Motivated by topological Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex can be mapped into without triple, quadruple, or, more generally, -fold points (image points with at least distinct preimages), for a given multiplicity . In particular, we are interested in maps that have no -Tverberg points, i.e., no -fold points with preimages in pairwise disjoint simplices of , and we seek necessary and sufficient conditions for the existence of such maps.
We present higher-multiplicity analogues of several classical results for embeddings, in particular of the completeness of the Van Kampen obstruction for embeddability of -dimensional complexes into , . Specifically, we show that under suitable restrictions on the dimensions (viz., if and for some ), a well-known deleted product criterion (DPC) is not only necessary but also sufficient for the existence of maps without -Tverberg points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick, by which pairs of isolated -fold points of opposite sign can be eliminated by local modifications of the map, assuming codimension .
An important guiding idea for our work was that sufficiency of the DPC, together with an old result of Özaydin on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the long-standing topological Tverberg conjecture, i.e., to construct maps from the -simplex to without -Tverberg points when not a prime power and . Unfortunately, our proof of the sufficiency of the DPC requires codimension , which is not satisfied for .
In a recent breakthrough, Frick found an extremely elegant way to overcome this “codimension obstacle” and to construct the first counterexamples to the topological Tverberg conjecture for all parameters with and not a prime power, by a clever reduction (using the constraints method of Blagojević–Frick–Ziegler) to a suitable lower-dimensional skeleton, for which the codimension restriction is satisfied and maps without -Tverberg points exist by Özaydin’s result and sufficiency of the DPC.
Here, we present a different construction (which does not use the constraint method) that yields counterexamples for , not a prime power.
Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems111Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948). An extended abstract of this paper appeared in Proc. 30th Annual Symposium on Computational Geometry (SoCG 2014) .
and Uli Wagner email@example.com
IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
July 15, 2019
- 1 Introduction
- 2 Preliminaries
- 3 The Higher-Multiplicity Whitney Trick
- 4 The Deleted Product Criterion for Tverberg Points
- 5 Counterexamples to the Topological Tverberg Conjecture in Dimension
Let be a finite simplicial complex111Throughout this paper, we will (ab)use the same notation for a simplicial complex (a collection of simplices) and its underlying polyhedron, relying on context to distinguish between the two when necessary. and let be a continuous map. Given an integer parameter , we say that is an -fold point or -intersection point of if , i.e., if there are pairwise distinct points such that .
Motivated by topological Tverberg-type problems (see below), an important topic in topological combinatorics, we are particularly interested in the following special type of -fold points. For a point in , we define its support as the smallest simplex222All simplices are considered closed, unless indicated otherwise. of that contains in its relative interior. We say that is a non-local -fold point or -Tverberg point of a map if it has preimages with pairwise disjoint supports, i.e., for pairwise disjoint simplices . Thus, when focussing on -Tverberg points, we ignore local -fold points that occur between images of simplices some of which share some vertices; we stress that being an -Tverberg point depends on the actual simplicial complex (the triangulation), not just on the underlying polyhedron.
The most basic case is that of (topological) embeddings, i.e., maps without double points.333Since is compact and is Hausdorff, a continuous map is an embedding iff it is injective. Finding conditions for a simplicial complex to be embeddable into — a higher-dimensional generalization of graph planarity — is a classical problem in geometric topology (see, e.g., [37, 45] for surveys) and has recently also become the subject of systematic study from a viewpoint of algorithms and computational complexity (see, e.g., [31, 30, 10]).
Generalizing classical results about embeddings, we are interested in necessary and sufficient conditions for the existence of maps without -Tverberg points, and in techniques that allow us to eliminate -fold points by local modifications of the map. In the present paper, we establish such results in the “critical case” (the smallest dimension for which a map in general position can have -fold points), assuming codimension ; see Theorems 6 and 17 below.
1.1 Topological Tverberg-Type Problems
The classical geometric Tverberg theorem , a cornerstone of convex geometry, can be rephrased as saying that if then any affine map from the -dimensional simplex to has an -Tverberg point. Bajmoczy and Bárány  and Tverberg [19, Problem 84] raised the question whether this remains true for arbitrary continuous maps:
Conjecture 1 (Topological Tverberg Conjecture).
