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Bounding Helly Numbers via Betti Numbers

Abstract

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If \(\mathcal{F}\) is a finite family of subsets of \(\mathbb{R}^{d}\) such that \(\tilde{\beta }_{i}\left (\bigcap \mathcal{G}\right ) \leq b\) for any \(\mathcal{G} \subsetneq \mathcal{F}\) and every 0 ≤ i ≤ ⌈d∕2⌉− 1 then \(\mathcal{F}\) has Helly number at most h(b, d). Here \(\tilde{\beta }_{i}\) denotes the reduced \(\mathbb{Z}_{2}\)-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these ⌈d∕2⌉ first Betti numbers allow for families with unbounded Helly number.

Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map \(C_{{\ast}}(K) \rightarrow C_{{\ast}}(\mathbb{R}^{d})\).

Dedicated to the memory of Jiří Matoušek, wonderful teacher, mentor, collaborator, and friend.

An extended abstract of this work was presented at the 31st International Symposium on Computational Geometry [21].

PP, ZP and MT were partially supported by the Charles University Grant GAUK 421511. ZP was partially supported by the Charles University Grant SVV-2014-260103. ZP and MT were partially supported by the ERC Advanced Grant No. 267165 and by the project CE-ITI (GACR P202/12/G061) of the Czech Science Foundation. UW was partially supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and SNSF-PP00P2-138948). Part of this work was done when XG was affiliated with INRIA Nancy Grand-Est and when MT was affiliated with Institutionen för matematik, Kungliga Tekniska Högskolan, then IST Austria.

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Notes

  1. 1.

    By definition, a homology cell is a topological space X all of whose (reduced, singular, integer coefficient) homology groups are trivial, as is the case if \(X = \mathbb{R}^{d}\) or X is a single point. Here and in what follows, we refer the reader to standard textbooks like [26, 42] for further topological background and various topological notions that we leave undefined.

  2. 2.

    The choice of \(\mathbb{Z}_{2}\) as the ring of coefficient ring has two reasons. On the one hand, we work with the van Kampen obstruction to prove certain non-embeddability results, and the obstruction is naturally defined either for integer coefficients or over \(\mathbb{Z}_{2}\) (it is a torsion element of order two). On the other hand, the Ramsey arguments used in our proof require working over a fixed finite ring of coefficients to ensure a finite number of color classes (cf. Claim 1).

  3. 3.

    We remark that this construction can be further improved (at the cost of simplicity). For example, for d = 3, it is possible to find a geometric simplicial complex \(\Gamma _{3}^{{\prime}}\) with six vertices (instead of five) with properties analogous to \(\Gamma _{3}\): Consider a simplex \(\Delta \subseteq \mathbb{R}^{3}\) with vertices v 1, v 2, v 3 and v 4. Let b the barycenter of this simplex and we set v 5 to be the barycenter of the triangle v 1 v 2 b and v 6 to be the barycenter of v 3 v 4 b. Finally, we set \(\Gamma _{3}^{{\prime}}\) to be the subdivision of \(\Delta\) with vertices v 1, , v 6 and with maximal simplices 1245, 1235, 3416, 3426, 5613, 5614, 5623, and 5624 where the label ABCD stands for conv { v A , v B , v C , v D }. One can check that this indeed yields a simplicial complex with the required properties. See the 1-skeleton of \(\Gamma _{3}^{{\prime}}\) in Fig. 1. We believe that an analogous example can be also constructed for d ≥ 4.

  4. 4.

    We also remark that our condition can be verified algorithmically since Betti numbers are easily computable, at least for sufficiently nice spaces that can be represented by finite simplicial complexes, say. By contrast, it is algorithmically undecidable whether a given 2-dimensional simplicial complex is 1-connected, see, e.g., the survey [50].

  5. 5.

    An open good cover is a finite family of open subsets of \(\mathbb{R}^{d}\) such that the intersection of any sub-family is either empty or is contractible (and hence, in particular, a homology cell).

  6. 6.

    The role of nerves is implicit in Amenta’s proof but becomes apparent when compared to an earlier work of Wegner [60] that uses similar ideas.

  7. 7.

    The result of Colin de Verdière et al. [11] holds in any paracompact topological space; Theorem 1 only subsumes the \(\mathbb{R}^{d}\) case.

  8. 8.

    This requires f and C 1, C 2, , C n to be generic in the sense that the number of minima of f over ∩ iI C i is bounded uniformly for I ⊆ {1, 2, , n}.

  9. 9.

