On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1n−k−1) ≤ (k+12k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.


Introduction
Given a closed surface M , it is a natural question to determine the maximum integer n such that the complete graph K n can be embedded (drawn without crossings) into M (e.g., n = 4 if M = S 2 is the 2-sphere, and n = 7 if M is a torus). This classical problem was raised in the late 19th century by Heawood [9] and Heffter [10] and completely settled in the 1950-60's through a sequence of works by Gustin, Guy, Mayer, Ringel, Terry, Welch, and Youngs (see [22,Ch. 1] for a discussion of the history of the problem and detailed references). Heawood already observed that if K n embeds into M then where χ(M ) is the Euler characteristic of M and b 1 (M ) = 2 − χ(M ) is the first Z 2 -Betti number of M , i.e., the dimension of the first homology group H 1 (M ; Z 2 ) (here and throughout the paper, we work with homology with Z 2 -coefficients). 1 Conversely, for surfaces M other than the Klein bottle, the inequality is tight, i.e., K n embeds into M if and only if (1) holds; this is a hard result, the bulk of the monograph [22] is devoted to its proof. (The exceptional case, the Klein bottle, has b 1 = 2, but does not admit an embedding of K 7 , only of K 6 .) The question naturally generalizes to higher dimension: Let ∆ (k) n denote the k-skeleton of the n-simplex, the natural higher-dimensional generalization of K n+1 = ∆  [23] and Flores [5] showed that ∆ (k) 2k+2 does not embed into R 2k (the case k = 1 corresponding to the nonplanarity of K 5 ). Somewhat surprisingly, little else seems to known, and the following conjecture of Kühnel [12, Conjecture B] regarding a generalized Heawood inequality remains unresolved: Conjecture 1 (Kühnel). Let n, k ≥ 1 be integers. If ∆ The classical Heawood inequality (1) and the Van Kampen-Flores Theorem correspond the special cases k = 1 and b k = 0, respectively. Kühnel states Conjecture 1 in slightly different form in terms of Euler characteristic of M rather than b k (M ). Our formulation is an equivalent form. The Z 2 -coefficients are not important in the statement of the conjecture but they are convenient for our further progress.
New result. Here, we prove an estimate in the spirit of the generalized Heawood inequality (2), with a quantitatively weaker bound. Note that our bound holds (at no extra cost) under weaker hypotheses.
A somewhat technical but useful relaxation is that instead of embeddings, we consider the following slightly more general notion (which also helps with setting up our proof method). Let K be a finite simplicial complex and let |K| be its underlying space (geometric 1 The inequality (1), which by a direct calculation is equivalent to n ≤ c(M ) := ( realization). We define an almost-embedding of K into a (Hausdorff) topological space X to be a continuous map f : |K| → X such that any two disjoint simplices σ, τ ∈ K have disjoint images, f (σ) ∩ f (τ ) = ∅. We stress that the condition for being an almost-embedding depends on the actual simplicial complex (the triangulation), not just the underlying space. That is, if K and L are two different complexes with |K| = |L| then a map f : |K| = |L| → X may be an almost-embedding of K into X but not an almost-embedding of L into X. Note also that every embedding is an almost-embedding as well. Our main result is as follows: As remarked above, this bound is weaker than the conjectured generalized Heawood inequality (2) and is clearly not optimal (as we already see in the special cases k = 1 or b k = 