Hardness of almost embedding simplicial complexes in $\mathbb R^d$

TL;DR

This paper proves that determining almost embeddability of simplicial complexes in certain dimensions is NP-hard, highlighting computational complexity barriers in topological embedding problems.

Contribution

It establishes the NP-hardness of recognizing almost embeddability for finite complexes in specific dimensions, extending previous embedding complexity results.

Findings

Existence of complexes not almost embeddable but with equivariant maps

NP-hardness of recognition problem for almost embeddability

Extension of techniques from embedding to almost embedding context

Abstract

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(\sigma)\cap f(\tau)=\emptyset$ whenever $\sigma,\tau$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Matou\v{s}ek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma and a generalized van Kampen--Flores theorem.

Equations

$$\tilde{K}:=\cup \{\sigma \times \tau :\sigma ,\tau \text{ are simplices of }K,\sigma \cap \tau =\varnothing \};$$

$$\tilde{f}(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$

$$E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Phi (x_1,\dots ,x_n)=\underset{s=1}{\overset{t}{\bigwedge }}(x_{n_{s1}}^{{\alpha }_{s1}}\vee x_{n_{s2}}^{{\alpha }_{s2}}\vee x_{n_{s3}}^{{\alpha }_{s3}}).$$

$$[n]:=\{1,\dots ,n\}.$$

$$F:=\left([k+\ell +3]\cup \{p\},\left(\genfrac{}{}{0ex}{}{[k+\ell +3]}{\le k+1}\right)\cup \left\{\{p\}\cup \sigma :\sigma \in \left(\genfrac{}{}{0ex}{}{[k+\ell +3]}{\le \ell +1}\right)\right\}\right).$$

$$P:=\{(q,r)\in ([t]\times [3]{)}^2:n_q=n_r,{\alpha }_q=0\text{ and }{\alpha }_r=1\}.$$

$$K(\Phi ):={\cup }_{s=1}^t{G}_s\bigcup _{\partial {\sigma }_q=a_{qr},\partial {\sigma }_r=b_{qr},(q,r)\in P}{\cup }_{(q,r)\in P}{T}_{qr}.$$

$$v(f)\in {\mathbb{Z}}_2$$

$$v(H):=\cup \{H(\sigma \times I)\cap H(\tau \times I):\sigma ,\tau \in K,\sigma \cap \tau =\varnothing ,\dim \sigma +\dim \tau =d\}$$

Full text

Theorem. Fix integers $d,k\geq 2$ such that $d=\frac{3k}{2}+1$.

(a) Assume that $P\neq NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb{R}^{d}$ but for which there exists an equivariant map $\widetilde{K}\to S^{d-1}$.

(b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^{d}$ is NP hard.

The proof is based on the technique from the Matoušek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma. The new part of our argument is a stronger ‘almost embeddings’ version of the generalized van Kampen–Flores theorem.

1. 1 Introduction
2. 2 Proof of Theorem 2
3. 3 The van Kampen number: proof of Lemma 5

1 Introduction

In this paper we study almost embeddings and equivariant maps of configuration spaces. They appear in studies of embeddings [FKT94, Sko08] as well as in topological combinatorics (for Tverberg-type problems see [BZ16, BBZ16, Sko16]). Almost embeddings also turned out to be a useful tool for studying Helly-type results on convex sets, implicitly in [Mat97] and explicitly in [GPP*+*15]. See definitions and more motivations below.

Throughout this paper, let $K$ be a finite simplicial complex.

A map $f\colon|K|\to\mathbb{R}^{d}$ is an almost embedding if $f(\sigma)\cap f(\tau)=\emptyset$ whenever $\sigma,\tau$ are disjoint simplices of $K$. (Existence of an almost embedding is obviously a necessary condition for existence of an embedding.)

The (simplicial) deleted product of $K$ is

[TABLE]

i.e., $\widetilde{K}$ is the union of products $\sigma\times\tau$ formed by disjoint simplices of $K$.

Suppose that $f:|K|\to\mathbb{R}^{d}$ is an almost embedding. Then the map $\widetilde{f}:\widetilde{K}\to S^{d-1}$ is well-defined by the Gauss formula

[TABLE]

We have $\widetilde{f}(y,x)=-\widetilde{f}(x,y)$; i.e., this map is equivariant with respect to the ‘exchanging factors’ involution $(x,y)\to(y,x)$ on $\widetilde{K}$ and the antipodal involution on $S^{d-1}$. Thus the existence of an equivariant map $\widetilde{K}\to S^{d-1}$ is a necessary condition for almost embeddability of $|K|$ in $\mathbb{R}^{d}$.