Let , , and . Then every continuous map has an -Tverberg point.
This was proved by Bajmoczy and Bárány  for , by Bárány, Shlosman, and Szűcs  for all primes , and by Özaydin  for prime powers ,444Further proofs of the prime power case were given by Volovikov , Živaljević , and Sarkaria . but the case of arbitrary has been a long-standing open problem, considered to be one of the most challenging in the area [29, p. 154].
There are numerous close relatives and other variants of (topological) Tverberg-type problems and results, e.g., the Colored Tverberg Problem [3, 4, 60, 59, 8] and generalized Van Kampen–Flores-type results [39, 53].
Here, we consider the following general problem:
Given a finite simplicial complex and parameters and , decide whether there exists a map without -Tverberg points.
In particular, we are interested in methods for proving the existence of such maps, (i.e., for showing that does not satisfy a topological Tverberg-type theorem with parameters and ).
For Problem 2, it suffices to consider maps that are piecewise-linear555Recall that is PL if there is some subdivision of such that is affine for each simplex of . (PL), since every continuous map can be approximated arbitrarily closely by a PL map, and if has no -Tverberg points, then the same holds for any map sufficiently close to .
Moreover, if or, more generally, if the deleted product of (see below) satisfies , then a simple codimension count shows that a PL map in general position has no -Tverberg points, so the problem is trivial. In the present paper, we focus on the first nontrivial case , for which a PL map in general position has a finite number of -Tverberg points.
The Deleted Product Criterion.
There is a well-known necessary condition for the existence of maps without Tverberg points, formulated in terms of the (combinatorial) deleted -fold product666Some authors prefer to work with deleted joins (which are again simplicial complexes) instead of deleted products as configuration spaces for Tverberg-type problems. However, it is known that deleted products provide necessary conditions that are at least as strong as those provided by deleted joins; see, e.g., [32, Sec. 3.3]. For further background on the broader configuration space/test map framework, see, e.g., [29, Ch. 6] or [58, 59]. of a complex , which is defined as
The deleted product is a regular polyhedral cell complex (a subcomplex of the cartesian product ), whose cells are products of pairwise disjoint simplices of .
Lemma 4 (Necessity of the Deleted Product Criterion).
Let be a finite simplicial complex, and let and be integers. If there exists a map without -Tverberg point then there exists an equivariant map777Here and in what follows, if and are spaces on which a finite group acts (all group actions will be from the right) then we will use the notation for maps that are equivariant, i.e., that commute with the group actions, for all and ).
where , and the symmetric group acts on both spaces by permuting components.888We remark that the action of is free on for all , but not free on unless is a prime.
We briefly recall the standard proof, which uses several notions that we will need later.
Given , one gets a map by . The map has no -Tverberg point iff avoids the thin diagonal
Moreover, is the unit sphere in the orthogonal complement , and there is a straightforward homotopy equivalence999First orthogonally project onto , and then radially retract the latter to . Concretely, , given by , where , , and . Both and are equivariant hence so is their composition
The -fold Van Kampen Obstruction.
Lemma 4 is an important tool for proving topological Tverberg-type results. Moreover, in many interesting cases, the existence of an equivariant map
can be decided using equivariant obstruction theory (for which the standard reference is [13, Sec. II.3]). In particular, in the case there is a single -dimensional equivariant cohomology class defined on that yields a complete criterion (see Sec. 4.1):
Suppose . Then there exists an equivariant map if and only if .
If , , and then is the classical Van Kampen obstruction to embeddability of into ([51, 44, 56]; see also  for a recent in-depth treatment and further references). Correspondingly, we call the -fold Van Kampen obstruction.
However, there is a caveat: Vanishing of the -fold Van Kampen obstruction implies the existence of an equivariant map as in (3), but it does not imply that is of the form as in (2), i.e., induced by a map without -Tverberg points; thus, if then it is unclear whether the deleted product criterion is incomplete and one needs more refined arguments to show that such a map does not exist, or whether does exist and a Tverberg-type theorem for is simply not true. A particularly pertinent example of this kind is a result of Özaydin [34, Theorem 4.2] (see Theorem 10 below), which was a major inspiration for our work.
Sufficiency of the deleted product criterion.