    There is also a version of the Van Kampen obstruction with integer coefficients, which in general yields more precise information regarding embeddability than the \(\mathbb{Z}_{2}\)-version, but we will not need this here. We refer to [37] for further background.

  10. 10.

    This observation was already used in [58] to study the (non-)embeddability of certain simplicial complexes. What we call a homological almost-embedding in the present paper corresponds to the notion of a homological minor used in [58].

  11. 11.

    We recall that a chain map γ: C D between chain complexes is simply a sequence of homomorphisms γ n : C n D n that commute with the respective boundary operators, γ n−1 C = D γ n .

  12. 12.

    If we consider augmented chain complexes with chain groups also in dimension − 1, then being nontrivial is equivalent to requiring that the generator of \(C_{-1}(K)\cong \mathbb{Z}_{2}\) (this generator corresponds to the empty simplex in K) is mapped to the generator of \(C_{-1}(\mathbf{R})\cong \mathbb{Z}_{2}\).

  13. 13.

    The induced chain map is defined as follows: We assume that we have fixed a total ordering of the vertices of K. For a p-simplex σ of K, the ordering of the vertices induces a homeomorphism \(h_{\sigma }: \vert \Delta _{p}\vert \rightarrow \vert \sigma \vert \subseteq \vert K\vert\). The image f (σ) is defined as the singular p-simplex fh σ .

  14. 14.

    Here, we use subscripts and superscripts on the boundary operators to emphasize which dimension and which chain complex they belong to; often, these indices are dropped and one simply writes φψ = ∂η + η∂.

  15. 15.

    We also recall that if f, gXY are (equivariantly) homotopic then the induced chain maps are (equivariantly) chain homotopic. Moreover, chain homotopic maps induce identical maps in homology and cohomology.

  16. 16.

    We stress that we work with the cellular chain complex for X.

  17. 17.

    We remark that a classical result due to Haefliger and Weber [25, 59] asserts that if dimK ≤ (2d − 3)∕3 (the so-called metastable range) then the existence of an equivariant map from \(\widetilde{K}\) to \(\mathbb{S}^{d-1}\) is also sufficient for the existence of an embedding \(K\hookrightarrow \mathbb{R}^{d}\) (outside the metastable range, this fails); see [ 49] for further background.

  18. 18.

    We stress that this does not mean that there is only one homotopy class of continuous maps \(\overline{K} \rightarrow \mathbb{R}\mathbb{P}^{\infty }\); indeed, there exist such maps that do not come from equivariant maps \(\widetilde{K} \rightarrow \mathbb{S}^{\infty }\), for instance the constant map that maps all of \(\overline{K}\) to a single point.

  19. 19.

    In fact, it is known that \(H^{{\ast}}(\mathbb{R}\mathbb{P}^{\infty })\) is isomorphic to the polynomial ring \(\mathbb{Z}_{2}[\xi ]\), that \(H^{{\ast}}(\mathbb{R}\mathbb{P}^{d})\cong \mathbb{Z}_{2}[\xi ]/(\xi ^{d+1})\), and that \(\overline{i}^{{\ast}}\) is just the quotient map.

  20. 20.

    Recall that a set is k-connected if it is connected and has vanishing homotopy in dimension 1 to k.

  21. 21.

    See Definition 10.

  22. 22.

    We could require that γ sends every vertex to a point in \(U_{\overline{\Psi (S)}}\), i.e. is a chain map induced by a map, and simply argue that since \(U_{\overline{\Psi (S)}}\) has at most b connected components, any b + 1 vertices of \(\Delta _{s}\) contains some pair that can be connected inside \(U_{\overline{\Psi (S)}}\). This argument does not, however, work in higher dimension. Since Sect. 4.2 is meant as an illustration of the general case, we choose to follow the general argument.

  23. 23.

    β is the chain map induced by β restricted to chains of dimension at most (k − 1).

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Acknowledgements

We would like to express our immense gratitude to Jiří Matoušek, not only for raising the problem addressed in the present paper and valuable discussions about it, but, much more generally, for the privilege of having known him, as our teacher, mentor, collaborator, and friend. Through his tremendous depth and insight, and the generosity with which he shared them, he greatly influenced all of us.

We further thank Jürgen Eckhoff for helpful comments on a preliminary version of the paper, and Andreas Holmsen and Gil Kalai for providing us with useful references.

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Goaoc, X., Paták, P., Patáková, Z., Tancer, M., Wagner, U. (2017). Bounding Helly Numbers via Betti Numbers. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_17

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