Theorem 1**.**

Fix integers $d,k\geq 2$ such that $d=\frac{3k}{2}+1$.

• $(a)$

Assume that $P\neq NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb{R}^{d}$ but for which there exists an equivariant map $\widetilde{K}\to S^{d-1}$.

• $(b)$

The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^{d}$ is NP hard.

The reader need not to know what NP hardness is: the essence of part (b) is explained by Theorem 2 below.

For $k=2$ part (a) is true even without $P\neq NP$ assumption, by [AMSW16, Theorem 1.5 and Proposition 1.7]. Part (a) follows by part (b) and the existence of a polynomial algorithm for checking the existence of equivariant maps [ČKV13]. Indeed, for fixed $d,k$ it is polynomial time decidable whether there exists an equivariant map $\widetilde{K}\to S^{d-1}$ [ČKV13]. Given that almost embeddabilty implies the existence of an equivariant map, part (b) implies part (a).

We discuss the possibility of removing the assumption $P\neq NP$ at the end of the introduction.

Remark*.*

The conclusions of Theorem 1 are in fact valid for each fixed integers $k,d$ such that $2\leq k\leq d\leq\frac{3k}{2}+1$ and $d\equiv 1\pmod{3}$. That is, we reflect only the interesting extremal cases in the statement of Theorem 1.

Indeed, for such integers $k,d$ let $k^{\prime}:=\frac{2(d-1)}{3}$. For a proof of part (a) with $(d,k)$ we take the complex $K^{\prime}$ from part (a) with parameters $(d,k^{\prime})$, and define the complex $K$ to be the disjoint union of $K^{\prime}$ and a $k$-simplex. Similarly, for a proof of part (b) with $(d,k)$ we add an isolated $k$-simplex to every $k^{\prime}$-complex. In both cases it is easy to check that the conclusion of Theorem 1 remains valid as for $d\geq k$ adding an isolated $k$-simplex does not affect neither almost embeddability to $\mathbb{R}^{d}$ nor the existence of an equivariant map to $S^{d-1}$.

Motivation and background.

A classical question in topology is to determine whether a simplicial complex $K$ embeds (topologically/piecewise linearly/linearly) in $\mathbb{R}^{d}$. It is easy to deduce that every $k$-dimensional simplicial complex embeds (even linearly) into $\mathbb{R}^{2k+1}$. Pioneering result in this area, known as the van Kampen–Flores theorem [vK33, Flo34, Sko14], states the existence of $k$-dimensional complexes that to do not embed into $\mathbb{R}^{2k}$ (for every integer $k$; even topologically).

In general, it is often very hard to determine whether a given complex $K$ embeds into $\mathbb{R}^{d}$. More precisely, this question subtly depends on the comparison of $k:=\dim K$ and $d$. For example, it is algorithmically undecidable to recognize whether a given $(d-1)$-complex embeds into $\mathbb{R}^{d}$ [MTW11], provided that $d\geq 5$. (This result follows from a celebrated theorem of Novikov on unrecognizability of the $d$-sphere [VKF74].)

Matoušek, the second author and Wagner [MTW11] proved that for each pair $(k,d)$ such that $4\leq d\leq\frac{3n}{2}+1$ it is NP hard to decide whether a $k$-dimensional simplicial complex PL embeds in $\mathbb{R}^{d}$. Theorem 1(b) is a version of this result for almost embeddability.

We describe the method from [MTW11] in detail (in order to prove our main results), up to one step in proof that we take directly from [MTW11]. We explicitly state the initial step of the proof (Theorem 2 below). Next, we slightly simplify the main construction (construction of $K(\Phi)$ in §2). We also present a simple proof of Lemma 7 below generalizing the van Kampen-Flores theorem (which is also one of the key tools for the result). For $\ell=k-1$ this lemma is proved in [vK33], for $\ell<k-1$ a weaker version of this lemma (when $f|_{S_{1}}$ is a PL embedding) is proved in [SS92, Lemma 1.4] using the Smith index. Thus this paper can serve as an exposition of the proof of the above result of [MTW11].

Theorem 1(b) is interesting on its own because almost embeddability is different from embeddability. Consider the following three properties of a finite simplicial complex $K$.