0). On the other hand, apart from applying more generally to almost-embeddings, the hypotheses of Theorem 2 are weaker than those of Conjecture 1 in that we do not assume the manifold M to be (k − 1)-connected. We conjecture that this connectedness assumption is not necessary for Conjecture 1, i.e., that (2) holds whenever ∆ (k) n almost-embeds into a 2k-manifold M . The intuition is that ∆ (k) n is (k − 1)-connected and therefore the image of an almost-embedding cannot "use" any parts of M on which nontrivial homotopy classes of dimension less than k are supported.
Previous work. The following special case of Conjecture 1 was proved by Kühnel [12,Thm. 2] (and served as a motivation for the general conjecture): Suppose that P is an ndimensional simplicial convex polytope, and that there is a subcomplex of the boundary ∂P of P that is k-Hamiltonian (i.e., that contains the k-skeleton of P ) and that is a triangulation of M , a 2k-dimensional manifold. Then inequality (2) holds. To see that this is indeed a special case of Conjecture 1, note that ∂P is a piecewise linear (PL) sphere of dimension n − 1, i.e., ∂P is combinatorially isomorphic to some subdivision of ∂∆ n (and, in particular, (n − 2)-connected). Therefore, the k-skeleton of P , and hence M , contains a subdivision of ∆ (k) n and is (k − 1)-connected.
In this special case and for n ≥ 2k + 2, equality in (2) is attained if and only if P is a simplex. More generally, equality is attained whenever M is a triangulated 2k-manifold on n + 1 vertices that is k + 1-neighborly (i.e., any subset of at most k + 1 vertices form a face, in which case ∆ (k) n is a subcomplex of M ). Some examples of (k + 1)-neighborly 2k-manifolds are known, e.g., for k = 1 (the so-called regular cases of equality for the Heawood inequality [22]), for k = 2 [15,14] (e.g., a 3-neighborly triangulation of the complex projective plane) and for k = 4 [3], but in general, a characterization of the higher-dimensional cases of equality for (2) (or even of those values of the parameters for which equality is attained) seems rather hard (which is maybe not surprising, given how difficult the construction of examples of equality is already for k = 1).
Proof technique. Our proof of Theorem 2 strongly relies on a different generalization of the Van Kampen-Flores Theorem, due to Volovikov [24], regarding maps into general manifolds but under an additional homological triviality condition: Note that the homological triviality condition is automatically satisfied if H k (M ; Z 2 ) = 0, e.g., if M = R 2k or M = S 2k . On the other hand, without the homological triviality condition, the assertion is in general not true for other manifolds (e.g., K 5 embeds into every closed surface different from the sphere, or ∆ (2) 8 embeds into the complex projective plane). Theorem 3 is only a special of the main result in [24]; it is obtained by setting j = q = 2, m = 2k, s = k + 1 and N = 2k + 2 in item 3 of Volovikov's main result (beware that k from Volovikov's condition "there exists a natural number k" is different from our k).
In addition, Volovikov [24] formulates the triviality condition in terms of cohomology, i.e., he requires that f * : 2k+2 ; Z 2 ) is trivial. However, since we are working with field coefficients and the (co)homology groups in question are finitely generated, the homological triviality condition (which is more convenient for us to work with) and the cohomological one are equivalent. 2 The key idea of our approach is to show that if n is large enough and f is a mapping from ∆  n with the following properties.

