Optimal bounds for a colorful Tverberg–Vrećica type problem
We prove the following optimal colorful Tverberg–Vrećica type transversal theorem: For prime and for any colored collections of points in , , , , , there are partition of the collections into colorful sets such that there is a -plane that meets all the convex hulls , under the assumption that is even or .
Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case (our optimal colored Tverberg theorem (2009)), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk–Ulam type theorem for -equivariant bundles that generalizes results of Volovikov (1996) and Živaljević (1999).
In their 1993 paper [TV93] H. Tverberg and S. Vrećica presented a conjectured common generalization of some Tverberg type theorems, some ham sandwich type theorems and many intermediate results. See [Ziv99] for a further collection of implications.
Conjecture 1.1 (Tverberg–Vrećica Conjecture).
Let and let be finite point sets in of cardinality . Then one can partition each into sets such that there is a -plane in that intersects all the convex hulls , , .
The Tverberg–Vrećica Conjecture has been verified for the following special cases:
(Tverberg’s theorem [Tve66]),
(Tverberg & Vrećica [TV93]),
for a weakened version was shown in [TV93] (one requires two more points for each ),
and are odd, and is an odd prime (Živaljević [Ziv99]),
(Vrećica [Vre08]), and
, , for some prime , and is even or (Karasev [Kar07]).
In this paper we consider the following colorful generalization of the Tverberg–Vrećica conjecture.
Let , and let be subsets of of cardinality . Let the be colored,
such that no color class is too large, . Then we can partition each into sets that are colorful in the sense that for all and find a -plane that intersects all the convex hulls .
The Tverberg–Vrećica Conjecture 1.1 is the special case of the previous conjecture when all color classes are given by singletons. The main result of this paper is the following special case.
Theorem 1.3 (Main Theorem).
Let be prime and such that is even or . Let be subsets of of cardinality . Let the be colored,
such that no color class is too large, . Then we can partition each into colorful sets and find a -plane that intersects all the convex hulls .
In Section LABEL:secNewColTVThm we will see that this theorem is quite tight in the sense that it becomes false if one single color class has elements and all the other ones are singletons.
Since we will prove Theorem 1.3 topologically it has a natural topological extension, Theorem LABEL:thmTopColoredTV.
Recently we had obtained the first case using equivariant obstruction theory [BMZ09]. In Section 2 we present two alternative proofs, based on the configuration space/test map scheme from [BMZ09]. The first one is more elementary and shorter; it uses a degree argument. The second proof puts the first one into the language of cohomological index theory. For this, we calculate the cohomological index of a join of chessboard complexes. This allows for a more direct proof of the case , which is the first of two keys for the Main Theorem 1.3.
The second key is a new Borsuk–Ulam type theorem for equivariant bundles. We establish it in Section 3, and prove the Main Theorem in Section LABEL:secNewColTVThm. The new Borsuk–Ulam type theorem can also be applied to obtain an alternative proof of Karasev’s above-mentioned result; see Section LABEL:secNewColTVThm. Karasev has also obtained a colored version of the Tverberg–Vrećica conjecture, different from ours, even for prime powers, which can also alternatively be obtained from our new Borsuk–Ulam type theorem.
2 The topological colored Tverberg problem revisited
In [BMZ09] we have shown the following new colored version of the topological Tverberg theorem. It is the special case of the Topological Main Theorem LABEL:thmTopColoredTV.
Theorem 2.1 ([Bmz09]).
Let be prime, , and . Let be an -dimensional simplex with a partition of the vertex set into “color classes” such that for all .
Then for every continuous map there are disjoint faces of that are colorful that is, such that
This implies the optimal colored Tverberg theorem (the Bárány–Larman conjecture) for primes minus one, even its topological extension. This conjecture being proven implies new complexity bounds in computational geometry; see the introduction of [BMZ09] for three examples.
In this section we present two new proofs of Theorem 2.1. The first one uses an elementary degree argument. The second proof puts the first one into the language of cohomological index theory, as developped by Fadell and Husseini [FH88]. Even though the second proof looks more difficult it actually allows for a more direct path, since it avoids the non-topological reduction of Lemma 2.2. This requires more index calculations, which however are valuable since they provide a first key step towards our proof of the Main Theorem 1.3 in Section LABEL:secNewColTVThm.
The configuration space/test map scheme
Suppose we are given a continuous map
and a coloring of the vertex set such that . We want to find a colored Tverberg partition .
As in [BMZ09] we construct a test-map out of . Let be the -fold join of . Since we are interested in pairwise disjoint faces , we restrict the domain of to the -fold -wise deleted join of , . (See [Mat03] for an introduction to these notions.) Since we are interested in colorful s, we restrict the domain further to the subcomplex . The space is known as the chessboard complex . We write
Hence we get a map
Let be the regular representation of and the orthogonal complement of the all-one vector . The orthogonal projection yields a map
The composition of this map with gives us the test-map ,
The pre-images of zero correspond exactly to the colored Tverberg partitions. Hence the image of contains if and only if the map admits a colored Tverberg partition. Suppose that is not in the image, then we get a map
into the representation sphere by composing with the radial projection map. We will derive contradictions to the existence of such an equivariant map.