(E) $K$ PL embeds into $\mathbb{R}^{d}$.

(AE) $K$ PL almost embeds in $\mathbb{R}^{d}$.

(EM) There exists an equivariant map $\widetilde{K}\to S^{d-1}$.

The conditions (AE) and (EM) appeared as ‘combinatorial’ or ‘algebraic’ counterparts of (E), useful to study ‘geometric’ condition (E). Theorem 1 indicates that the condition (AE) is closer to (E) than to (EM), from algorithmic point of view. More precisely, we have

[TABLE]

Here the straight arrows are clear and explained above, and the curved arrow is a theorem of Weber [Web67]; see also [Sko08, §5]. For every pair $(k,d)$ such that ‘$2d\geq 3k+3$ or $d\leq 2$’ does not hold, i.e. such that $3\leq d\leq\frac{3k}{2}+1$, we have $(EM)\not\Rightarrow(E)$ [SS92], [FKT94], [SSS98], [GS06]. Moreover, for every pair $(k,d)$ such that $4\leq d\leq\frac{3k}{2}+1$ we have $(AE)\not\Rightarrow(E)$ [SS92], [SSS98]. See [Sko08, §5, §7] for a survey. By [AMSW16, Theorem 1.5 and Proposition 1.7] $(EM)\not\Rightarrow(AE)$ for $d=2k=4$. Theorem 1(a) shows (modulo $P\neq NP$) that for every pair $(k,d)$ such that $4\leq d=\frac{3k}{2}+1$ $(EM)\not\Rightarrow(AE)$.

It might be also interesting to compare algorithmic complexity (of embeddability and almost embeddability) with the ‘geometric’ refinement complexity introduced in [FK14]. Although it is not directly related to the results in this paper, let us also consider embeddability and almost embeddability of $k$-complexes to $\mathbb{R}^{2k}$ when $k\geq 3$. Then there is a quite noticeable gap between the two complexities for embeddability, which is polynomial time solvable whereas the refinement complexity grows exponentially [FK14]. V. Krushkal kindly informed us that this gap is also present for almost embeddability by generalizations of Proposition 4.1 and ‘Proof of the bound (4.1) on refinement complexity’ from [FK14].111However, it is not an aim of this paper to provide the details.

Proof technique.

We prove Theorem 1 by applying the technique of [MTW11] (which builds on a construction222Let us emphasize that [SS92], [FKT94], [SSS98], and [MTW11] slightly differ in technical details. In particular, the examples in [SS92] and [SSS98] were built with the aim to be almost-embeddable, thus we could not use them immediately without a modification. in [SS92], [FKT94], [SSS98]; see [Sko08, §5, §7] for a survey) and the Singular Borromean Rings Lemma 4 [AMSW16]. The new part of our argument is the ‘almost embeddings’ version (Lemma 7) of the generalized van Kampen–Flores theorem. That is, of [SS92, Lemma 1.4], [FKT94, Lemma 6], [SSS98, Lemma 1.1], [Sko08, Lemma 7.2], and [MTW11, Lemmas 4.1 and  5.1(i)]. Our version is stronger because we do not assume that $f|_{S_{j}}$ is an embedding. In spite of this, our proof (presented in Section 3) is simpler, cf. [MTW11, Remark at the end of 5.1].

Let us emphasize that this passage from embeddability to almost embeddability, being not hard, is not entirely trivial. Although new proofs of ‘almost embeddings’ analogues of main lemmas are simpler, they require certain change of the viewpoint. The Borromean Rings Lemma 4 for almost embeddings is only proved for $k=2l$ [AMSW16], not for $k\geq 2l$ as for embeddings. Also recall that almost embeddability does not in general imply embeddability [SS92, SSS98], cf. ‘Motivation and background’.

Formally, Theorem 1 follows by NP-hardness of recognition of 3-SAT problem and the following result.

A $3$-CNF formula in variables $x_{1},\dots,x_{n}$ is

[TABLE]

Here $n_{si}\in\{1,\ldots,n\}$, $\alpha_{si}\in\{0,1\}$ and $x^{0}=\neg x$, $x^{1}=x$.

Theorem 2**.**

Let $d,k\geq 2$ be fixed integers such that $d=\frac{3k}{2}+1$. Then to each $3$-CNF formula $\Phi$ there corresponds, by a polynomial algorithm (in the size of the formula $\Phi$), a finite $k$-dimensional complex $K(\Phi)$ such that $K(\Phi)$ is almost embeddable in $\mathbb{R}^{d}$ if and only if $\Phi$ is satisfiable (i.e., if the Boolean function $\mathbb{Z}_{2}^{n}\to\mathbb{Z}_{2}$ corresponding to $\Phi$ is not identically zero).