This raises the question whether there exists a converse to Lemma 4, at least under some suitable additional hypotheses.
For the classical case , this is known to be the case, under suitable restrictions on the dimensions. A fundamental result of this type was first stated by Van Kampen  (albeit with a lacuna in the proof ), and complete proofs were later provided by Shapiro  and by Wu . It is convenient for us to separate the statement in two parts: a first one concerning maps without -Tverberg points (also called almost-embeddings), and a second one concerning embeddings.
Theorem 6 (Van Kampen–Shapiro–Wu).
Let be a simplicial complex, .
There exists an almost-embedding if and only if there exists an equivariant map .
If there an almost-embedding then there exists an embedding ; moreover, can be taken to be piecewise-linear.
Our main result is a generalization of 1 to -Tverberg points.101010Generalizing 2 to -fold points that may be local, i.e., whose preimages are not pairwise disjoint, turns out to be more subtle; we plan to treat this in a follow-up paper.
Theorem 7 (Sufficiency of the Deleted Product Criterion for Tverberg Points).
Suppose , , and . If is a finite -dimensional simplicial complex, then there exists a map without -Tverberg point iff there exists an equivariant map (equivalently, iff ).
The proof of Theorem 7 will be presented in Section 4 (see the beginning of that section for an overview). The proof is structured along the lines of the classical proof of 1 (see  for a very accessible account of the latter) and based on appropriate higher-multiplicity generalizations of the corresponding tools, in particular -fold Van Kampen finger moves (Section 4.2) and an -fold Whitney trick (Theorem 17).
The codimension restriction is crucial for many steps of the proof of Theorem 7. In the classical case of embeddings, it is known that Theorem 6 fails for (see ) but holds for (embeddings of graphs in the plane), even under slightly weaker assumptions; the latter fact is equivalent to the Hanani–Tutte Theorem [11, 48]. It would be interesting to know if either of these facts generalize to higher multiplicities; see Section 1.3 for a more detailed discussion of these and related open questions.
For embeddings, there is a far-reaching generalization of Theorem 6: The Haefliger–Weber Theorem [20, 55] (see also  for a modern survey and extensions) guarantees that in the so-called metastable range , an -dimensional complex embeds (piecewise-linearly) into if and only if there is an equivariant map . In a subsequent paper, we plan to present a generalization of this to -Tverberg points, which works in a corresponding -metastable range .
Vanishing of the generalized Van Kampen obstruction amounts to the solvability of a certain system of inhomogeneous linear equations over the integers (see Section 4.2). As a consequence, we have the following:
There is an algorithm which, under the assumptions of Theorem 7 , decides whether a given input complex admits a map into without -Tverberg points. If the parameters and are fixed, the algorithm runs in polynomial time in the size (number of simplices) of .
Özaydin’s and Frick’s work: counterexamples to the topological Tverberg conjecture.
As mentioned above (see also the discussion in ), an important motivation for our work was the following result by Özaydin [34, Theorem 4.2]. For every , let denote an -dimensional, -connected free -cell complex. Such complexes exist for all : e.g., one can take the -fold join , where is considered as a -dimensional complex and acts on itself by right multiplication. They have the universal property that every free -cell complex of dimension maps equivariantly into (see [29, Sec. 6.2]).
Theorem 10 (Özaydin).
Let and . There exists an equivariant map
if and only if is not a prime power.
Hence, by the universal property of , there exists an equivariant map
whenever is not a prime power and is a simplicial complex such that ; in particular, this applies if or if .111111On the other hand, Bárány et al. [5, Lemma 1] showed that is -connected for , hence is of the type . Thus, for prime powers , there is no equivariant map , by Theorem 10 (and hence that the topological Tverberg conjecture holds in this case).
Inspired by this and by the analogy with the classical theorems on embeddability, one of the guiding ideas for our work was that combining Özaydin’s result and sufficiency of the deleted product criterion for -Tverberg points might yield an approach to constructing counterexamples to the topological Tverberg conjecture if is not a prime power.
Unfortunately, our proof of Theorem 7 requires codimension , which is not satisfied for (one can replace by its -skeleton without loss of generality, but the problem persists).