The induced map on the geometric realizations
The value n 0 can be taken as s k b(s − 2k) + 2s − 2k + 1.
Therefore, if s ≥ 2k + 2, then f • g cannot be an almost-embedding by Volovikov's theorem. We deduce that f is not an almost-embedding either, and Theorem 2 immediately follows. This deduction requires the following lemma as in general, a composition of two almost-embeddings needs not be an almost-embedding.

Lemma 5.
Let K and L be simplicial complexes and X a topological space. Suppose g is an almost-embedding of K into |L| and f is an almost-embedding of L into X. Then f • g is an almost-embedding of K into X, provided that g is the realization of a simplicial map g simp from some subdivision K of K to L.
We prove Lemma 4 in Section 4 thus completing the proof of Theorem 2. Before that, in Section 3 we first present a simpler version of that proof that introduces the main ideas in a simpler setting, and yields a weaker bound for n 0 (see Equation (4)). Further related questions and problems will be discussed in Section 5.

Preliminaries
We begin by fixing some terminology and notation. We will use card(U ) to denote the cardinality of a set U . ϑ aϑ aϑ * ∂ϑ We also recall that the stellar subdivision of a maximal face ϑ in a simplicial complex K is obtained by removing ϑ from K and adding a cone a ϑ * (∂ϑ), where a ϑ is a newly added vertex, the apex of the cone (see the figure on the left).
Throughout this paper we only work with homology groups and Betti numbers over Z 2 , and for simplicity, we will for the most part drop the coefficient group Z 2 from the notation. Moreover, we will need to switch back and forth between singular and simplicial homology. More precisely, if K is a simplicial complex then H * (K) will mean the simplicial homology of K, whereas H * (X) will mean the singular homology of a topological space X. In particular, H * (|K|) denotes the singular homology of the underlying space |K| of a complex K. We use analogous conventions for C * (K), C * (X) and C * (|K|) on the level of chains, and likewise for the subgroups of cycles and boundaries, respectively. 3 Given a cycle c, we denote by [c] the homology class it represents.
A mapping h : |K| → X induces a chain map h sing : C * (|K|) → C * (X) on the level of singular chains; see [8,Chapter 2.1]. There is also a canonical chain map ι K : The three chain maps mentioned above also induce maps h sing * , (ι K ) * , and h * on the level of homology satisfying h * = h sing * • (ι K ) * . We also need a technical lemma saying that our maps compose, in a right way, on the level of homology.
that behaves as an almost-embedding, in the sense that whenever σ and σ are disjoint k-faces of ∆ s , ϕ(σ) and ϕ(τ ) have disjoint supports, and such that for every (k + 1)-face τ of ∆ s the homology class [(f • ϕ)(∂τ )] is trivial. We then use ϕ to design a subdivision D of ∆ (k) s and a simplicial map g simp : D → ∆ (k) n that induces a map g : |D| → |∆ (k) n | with the 3 We remark that throughout this paper, we will only work with spaces that are either (underlying spaces of) simplicial complexes or topological manifolds. Such spaces are homotopy equivalent to CW complexes [20,Corollary 1], and so on the matter of homology, it does not really matter which (ordinary, i.e., satisfying the dimension axiom) homology theory we use as they are all naturally equivalent for CW complexes [8,Thm. 4.59]. However the distinction between the simplicial and the singular setting will be relevant on the level of chains. desired properties: g is an almost-embedding and (f • g) * ([∂τ ]) is trivial for all (k + 1)-faces τ of ∆ s . Since the cycles ∂τ , for (k + 1)-faces τ of ∆ s , generate all k-cycles of ∆ The purpose of this section is to give a first implementation of the above strategy that proves Lemma 4 with a bound of In Section 4 we then improve this bound to s k b(s − 2k) + 2s − 2k + 1 at the cost of some technical complications.

Construction of ϕ
For every face ϑ of ∆ s of dimension at most k − 1 we set ϕ(ϑ) = ϑ. We then "route" each σ i by mapping it to its stellar subdivision with an apex The picture on the left shows the case k = 1, the support of z(σ i , u) is dashed on the left, and the support of the resulting ϕ(σ i ) is on the right. We ensure that ϕ behave as an almost-embedding by using a different apex u ∈ U for each σ i . The difficulty is to choose these m apices in a way that [f (ϕ(∂τ ))] is trivial for every (k + 1)-face τ of ∆ s . To that end we associate to each u ∈ U the sequence and we denote by v i (u) the ith element of v(u). We work with Z 2 -homology, so H k (M ) m is finite; more precisely, its cardinality equals 2 bm . From n ≥ n 0 = (m − 1)2 bm + s + 1 we get that card(U ) ≥ (m − 1) card(H k (M ) m ) + 1. The pigeonhole principle then guarantees that there exist m distinct vertices u 1 , u 2 , . . . , u m of U such that v(u 1 ) = v(u 2 ) = · · · = v(u m ). We use u i to "route" σ i and put We finally extend ϕ linearly to C * ∆ Before proving the lemma, we establish a simple claim that will be also useful later on.