The analogue of Theorem 2 for embeddability is proved in [MTW11, §4, §5]. Our complex $K(\Phi)$ slightly differs from the complex constructed in [MTW11, §4.2, §5.2], which we call $K^{\prime}(\Phi)$.333We make this minor change to simplify the construction of $K(\Phi)$ and proof of the ‘only if’ part, which, however, would work for $K^{\prime}(\Phi)$ as well. For the ‘if’ part of Theorem 2 we need the following lemma.

Lemma 3** (proof is sketched in Section 2).**

The complex $K(\Phi)$ constructed in Section 2 is obtained from the complex $K^{\prime}(\Phi)$ constructed in [MTW11, §4.2, §5.2] by several contractions of edges and several compressions of $S^{\ell-1}\times I$. (Compression of a subcomplex identified with $S^{\ell-1}\times I$ is contracting to a point each segment $x\times I$, $x\in S^{\ell-1}$.)

Proof of Theorem 2: the ‘if’ part.

Since the formula $\Phi$ is satisfiable, the complex $K^{\prime}(\Phi)$ PL embeds into $\mathbb{R}^{d}$ [MTW11, Section 5].444The converse is also true but is not used here. Recall that the quotient of a PL manifold by a map with collapsible point-inverses is PL homeomorphic to the same manifold [Coh67]. Hence contracting an edge and compressing $S^{\ell-1}\times I$ keep PL embeddability.555The analogous claim is not true for ‘decontractions’. Thus by Lemma 3 $K(\Phi)$ also PL embeds into $\mathbb{R}^{d}$. ∎

Therefore it suffices to prove the converse: if $K(\Phi)$ almost-embeds in $\mathbb{R}^{d}$, then $\Phi$ is satisfiable. This is a strengthening of the analogous fact from [MTW11]. We use the same idea as [MTW11] but need to replace two key lemmas [MTW11, Lemmas 5.1(i) and 5.3(i)] by suitable analogues for almost embeddings. These analogues are Lemma 5 and the Singular Borromean Rings Lemma 4 [AMSW16, Lemma 1.9] below, respectively.

Let $T:=S^{\ell}\times S^{\ell}$ be the $2\ell$-dimensional torus with meridian $a:=S^{\ell}\times\cdot$ and parallel $b:=\cdot\times S^{\ell}$. See well-known definition of ‘linked modulo 2’, e.g., in [ST80, §77] or in [Sko, §2.2 ‘Linking modulo 2’].

Lemma 4** (Singular Borromean Rings).**

For each $k=2\ell$ let $S^{k}_{a}$ and $S^{k}_{b}$ be copies of $S^{k}$. Then there is no PL map $f\colon T\sqcup S^{k}_{a}\sqcup S^{k}_{b}\to\mathbb{R}^{k+\ell+1}$ such that

(a) the $f$-images of the components are pairwise disjoint;

(b) $f(S^{k}_{a})$ is linked modulo 2 with $f(a)$ and is not linked modulo 2 with $f(b)$;

*(c) $f(S^{k}_{b})$ is linked modulo 2 with $f(b)$ and is not linked modulo 2 with $f(a)$. *

On the assumption $P\neq NP$.

A reader could expect that analyzing the algorithm in [ČKV13] and the proof of Theorem 1(b) would yield a direct construction of an example of Theorem 1(a), without the assumption $P\neq NP$. Here we discuss the difficulties that appear in this analysis.

Let us consider a 3-CNF formula $\Phi$ and let $d$ and $k$ be fixed, $d=\frac{3k}{2}+1$. If $\Phi$ is satisfiable, then $(AE)$ holds for $K(\Phi)$ by Theorem 2 and therefore $(EM)$ holds for $K(\Phi)$ as well.

If $\Phi$ is not satisfiable, then $(AE)$ does not hold for $K(\Phi)$ by Theorem 2 but we do not know whether $(EM)$ holds for $K(\Phi)$. However, if $(EM)$ did not hold for every $\Phi$ which is not satisfiable, then we would deduce that it is NP-hard to recognize whether a given simplicial complex $K$ satisfies $(EM)$. On the other hand, this is a polynomial time solvable problem due to [ČKV13]. This would be only possible if $P=NP$.