In a recent breakthrough, following the announcement of our work in the extended abstract , Frick  found a very elegant way to overcome this codimension obstacle and to construct the first counterexamples to the topological Tverberg conjecture. Specifically, Frick proves that for every that is not a prime power, there exists a map without -Tverberg points, where ; in particular, there exists a map without -Tverberg point. It is known that this implies that there are counterexamples for all , see [12, Proposition 2.5].
Frick’s argument exemplifies the constraint method of Blagojević–Frick–Ziegler  and builds the counterexample from a map without -Tverberg points, where the existence of follows from Özaydin’s result (Theorem 10) and ours (Theorem 7).
Here, we present a different construction (which does not use the constraint method) that yields counterexamples in dimension ; this seems to be the natural limit for counterexamples constructed using the -fold Whitney trick, due to the codimension 3 requirement for the latter.
Suppose is not a prime power and let . Then there exists a map without -Tverberg points.
The smallest counterexample obtained in this way is a map without any -Tverberg point.
The proof of Theorem 11 will be given in Section 5. It is based on three ingredients: Özaydin’s result (Theorem 10), our higher-multiplicity Whitney trick (Theorem 17 below), and a particular kind of PL map that we will call prismatic (see Definition 48).
In principle, the proofs of Theorems 7 and 11 are constructive and do not require explicit knowledge of Özaydin’s equivariant map (4); the existence of this map enters only in terms of the equivalent condition that the relevant obstruction vanishes. In each case, we start with an arbitrary map (respectively, with a prismatic map) that may have -Tverberg points and then construct the desired map through a finite sequence of -fold Finger moves, followed by a finite number of applications of the -fold Whitney trick. It is an interesting question how complicated the final PL map in Theorem 11 needs to be; see the discussion in Section 1.3 (3).
The key property of prismatic maps is that we will be able to ensure that all their Tverberg points are of the same type , in the following sense:
Definition 13 (Tverberg Partitions and Type).
Let , , , and let be a PL map in general position. Suppose is an -Tverberg point of and , . The vertex sets of the simplices form a partition of the vertex set of , hence and (by general position) for . Somewhat abusing terminology, we say that form a Tverberg partition for , and we call the multiset of dimensions the type of this Tverberg partition and of the Tverberg point .
As a byproduct of the proof of Theorem 11, we obtain the following result (where denotes the multiset containing the element with multiplicity ):
Suppose , , and . Then there exists an affine map such that all -Tverberg points of are of the same type , where .
It is also well-known that for every and , there are affine maps121212 Specifically, such an affine map is given by the point configuration in (the images of the vertices) consisting of small clusters of points centered at the vertices of a -simplex, plus one point at the barycenter of the simplex. all of whose Tverberg points are of type .This raises the question whether we can generally construct (affine) maps all of whose Tverberg points are of a specified type:
Let and . Suppose we are given integers such that . Does there exist an affine map such that all -Tverberg points of are of the same type ?
1.2 A Higher-Multiplicity Whitney Trick
Our main tool to deal with intersections of higher multiplicity is a Whitney trick for -fold points (Theorem 17 below).
The classical Whitney trick (more precisely, its piecewise-linear version, see, e.g., [54, p. 179] or [38, Lemma 5.12]) allows one to eliminate a pair of isolated double points of opposite sign (see Section 2.2 for the definition of intersection signs) of a PL map by an ambient PL isotopy fixed outside a small ball, provided the codimension is at least .
Here and in what follows, an ambient PL isotopy of is a PL homeomorphism that preserves the -component and thus gives rise to a family of PL homeomorphisms , (see Section 2.1 for more background on isotopies).
Theorem 16 (Whitney Trick).
Suppose that and are connected, orientable PL manifolds, possibly with boundary, of respective dimensions of respective dimensions and , , and that
is a PL map in general position defined on their disjoint union.
If are two double points of opposite sign131313We remark that the sign of a double point depends on the choice of orientations of the and of , but if the are connected then the condition of having opposite signs is independent of such a choice. and if , , then there exists a PL ambient isotopy of such that
Moreover, the isotopy can be chosen to be local, in the following sense: Given any closed polyhedron of dimension and with , there exists a PL ball disjoint from such that is fixed outside of .