S o C G ' 1 5
Proof of Lemma 7. The map ϕ is the identity on -chains with ≤ k − 1 and Equation (5) immediately implies that ∂ϕ(σ) = ∂σ for every k-simplex σ. It follows that ϕ is a chain map. Now let τ be a (k + 1)-simplex of ∆ s and let σ i1 , . . . , σ i k+2 be its k-faces. We have The u i 's are chosen in such a way that the homology class f z(σ ij , u ) = v ij (u ) is independent of the value . When passing to the homology classes in the above identity, we can therefore replace each u ij with u 1 , and obtain, This class is trivial by Claim 8.
Here is the idea behind the proof with k = 1 and u i1 = u 1 (same colors represent same homology classes; the class on the right is trivial, because each edge appears twice):

Construction of D and g
The definition of ϕ, an in particular Equation (5), suggests to construct our subdivision D of ∆  On principle, we would like to derive Condition 2 of Lemma 4 by observing that g 'induces' a chain map from C * (∆ n ) that coincides with ϕ. Making this a formal statement is thorny because g, as a continuous map, naturally induces a chain map g on singular rather than simplicial chains. We can't use directly g simp either, since we are interested in a map from C * (∆ n ) denotes the (simplicial) chain map induced by g simp . We thus have in homology f * • ϕ * = f * • (g simp ) * • ρ * and since ρ * is an isomorphism and f * • ϕ * is trivial, Lemma 7 yields that f * • (g simp ) * is also trivial. Since f * • (g simp ) * = (f • g) * by Lemma 6, (f • g) * is trivial as well. This concludes the proof of Lemma 4 with the weaker bound.

Proof of Lemma 4
We now prove Lemma 4 with the bound claimed in the statement, namely Let k, b, s be fixed integers. We consider a 2k-manifold M with kth Betti number b, a map f : |∆ (k) n | → M , and we assume that n ≥ n 0 . The proof follows the same strategy as in Section 3 : we construct a chain map ϕ : C * (∆ When constructing ϕ, we refine the arguments of Section 3 to "route" each k-face using not only one, but several vertices from U ; this makes finding "collisions" easier, as we can use linear algebra arguments instead of the pigeonhole principle. This comes at the cost that when upgrading g, we must content ourselves with proving that it is an almost-embedding. This is sufficient for our purpose and has an additional benefit: the same group of vertices from U may serve to route several k-faces provided they pairwise intersect in ∆