Thus, we have good reasons to expect that for any choice of $k$ and $d$ with $d=\frac{3k}{2}+1$, there is a non-satisfiable formula $\Phi$ such that $(EM)$ holds for $K(\Phi)$. Actually, we conjecture that $(EM)$ holds for $K(\Phi)$ for every $\Phi$.

For a proof of Theorem 1(a), without the assumption $P\neq NP$, it would be fully sufficient to exhibit a single non-satisfiable $3$-CNF formula $\Phi$ such that $(EM)$ holds for $K(\Phi)$. In fact, the construction in Theorem 2 makes also sense for the simplest non-satisfiable $1$-CNF formula $\Phi_{\operatorname{NO}}=x_{1}\wedge\neg x_{1}$ (in the definition of the clause gadget $G$ in Section 2 only a single simplex is removed instead of three simplices). Let $K_{\operatorname{NO}}(k,d):=K(\Phi_{\operatorname{NO}})$ for given parameters $d$ and $k$ (here we want to emphasize the dependence on $k$ and $d$). The complex $K_{\operatorname{NO}}(2,4)$ is essentially the complex constructed by Freedman, Krushkal and Teichner [FKT94] (up to a minor modification), and we know that $(EM)$ holds in this case (as we discussed below the statement of Theorem 1).

For few other values of $k$ and $d$ we could, in principle, run the algorithm of [ČKV13] on $K_{\operatorname{NO}}(k,d)$ (unfortunately, it is not implemented666That is, the code is not written.). However, we do not know how to verify $(EM)$ for infinitely many values of $k$ and $d$ we are interested in; the dependence of the algorithm from [ČKV13] on $k$ and $d$ is somewhat complicated. The algorithm in [ČKV13] is based on the obstruction theory. As far as we know there are no other tools developed, besides the obstruction theory, that would allow us to verify $(EM)$ for our examples ‘by hand’.

2 Proof of Theorem 2

First we define building blocks for the complex $K(\Phi)$ (most importantly, clause gadgets) and prove their properties.

Take any integers $0\leq\ell<k$. We suppress dependence on $k,\ell$. For an integer $n$ denote

[TABLE]

Definition of an auxiliary complex $F$.

Complex $F$ has the vertex set $[k+\ell+3]\cup\{p\}$. The simplices are

• •

complete $k$-skeleton on $[k+\ell+3]$, and

• •

all the simplices of dimension at most $\ell+1$ that contain $p$.

In other words,

[TABLE]

Here ${n\choose\leq m}$ is the set of all subsets of $[n]$ having at most $m$ elements.

Definitions of $\sigma_{j}$, $S_{j}$ and clause gadget $G$.

In this definition $j$ is any element of $[3]$.

Set $\sigma_{j}$ to be the simplex with vertex set $\{p\}\cup[\ell+2]-\{j\}$. Then $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ are three $(\ell+1)$-simplices containing $p$.

Set $S_{j}$ to be the union of all $k$-simplices with vertices in $[k+\ell+3]$ that do not intersect $\sigma_{j}$. Clearly, this union is homeomorphic to the $k$-sphere.777This was called a complementary sphere in [MTW11].

Finally, we define the clause gadget as $G:=F-\sigma_{1}-\sigma_{2}-\sigma_{3}$.

Lemma 5**.**

For any $\ell<k$ and general position PL almost-embedding $f:|G|\to\mathbb{R}^{k+\ell+1}$ there is $i\in[3]$ such that $f(\partial\sigma_{i})$ is linked modulo 2 with $f(S_{i})$.

For $k=2\ell=2$ Lemma 5 is proved in [FKT94, proof of Lemma 8]. Cf. [AMSW16, Remark 2.3.b].

Proof of Theorem 2: construction of $K(\Phi)$.

Recall the notation for $3$-CNF formula $\Phi$ given before Theorem 2. The ‘multiple’ $x_{n_{s1}}^{\alpha_{s1}}\vee x_{n_{s2}}^{\alpha_{s2}}\vee x_{n_{s3}}^{\alpha_{s3}}$ is the $s$-th clause of $\Phi$. The ‘summand’ $x_{n_{si}}^{\alpha_{si}}$ is the $i$-th literal of the $s$-th clause. Without loss of generality, we may assume that no clause (‘multiple’) contains both $x_{m}$ and $\neg x_{m}$ for some $m$ (otherwise such a clause would be redundant). Denote

[TABLE]

This is the set of all pairs $(q,r)=((q_{1},q_{2}),(r_{1},r_{2}))$ such that for some $m$

$\bullet$ the $q_{2}$-th literal (‘summand’) of the $q_{1}$-th clause (‘multiple’) is $x_{m}$, and

$\bullet$ the $r_{2}$-th literal (‘summand’) of the $r_{1}$-th clause (‘multiple’) is $\neg x_{m}$.