Figure 1 illustrates this in a low-dimensional situation. The idea of the trick is to “push” upwards until the two intersections points and disappear, while keeping the boundary of fixed. In low codimensions, doing this might require passing over some obstacles and/or introducing new double points, but if , then these problems can be avoided.141414The hypotheses for of the Whitney trick can be weakened, e.g., one of the can be allowed to have dimension , but then one needs to impose additional technical conditions like local flatness and simple connectivity of the complement ; see, e.g., [38, Lemma 5.12].
In the present paper, we prove the following analogue of Theorem 16 for -fold points:
Theorem 17 (Higher-Multiplicity Whitney Trick).
Let , and let be connected, orientable PL manifolds151515We are mostly interested in the case that each is a simplex, but the proof of the more general case comes at no extra cost., of respective dimensions , such that
be a PL map in general position defined on their disjoint union, and suppose that
are two -fold points of opposite intersection signs (see Section 2.2).
Then there exist PL ambient isotopies of such that
Moreover, these isotopies can be chosen to be local, in the following sense: Given any closed polyhedron of dimension and with , there exists a PL ball disjoint from such that is fixed outside of , .
As another application of these ideas, we also have the following generalization of the classical result of Whitney that -dimensional manifolds embed into :
Let , , and let a PL manifold of dimension . Then there exists a PL map without -fold points.
1.3 Future Work and Open Problems
Codimension . Theorem 6 fails for : Freedman, Krushkal and Teichner  constructed examples of finite -dimensional complexes whose Van Kampen obstruction vanishes but which are not embeddable into . (More generally, for every pair with , there are counterexamples [28, 43, 42, 18] that show that the deleted product criterion is insufficient for embeddabbility of -complexes into .) We suspect that similar counterexamples to Theorem 7 exist for .
The Planar Case and Hanani–Tutte. On the other hand, Theorem 6 remains true for (embeddings of graphs in the plane), even under the slightly weaker assumption that the Van Kampen obstruction vanishes modulo . This is essentially the Hanani–Tutte Theorem [11, 48], which guarantees that a graph is planar iff it can be drawn in the plane such that any pair of vertex-disjoint edges cross an even number of times. The classical proofs of that theorem rely on Kuratowski’s Theorem, but more recently [35, 36], more direct proofs have been found that do not use forbidden minors (an earlier attempt at a Whitney-trick for graphs in the plane  contained an error; see [46, p. 17]). It would be very interesting to know whether there is an analogue of the Hanani–Tutte theorem for Tverberg-type problems in . In particular, in light of Özaydin’s result, this would be an approach to completely settling the non-prime power case of the topological Tverberg conjecture by constructing counterexamples for . We plan to investigate this in a future paper.
Complexity of Maps without Tverberg Points. It is an interesting question how complicated the counterexamples to the topological Tverberg conjecture need to be. For and , Freedman and Krushkal have constructed examples of -dimensional complexes with simplices such that admits a PL embedding into (equivalently, ), but any subdivision of that supports such a PL embedding requires at least simplices, where is a constant depending on . Complementing this, they also showed that there is always a suitable subdivision with at most simplices, for any . It would be interesting to know whether there are similar bounds for maps without -Tverberg points, , , .
We would like to thank the anonymous referees of the extended abstract  for helpful comments and remarks.
Moreover, we would like to thank Florian Frick, Gil Kalai, Arkadiy Skopenkov, and Günter Ziegler for detailed comments on a preliminary draft of this paper, and Pavle Blagojević for asking us to clarify the role of the full symmetric group (Remark 45).
Furthermore, U.W. would like to express his gratitude to Jiří Matoušek, Eran Nevo, and Martin Tancer for years of fruitful collaboration on algorithmic and combinatorial aspects of the embeddability problem and many discussions on the classical Van Kampen obstruction; without this background, the work presented here would not have been undertaken.
2.1 Tools from Piecewise-Linear Topology
In this subsection (which readers may want to skip or just skim through at first reading), we collect, for ease of reference, a number of basic notions and results from piecewise-linear (PL) topology that we will use repeatedly throughout the paper
For a very readable and compact introduction to the area, see the survey article . For more details see, e.g., the textbook  or the lecture notes . We refer the reader to any of these sources for much of the basic terminology, such as PL manifolds and regular neighborhoods. A polyhedron will always mean the underlying polyhedron of some geometric simplicial complex in some .