Definition of multipoints and the map v.
As we said we plan to route k-faces of ∆ s through collections of vertices from U , we will call these collections multipoints. It turns out that this is useful for our needs only if these multipoints have an odd cardinality. In order to easily proceed with later computations, we define multipoints as vectors rather than subsets of U as below.
Let C 0 (U ) denote the Z 2 -vector space of formal linear combinations of vertices from U . A multipoint is an element of C 0 (U ) with an odd number of non-zero coefficients. The multipoints form an affine subspace of C 0 (U ) which we denote by M. The support, sup(µ), of a multipoint µ ∈ M is the set of vertices v ∈ U with non-zero coefficient in µ. We say that two multipoints are disjoint if their supports are disjoint.
For any k-face σ i and any multipoint µ we define: Now, we proceed as in Section 3 but replace the unused points by the multipoints of M and the cycles z(σ i , u) with the cycles z(σ i , µ). Since Z 2 is a field, H k (M ) m is a vector space and we can replace the sequences v(u) of Section 3 by the linear map v : Finding collisions. The following lemma takes advantage of the vector space structure of H k (M ) m to find disjoint multipoints µ 1 , µ 2 , . . . to route the σ i 's more effectively than by simple pigeonhole.
(This is, in a sense, a simple affine version of Tverberg's theorem.) The statement is obvious for r = 1, so assume that r ≥ 2 and that the statement holds for r − 1. Let A denote the affine hull of {ψ(v s+2 ), ψ(v s+3 ), . . . , ψ(v n+1 )} and let I r denote a minimal cardinality subset of U such that the affine hull of {ψ(v) : v ∈ I r } equals A. Since dim A ≤ d the set I r has cardinality at most d + 1. The cardinality of U \ I r is at least (d + 1)(r − 2) + 1 so we can apply the induction hypothesis for r − 1 to U \ I r . We thus obtain r − 1 disjoint subsets I 1 , I 2 , . . . , I r−1 whose images under ψ have affine hulls with non-empty intersection. Since the affine hull of ψ(U \ I r ) is contained in the affine hull of ψ(I r ), the claim follows. Now, let a ∈ V be a point common to the affine hulls of ψ(I 1 ), ψ(I 2 ), . . . , ψ(I r ). Writing a as an affine combination in each of these spaces, we get where J j ⊆ I j and |J j | is odd for any j ∈ [r]. Setting µ j = u∈Jj u finishes the proof.
Proof. By Claim 8 we know that (7) is true for points. For a multipoint µ, we get (7) as a linear combination of equations for the points in sup(µ) (using that card(sup(µ)) is odd).
Proof. Let τ be a (k + 1)-face of ∆ s and let σ i1 , . . . , σ i k+2 denote its k-faces. For any multipoint µ, Claim 10 implies Coloring hypergraphs to reduce the number of multipoints used. We could now apply Lemma 9 with r = m to obtain one multipoints per k-face, all pairwise disjoint, to proceed with our "routing". As mentioned above, however, we only need that ϕ is an almost-embedding, so we can use the same multipoint for several k-faces provided they pairwise intersect. Optimizing the number of multipoints used reformulates as the following hypergraph coloring problem: Assign to each k-face σ i of ∆ s some color c(i) ∈ N such that card{c(i) : 1 ≤ i ≤ m} is minimal and disjoint faces use distinct colors.
This question is classically known as Kneser's hypergraph coloring problem and an optimal solution uses s − 2k + 1 colors [17,18]. Let us spell out one such coloring (proving its optimality is considerably more difficult, but we do not need to know that it is optimal). For every k-face σ i we let min σ i denote the smallest index of a vertex in σ i . When min σ i ≤ s−2k we set c(i) = min σ i , otherwise we set c(i) = s − 2k + 1. Observe that any k-face with color c ≤ s − 2k contains vertex v c . Moreover, the k-faces with color s − 2k + 1 consist of k + 1 vertices each, all from a set of 2k + 1 vertices. It follows that any two k-faces with the same color have some vertex in common.
Defining ϕ. We are finally ready to define the chain map ϕ : Recall that we assume that n ≥ n 0 = ( s k b + 1)(r − 1) + s + 1. Using the bound of Lemma 11 we can apply Lemma 9 with r = s − 2k + 1, obtaining s − 2k + 1 multipoints µ 1 , µ 2 , . . . , µ s−2k+1 ∈ M. We set ϕ(ϑ) = ϑ for any face ϑ of ∆ s of dimension less than k. We then "route" each k-face σ i through the multipoint µ c(i) by putting where c(i) is the color of σ i in the coloring of the Kneser hypergraph proposed above. We finally extend ϕ linearly to C * (∆ s ). We need the following analogue of Lemma 7.
The proof of Lemma 12 is very similar to the proof of Lemma 7; it just replaces points with multipoints and Claim 8 with Claim 10.
We next argue that ϕ behaves like an almost embedding.
Proof. Since ϕ is the identity on chains of dimension at most (k − 1), the statement follows if neither face has dimension k. For any k-chain σ i , the support of ϕ(σ i ) uses only vertices from σ i and from the support of µ c(i) . Since each µ c(i) has support in U , which contains no vertex of ∆ s , the statement also holds when exactly one of ϑ or η has dimension k. When both ϑ and η are k-faces, their disjointness implies that they use distinct µ j 's, and the statement follows from the fact that distinct µ j 's have disjoint supports. Figure 1 Examples of subdivisions for k = 1 and = 3 (left) and for k = 2 and = 5 (right).

Construction of D and g
We define D and g similarly as in Section 3, but the switch from points to multipoints requires to replace stellar subdivisions by a slightly more complicated decomposition.