(In other words, this is the set of all pairs of (pairs of) indices of literals in conflict.)

Take copies $G_{1},\ldots,G_{t}$ of (the clause gadget) $G$. Denote by $\sigma_{q}=\sigma_{(q_{1},q_{2})}$ the simplex $\sigma_{q_{2}}$ in the copy $G_{q_{1}}$ . Take a triangulation of $k$-torus $T$ extending triangulations of its meridian and parallel $a$ and $b$ as boundaries of $(\ell+1)$-simplices. For each $(q,r)\in P$ take a copy $T_{qr}\supset a_{qr},b_{qr}$ of $T\supset a,b$. Set

[TABLE]

That is, this complex is obtained from the copies $G_{s}$ and $T_{qr}$, by identifying the $\ell$-spheres $\partial\sigma_{q}$ and $a_{qr}$, and the $\ell$-spheres $\partial\sigma_{r}$ and $b_{qr}$, for each $(q,r)\in P$.

Recall that $k$ and $\ell$ are fixed. Then each of the complexes $G$ and $T$ can be built by a constant-time algorithm. Hence $K(\Phi)$ is obtained from $\Phi$ by a polynomial algorithm in $n$ and $t$ (i.e. in the size of the formula). ∎

Sketch of a proof of Lemma 3.

The construction of $K^{\prime}(\Phi)$ [MTW11, §4.2, §5.2] is different from the above construction of $K(\Phi)$ by the following details:

$\bullet$ the torus $T$ is replaced by a polyhedron $X$ containing ‘parallel’ and ‘meridian’ $a$ and $b$, and an edge whose contraction yields $T$;

$\bullet$ the simplices $\sigma_{i}$ in the definitions of $G$ and of $K(\Phi)$ are replaced by $k$-disks $\omega_{i}\subset\mathop{Int}\sigma_{i}$.

Thus $K(\Phi)$ is obtained from $K^{\prime}(\Phi)$ by contracting the edge in each copy of $T$, and by compressing $\sigma_{j}-\mathop{Int}\omega_{j}\cong S^{\ell-1}$ in each copy of $G$ and for each $j\in[3]$. ∎

Proof of Theorem 2: the ‘only if’ part.

Assume that there is a PL almost embedding $f:|K(\Phi)|\to\mathbb{R}^{k+\ell+1}$. By Lemma 5 for every $s\in[t]$ there is $i(s)\in[3]$ such that the $f$-images of the spheres $S_{si(s)},\partial\sigma_{si(s)}\subset|G_{s}|$ are linked modulo 2.

Let us assume, for contradiction, that $\Phi$ is not satisfiable; that is, $\Phi\equiv 0$. (The reader not so familiar with literals, clauses and conflicts, may wish to skip the next paragraph and check rather the footnote in the following one.)

The function $i$ selects one literal (‘summand’) in each clause (‘multiple’). Then two selected literals (‘summands’) must be in conflict.

This means that there are $q_{1},r_{1}\in[t]$ and $m\in[n]$ such that the $i(q_{1})$-th literal (‘summand’) of the $q_{1}$-th clause (‘multiple’) is $x_{m}$ and the $i(r_{1})$-th literal (‘summand’) of the $r_{1}$-th clause (‘multiple’) is $\neg x_{m}$.888Indeed, in the opposite case for each $m\in[n]$ there is $\alpha(m)\in\{0,1\}$ such that $\alpha_{si(s)}=\alpha(n_{si(s)})$ for each $s\in[t]$. So we can take $x_{n_{si(s)}}:=\alpha(n_{si(s)})$ for each $s\in[t]$ and then extend this to a satisfying assignment. That is, $(q,r):=((q_{1},i(q_{1})),(r_{1},i(r_{1})))\in P$.