The first proof establishes a key special case of Theorem 2.1, which implies the general result by the following reduction.
Lemma 2.2 ([Bmz09]).
It suffices to prove Theorem 2.1 for , and .
For the elementary proof of this lemma see [BMZ09, Reduction of Thm. 2.1 to Thm. 2.2]. This lemma is the special case of Lemma LABEL:lemElementaryReductionLemma, which we prove later.
Therefore we can assume that and . Let be the restriction of to . The chessboard complex is a connected orientable pseudo-manifold, hence is one as well. The dimensions coincide. Thus we can talk about the degree . Here we are not interested in the actual sign, hence we do not need to fix orientations. Since is a free -space and is -connected, the degree is uniquely determined modulo : This is because is unique up to -homotopy on the codimension one skeleton of , and changing on top-dimensional cells of has to be done -equivariantly, hence it affects by a multiple of .
To determine , we let be the affine map that takes the vertices in to and the vertices in to , where is the th standard basis vector of . The singleton does not matter, we can choose it arbitrarily in . Let be the normalization of the point . The pre-image is exactly the set of barycenters of the top-dimensional faces of . With we mean the full subcomplex of . One checks that all pre-images of have the same pre-image orientation. This was essentially done in [BMZ09] when we calculated that . Hence
Alternatively one can take any map , show that its degree is by a similar pre-image argument in dimension , and deduce that
First proof of Theorem 2.1.
Since , is not null-homotopic. Thus does not extend to a map with domain . Therefore the test-map of (2) does not exist. ∎
The degree is even uniquely determined modulo . To see this one uses the -equivariance of and the fact that is given uniquely up to -homotopy on the non-free part, which lies in the codimension one skeleton of . The latter can be shown with the modified test-map from [BMZ09]. This might be an ansatz for the affine version of Theorem 2.1 for non-primes .
Let in the following denote Čech cohomology with -coefficients, where is prime. The equivariant cohomology of a -space is defined as
where is a contractible free -CW complex and . The classifying space of is . If is furthermore a projection to a trivial -space , we denote the cohomological index of over , also called the Fadell–Husseini index [FH88], by
If is a point then one also writes and .
The cohomological index has the four properties
Monotonicity: If there is a bundle map then
Additivity: If is excisive, then
Subbundles: From the continuity of Čech cohomology it follows that if there is a is a bundle map and a closed subbundle then
The first two properties imply the other two. For more information about this index theory see [FH87] and [FH88].
If is odd then the cohomology of as a -algebra is
where and . If is even, then and , .
Let be an -dimensional connected free -CW complex and let be an -dimensional -connected free -CW complex. If there is a -map that induces an isomorphism on , then
The two fiber bundles and induce two Leray–Serre spectral sequences and , and induces a morphism between them; see Figure 1.
The two th rows and at the -pages are identified by . At the -pages both spectral sequences have to satisfy whenever . This is because is free, hence , which is zero in degrees . The analog holds for . Therefore, in the elements in the bottom row must be hit by some differential. These differentials can come only from the th row at the -page (this argument even gives us the -module structure of the th row). Hence there is a non-zero transgressive element , that is, . Let . Then . Therefore survives at least until . Analogously, the whole th row survives until . We know that all elements in have to die eventually, so they do it exactly on page . Thus these are exactly the elements that make the part of the bottom row to zero.
Hence no non-zero differential can arrive at the bottom row on an earlier page of . This is because any element with would imply for any . Thus the whole bottom row of survives until . At the -page we know exactly the differentials, since we know them for the other spectral sequence .
Therefore at , the bottom row is . Everything else in has become zero. Since the index defining map is the bottom edge homomorphism, we get
We apply this theorem to the above maps and .
The -index of is
and the -index of is
Using this corollary we can compute the index of more general joins of chessboard complexes.
Let and let . Let . Then
Let and . Then and . We calculate . There is an inclusion . This implies
Since is free, , hence
It is interesting to note that the last argument of the proof would fail for odd if were odd, due to the relation in .
Second proof of Theorem 2.1.
By the monotonicity of the index, a test-map according to (2) would imply that
This is a contradiction since and , as is an -dimensional free -sphere. ∎
On the surface this proof seems to be a more difficult reformulation of the first proof. However, its view point is essential for the transversal generalization, since we do not rely on the geometric tools of the Reduction Lemma 2.2 anymore, and such a reduction lemma does not seem to exist for the Tverberg–Vrećica type transversal theorem. Thus we use the more general configuration space (1) instead.
3 A new Borsuk–Ulam type theorem
In this section we prove the second topological main step towards the proof of Theorem 1.3. This is the following Borsuk–Ulam type theorem. It will be applied in combination with the subsequent intersection lemma LABEL:lemIntersectionLemma.
Theorem 3.1 (Borsuk–Ulam type).
be an elementary abelian group in particular, is a prime,
a -CW-complex with index ,
a connected, trivial -space,
a -vector bundle all fibers carry the same -representation,
the fixed-point subbundle of ,
its -invariant orthogonal complement subbundle ,
be the fiber of the sphere bundle .
acts trivially on that is, is orientable if , and
we are given a -bundle map ,