The subdivision D.
We define D so that it coincides with ∆ s on the faces of dimension at most (k − 1) and decomposes each face of dimension k independently. The precise subdivision of a k-face σ i depends on the cardinality of the support of the multipoint µ c(i) used to "route" σ i under ϕ, but the method is generic and spelled out in the next lemma; refer to Figure 1. Proof. This proof is done in the language of geometric simplicial complexes (rather than abstract ones).
The case = 1 can be done by a stellar subdivision and labelling the added apex x 1 . The case k = 1 is easy, as illustrated in Figure 1 (left). We therefore assume that k ≥ 2 and build our subdivision and labelling in four steps: We start with the boundary of our simplex σ where each vertex w i is labelled by itself. Let ϑ be the (k − 1)-face of ∂σ opposite vertex w 2 , ie labelled by w 1 , w 3 , w 4 , · · · w k+1 . We create a vertex in the interior of σ, label it w 2 , and construct a new simplex σ as the join of ϑ and this new vertex; this is the dark simplex in Figure 1 (right). We then subdivide σ by considering − 1 distinct hyperplanes passing through the vertices of σ labelled w 3 , w 4 , . . . , w k+1 and through an interior points of the edge of σ labelled w 1 , w 2 . These hyperplanes subdivide σ into smaller simplices. We label the new interior vertices on the edge of σ labelled w 1 , w 2 by alternatively, w 1 and w 2 ; since is odd we can do so in a way that every sub-edge is bounded by two vertices labelled w 1 , w 2 . We operate a stellar subdivision of each of the smaller simplices subdividing σ , and label the added apices x 1 , x 2 , . . . , x . This way we obtain a subdivision S of σ .
We finally consider each face η of S subdividing ∂σ and other than ϑ and add the simplex formed by η and the (original) vertex w 2 of σ. These simplices, together with S , form the desired subdivision S of σ.
It follows from the construction that no face of ∂σ was subdivided.
Property 1 is enforced in the first step and preserved throughout. We can ensure that Property 2 holds in the following way. First, we have that any k-simplex of S contains a vertex x j for some j ∈ [ ]. Next, if we consider a k-simplex of S which is not in S it is a join of a certain (k − 1)-simplex η of S , with η ⊂ ∂σ , and the vertex w 2 of σ. However, the only such (k − 1)-simplex labelled by w 1 , w 3 , w 4 , . . . , w k+1 is ϑ, but the join of ϑ and w 2 does not belong to S.
Properties 3 and 4 are enforced by the stellar subdivisions of the third step, and no other step creates, destroys or modifies any simplex involving a vertex labelled x i .
The subdivision D of ∆ (k) s is now defined as follows. First, we leave the (k − 1)-skeleton untouched. Next, for each k-simplex σ i we let i denote the number of points in the support of µ c(i) ; since we work with Z 2 coefficients, i is odd. We then compute some subdivision S(i) of σ i using Lemma 14 with := i .
We let ρ : s ) → C * (D) denote the map that is the identity on ∆ (k−1) s and that maps each σ i to the sum of the k-dimensional simplices of S(i). This maps induces an isomorphism ρ * in homology.

Related Questions and Outlook
While we consider Conjecture 1 natural and interesting in its own right, there are a number of connections to other problems that are worth mentioning and provide additional motivation.

Topological Helly-Type Theorems for subsets of Manifolds
In [6, Thm. 1], we use the Van Kampen-Flores Theorem to prove the following topological Helly-type theorem for finite families F of subsets of R d , under the assumption that for every proper subfamily G F, the More precisely, our proof heavily relies the fact that the Van Kampen-Flores Theorem also applies to the following generalization of almost-embeddings: We define a homological almost-embedding of a finite simplicial complex K into a topological space X as a chain map ϕ from the simplicial chain complex C * (K; Z 2 ) to the singular chain complex C * (X; Z 2 ) with the properties that (i) for every vertex of K, ϕ(v) consists of an odd number of points in X and (ii) for any pair σ, τ of disjoint simplices of K, the image chains ϕ(σ) and ϕ(τ ) have disjoint underlying point sets.
One can show that Volovikov's theorem, and consequently Theorem 2 extend to homological almost-embeddings; we plan to discuss this in more detail in the full version of the present paper. Thus, (3) holds whenever ∆ (k) n homologically almost-embeds into M . As a consequence, the Helly-type result in [6, Thm. 1] generalizes (with an appropriate change in the constants) to families of subsets of an arbitrary d-dimensional manifold.