2.1.1 Isotopies, Ambient Isotopies, and Unknotting
One of the facts that make working in codimension at least easier is that isotopic embeddings are also ambient isotopic, see below. This fails in codimension ; for instance, any two PL knots (embeddings of ) in are isotopic, but not necessarily ambient isotopic.
Let be a polyhedron, and let be a PL manifold. A (PL) isotopy of in is a PL embedding that is level-preserving, i.e., such that for all . An isotopy determines embeddings by for and .
An isotopy is fixed on a subspace if for all and . An isotopy is allowable if for some closed subpolyhedron .
Two embeddings are (allowably) isotopic (keeping fixed) if there is an (allowable) isotopy (fixed on ) of in such that and .
An ambient PL isotopy of of is a level-preserving PL homeomorphism such that is the identity on . Two PL embeddings are ambient isotopic (keeping fixed) if there is an ambient isotopy of , fixed on , with . An ambient isotopy of extends an isotopy of in if for all .
Let and be PL manifolds, possibly with boundary. A PL embedding is proper if . An isotopy is proper if it is proper as an embedding.
From isotopy to ambient isotopy.
Theorem 19 (Hudson [23, Thm 1]).
Let and be PL manifolds, compact, and let be a proper isotopy of in , fixed on . If , then there is an ambient isotopy of , fixed on , that extends .
We will also need the following result concerning embeddings of compact polyhedra:161616In , the result is stated in a stronger form: The conclusion remains true under the weaker hypothesis that and are allowably concordant keeping fixed. (The notion of an allowable concordance between and fixing is a generalization of an allowable isotopy fixing , where the requirement that preserve levels is relaxed to the conditions for and for , see [25, Section 1].)
Proposition 20 (Hudson [25, Corollary 1.3]).
Let be a compact polyhedron and let be a PL manifold. Let be allowably isotopic embeddings keeping fixed, with . If , then and are ambient isotopic keeping fixed.
Unknotting of balls and spheres.
A (PL) -manifold pair is a pair of PL manifolds and of dimensions and , respectively such that properly.
A pair of PL balls (respectively, a pair of PL spheres), , is unknotted if it is PL homeomorphic to the standard ball pair (respectively, to the standard sphere pair .)
Theorem 21 (Zeeman [57, Ch. IV, Theorem 9]).
If then every PL ball pair and every PL sphere pair are unknotted.
We will also need the following relative version:
Corollary 22 (Zeeman [57, Ch. IV, Corollary 1, p. 16]).
If , then any two proper embeddings that agree on are ambient isotopic, keeping fixed.
From homotopy to ambient isotopy.
Theorem 23 (Zeeman [57, Ch X, p 198, Thm 10.1]).
Let and be compact manifolds of dimensions and , respectively, and let be two proper embeddings. Suppose that is homotopic to relative to . Then if , is -connected, and is -connected, then and are ambient isotopic keeping fixed.
Theorem 24 (Irwin [57, Ch. VIII, p. 4, Thm. 23]).
Assume is compact and let be a continuous map such that is a piecewise-linear embedding of in . Then is homotopic to a proper embedding keeping fixed provided
2.1.2 General Position and Transversality
There are many variants of general position. For the purposes of studying -fold points and -Tverberg points, the following definitions are convenient.
General position in .
A collection of affine subspaces of is in general position if for every and pairwise distinct ,
A set of points in is in general position if, for every and pairwise disjoint subsets , the affine hulls , , are in general position.171717Note that this is stronger than requiring that every subset of at most points in is affinely independent; e.g. the vertices of a regular hexagon are not in general position in the stronger sense.
A collection of convex polyhedra in is in general position if are in general position for every choice of nonempty faces , .
If is a simplicial complex and is a simplexwise-linear map, then we say that is in general position if the images of the vertices of are pairwise distinct and in general position. A PL map is in general position if there is some subdivision of such that is simplexwise-linear and in general position as a map .
If is a finite simplicial complex and is a continuous map then, by a simple compactness and perturbation argument, for every , there exists a PL map in general position such that .
General position in PL manifolds.
