler. Tverberg’s Theorem at 50: Ex-tensions and Counterexamples,
Notices Amer. Math. Soc. , vol. 63, no. 7, 2016.[BL92] I. Bárány and D.G. Larman. A colored version of Tverberg’s theorem.
J. LondonMath. Soc. , 45:314–320, 1992.[BS] I. Bárány, P. Soberon. Tverberg’s theorem is 50 years old: a survey,
Bull. Amer.Math. Soc.
55 no. 4, pp. 459–492, 2018.[BFZ1] P.V.M. Blagojević, F. Frick, G.M. Ziegler. Tverberg plus constraints,
Bull. Lond.Math. Soc. , 46:953–967, 2014.[BFZ2] P.V.M. Blagojević, F. Frick, and G. M. Ziegler, Barycenters of polytope skeletaand counterexamples to the topological Tverberg conjecture, via constraints,
J.Eur. Math. Soc. (JEMS) , 21 (7), 2107–2116 (2019).[BMZ] P.V.M. Blagojević, B. Matschke, G. M. Ziegler, Optimal bounds for the col-ored Tverberg problem,
J. Eur. Math. Soc. (JEMS) , 2015, 17:4, 739–754, arXiv:0910.4987.[BZ] P.V.M. Blagojević, G. M. Ziegler, Beyond the Borsuk–Ulam theorem: the topo-logical Tverberg story, in “A journey through discrete mathematics” , M. Loebl, J.Nešetřil, R. Thomas (Eds.), Springer 2017, 273–341.16Cha] M.K. Chari. On discrete Morse functions and combinatorial decompositions.
Discrete Math. , 217(1-3):101–113, 2000.[Fo02] R. Forman, A user’s guide to discrete Morse theory,
Sém. Lothar. Combin.
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 , to appear in
Israel J. Math. arXiv:1812.00366 [math.CO].[JPZ-2] D. Jojić, G. Panina, R. Živaljević,
Splitting necklaces, with constraints ,arXiv:1907.09740 [math.CO].[JPVZ] D. Jojić, S.T. Vrećica, G. Panina, R. Živaljević,
Generalized chessboard complexesand discrete Morse theory , submitted.[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 preprint).[KB] A.M. Kushkuley, Z.I. Balanov.
Geometric Methods in Degree Theory for Equiv-ariant Maps , Lecture Notes in Math. 1632, Springer 1996.[MW14] I. Mabillard, U. Wagner. Eliminating Tverberg points, I. An analogue of theWhitney trick. In
Proc. 30th Ann. Symp. on Computational Geometry (SoCG),Kyoto 2014 (ACM, 2014), pp. 171–180.[MW15] I. Mabillard, U. Wagner. Eliminating Higher-Multiplicity Intersections, I. A Whit-ney Trick for Tverberg-Type Problems, arXiv:1508.02349, August 2015.[MW16] I. Mabillard, U. Wagner. Eliminating Higher-Multiplicity Intersections, II. TheDeleted Product Criterion in the r -Metastable Range, (SoCG’16), LIPIcs. Leibniz Int. Proc. Inform.,2016. arXiv:1601.00876 [math.GT], January 2016.17M03] J. Matoušek. Using the Borsuk-Ulam Theorem. Lectures on Topological Methodsin Combinatorics and Geometry . Universitext, Springer-Verlag, Heidelberg, 2003(Corrected 2nd printing 2008).[Sar] K.S. Sarkaria, A generalized van Kampen-Flores theorem,
Proc. Amer. Math.Soc.
11 (1991), 559–565.[Sk18] A. B. Skopenkov. A user’s guide to the topological Tverberg conjecture,
Russ.Math. Surv. , 2018, 73 (2), 323–353.[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.[VŽ11] S. Vrećica, R. Živaljević. Chessboard complexes indomitable.
J. Combin. TheorySer. A , 118(7):2157–2166, 2011.[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
Hand-book of 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.