Extremal Problems for Embeddings
Closely related to the classical Heawood inequality is the well-known fact that for a (simple) graph embedded into a surface M , the number of edges of G is at most linear in the number of vertices of G (see, e.g., [22,Thm. 4.2]). More specifically, if G embeds into a surface M with first Z 2 -Betti number b 1 (M ), and if f 1 (G) and f 0 (G) denote the number of vertices and of edges of G, respectively, then f 1 (G) ≤ 3f 0 (G) − 6 + 3b 1 (M ).
Note that this immediately implies (1) when applied to G = K n .
This question also naturally generalizes to higher dimensions:

Conjecture 16.
Let M be a 2k-dimensional manifold with kth Z 2 -Betti number b k (M ). If K is a finite k-dimensional simplicial complex that embeds into M then where f i denotes the number of i-dimensional faces of K, −1 ≤ i ≤ k, and C is a constant that depends only on k and on b k (M ). 5 The special case M = R 2k of the problem was first raised by Grünbaum [7] more than forty years ago, and has since then been rediscovered and posed independently by a number of authors (see, e.g., Dey [4], where the problem is motivated by the question of counting triangulations of higher-dimensional point sets), and the problem remains wide open even in that case. Moreover, there is a beautiful conjecture, due to Kalai and Sarkaria [11,Conjecture 27] that gives a necessary condition for embeddability into R 2k in terms of algebraic shifting and would, in particular, imply that the constant C in Conjecture 16 can be taken to be k + 2 if M = R 2k .
The aforementioned extension of Theorem 2 to homological almost-embeddings together with [25,Thm. 7] imply the following result for random complexes: Corollary 17. Let X k (n, p) denote the Linial-Meshulam model [16,19] of k-dimensional random complexes on n vertices. 6 Given integers k ≥ 1 and b ≥ 0, there exists a constant C = C(k, b) with the following property: If M is a 2k-dimensional manifold with Z 2 -Betti number b k (M ) ≤ b and if p ≥ C/n then asymptotically almost surely, X k (n, p) does not embed into M .
This generalizes [25,Thm. 2] and can be viewed as evidence for Conjecture 16 (in a sense, it shows that the conjecture holds for "almost all complexes").
The arguments in [25,Thm. 7] are based on the following notion closely related to homological almost-embeddings: If K and L are simplicial complexes, we say that K is a homological minor of K if there is a chain map ϕ from the simplicial chain complex C * (K; Z 2 ) into the simplicial chain complex of C * (K; Z 2 ) that satisfies conditions (i) and (ii) in the definition of a homological almost-embedding (one might call ϕ a simplicial homological almost-embedding). In [25,Conj. 6], we propose a conjectural generalization of Mader's theorem to the extent that a finite k-dimensional simplicial complex K contains ∆ (k) t as a homological minor provided that f k (K) ≥ C · f k−1 (K) for some suitable constant C = C(k, t). If true, this conjecture, together with the extension of Theorem 2 to homological almost-embeddings, would imply Conjecture 16.
We remark that Conjecture 1 is also closely related to the combinatorial theory of face numbers of triangulated spheres and manifolds, in particular the Generalized Lower Bound Theorem for polytopes (which is the main ingredient in Kühnel's proof of his special case) and conjectured generalizations thereof to triangulated spheres and manifolds. A detailed discussion of these questions goes beyond the scope of this extended abstract, and we refer the reader to [12] and [13,Ch. 4].