Defining general position without reference to a particular triangulation and, more generally, for maps into PL manifolds other than , is more involved. We follow the presentation [57, Ch. VI], which is very suitable for dealing with -fold points.
Let be a PL map from a polyhedron to a PL manifold. For , let us say that a point is -singular if it is the preimage of an -fold image point of , i.e., if . The (closed) -singular set is defined as the closure of the set of -singular points of . Each is a subpolyhedron of ([57, Ch. VI, Lemma 31, p. 19]). The set is also sometimes simply called the singular set of and denoted .
Suppose and . Then a PL map is said to be in general position if for every . If is a subpolyhedron then is said to be in general position for the pair if and are both in general position and, if then for every .
Theorem 25 ([57, Ch. VI, Theorem 18, p. 27]).
Let be a PL map, , and let be a subpolyhedron. If is in general position then for every there exists a map that is in general position for the pair , and are homotopic through an -small homotopy that keeps fixed.
We will also need the following version of being in general position with respect to a given polyhedron:
Theorem 26 ([57, Ch. VI, Theorem 15, p. 7]).
Let be a PL manifold of dimension , and let and be polyhedra. Given an embedding such that , for every there is an embedding such that is in general position with respect to , in the sense that
and and are ambient isotopic through an -small ambient isotopy fixing and .
Suppose that are properly embedded PL submanifolds of a PL manifold , , , and . We say that the are mutually transverse (or that they intersect transversely) if they locally intersect like affine subspaces in general position.
More precisely, the intersect transversely at a point [respectively, ] if there is a neighborhood of in and a PL homeomorphism [respectively, ] such that the images , , are affine subspaces in general position [respectively, intersections of such subspaces with the upper halfspace ]. The are mutually transverse if they intersect transversely at every . (In particular, if , then this implies that . )
In general, transversality for PL manifolds is much more subtle than the corresponding theory in the smooth case, see e.g., the discussion in .181818A particularly striking fact is the failure of relative PL transversality: Hudson  showed that for every with , , there are transverse PL spheres which can not be extended to transverse embeddings of balls .
In the present paper, we will only use the following simple fact: If are pairwise disjoint PL manifolds, , , and if is a PL map in general position, then the images are mutually transverse at every -fold point (necessarily an -Tverberg point) of ; indeed, for suitable subdivisions of the on which is simplexwise linear, there are simplices of the subdivisions, , such that the images are linear -simplices in general position whose relative interiors intersect exactly at . All operations that we will perform will preserve transversality of the intersections.
2.2 Oriented Intersections and Intersection Signs
In this subsection, we review the induced orientation on the intersection of oriented simplices in general position in and the resulting intersection product on piecewise-linear chains (this is a particular case of Lefschetz intersection theory ). We first fix the notation and state the basic properties that we will need later (Lemmas 27 and 28). The definition and the proofs of the two lemmas, which boil down to elementary linear algebra, are included here for the sake of completeness but are deferred until the end of this subsection, and the reader may wish to skip them at first reading.
Let be oriented simplices or, more generally, convex polyhedra in general position in , , (see Figure 2 for an illustration in the case , ).
Then the intersection is either empty or a convex polyhedron of dimension . In the latter case, given orientations of the ambient space and of each , we can define (see Definition 29 below) an induced orientation on
which depends on the order of the and on the choices of the orientations. We will also speak of the oriented intersection of the in , and occasionally write to stress dependence of the orientation on that of the ambient space. If the dimensions satisfy
then the intersection is either empty, or it consists of a single point that lies in the relative interior of each , and the induced orientation amounts to associating an (-fold) intersection sign in to , denoted by
or by , if we want to stress the ambient space.
The following lemma summarizes several properties that we will need in this paper.
Suppose we have chosen an orientation of , and let be oriented simplices in general position in , , .
Orientation reversal: Reversing the orientation of one (denoted by ) also reverses the orientation of the intersection,
If we reverse the orientation of (denoted by ) then the orientation of the intersection changes by a factor of ,
Skew commutativity: For pairwise oriented intersections,
Thus, in general, if then
where is the set of inversions of .
Restriction: Consider the oriented pairwise intersections as oriented convex subpolytopes of (the affine hull of) . If we compute the -fold oriented intersection of these within