Then $\partial\sigma_{q}=a_{qr}$ and $\partial\sigma_{r}=b_{qr}$. Since $f$ is an almost embedding, the $f$-images of $S_{q}$, $S_{r}$ and $T_{qr}$ are pairwise disjoint. The $\ell$-sphere $b_{qr}$ bounds the disk $v*b_{qr}$ outside $S_{q}$, where $v$ is any vertex of $G_{r_{1}}$ outside $\partial\sigma_{r_{1}i}$ for each $i\in[3]$. Hence $f(S_{q})$ is unlinked modulo 2 with $f(b_{qr})$. Analogously $f(S_{r})$ is unlinked modulo 2 with $f(a_{qr})$. Since $k=2\ell$, all this contradicts the Singular Borromean Rings Lemma 4 applied to the restriction of $f$ to $S_{q}\sqcup S_{r}\sqcup T_{qr}$. ∎

3 The van Kampen number: proof of Lemma 5

For a general position PL map $f\colon|K|\to\mathbb{R}^{d}$ of a finite $k$-complex define the van Kampen number

[TABLE]

to be the parity of the number of points $x\in\mathbb{R}^{d}$ such that $x\in f(\sigma)\cap f(\tau)$ for some disjoint simplices $\sigma,\tau\in K$ with $\dim\sigma+\dim\tau=d$. (For an exposition and another applications of the van Kampen number see [Sko17], [Sko, §1].)

Lemma 6**.**

Let $d$ be an integer and $K$ a finite complex such that for every pair $\sigma,\tau$ of disjoint $s$- and $t$-simplices in $K$ with $s+t=d-1$ the following two numbers have the same parity:

$\bullet$* the number of $(s+1)$-simplices $\nu$ containing $\sigma$ and disjoint with $\tau$;*

$\bullet$* the number of $(t+1)$-simplices $\mu$ containing $\tau$ and disjoint with $\sigma$.*

Then $v(f)$ is independent of a general position PL map $f\colon|K|\to\mathbb{R}^{d}$.

For $d=2$ and $K=K_{5}$ this corresponds to well-known proof of the non-planarity of $K_{5}$ [Sko, Lemma 3.4], [BE01, §5]. For the general case the proof is analogous.

Proof of Lemma 6.

Lemma 6 follows analogously to [Sha57, Lemma 3.5] (by interpreting $v(f)$ as an obstruction to the existence of certain equivariant map).

A direct proof is as follows. Take a general position PL homotopy $H:|K|\times I\to\mathbb{R}^{d}\times I$ between general position PL maps $H_{0},H_{1}:|K|\to\mathbb{R}^{d}$. Then

[TABLE]

is a graph. For each $i=0,1$ the vertices of this graph in $\mathbb{R}^{d}\times i$ are exactly the points $x$ from the definition of $v(H_{i})$. So $v(H_{i})$ equals to the number modulo 2 of vertices of this graph in $\mathbb{R}^{d}\times i$. This graph also has vertices in $\mathbb{R}^{d}\times(0,1)$ corresponding to pairs $(\nu,\tau)$ and $(\sigma,\mu)$ from the bullet points of the lemma. (There could be connected components of $v(H)$ containing no such vertices; there could be some other vertices of even degree, e.g. vertices of degree 2 coming from double points of $H$ or vertices of degree 4 coming from triple points of $H$.)

Analogously to [Hud69, Lemma 11.4] any vertex of this graph

$\bullet$ contained in $\mathbb{R}^{d}\times i$ has odd degree.

$\bullet$ contained in $\mathbb{R}^{d}\times(0,1)$ has even degree (by the bullet points of the lemma).

Hence $v(H_{0})=v(H_{1})$. ∎

Lemma 7**.**

For any $\ell<k$ and general position PL map $f\colon|F|\to\mathbb{R}^{k+\ell+1}$ we have $v(f)=1\in\mathbb{Z}_{2}$.

Proof.

For some $f$ Lemma 7 was proved in [SS92, Lemma 1.1]. Then Lemma 7 follows for any $f$ by Lemma 6 after we verify the assumptions of the lemma.

Take any pair $\sigma,\tau$ of disjoint $s$- and $t$-simplices in $F$ such that $s+t=k+\ell$. Without loss of generality we assume that $s\leq t$. Since $t\leq k$, we obtain $s\geq\ell$. Among $(k+1)+(\ell+1)+2$ vertices of $F$ there are exactly two which are not contained in $\sigma\cup\tau$. We distinguish two cases.

• •

Case $s=\ell$. In this case $t=k$. Since $\dim F=k$, the simplex $\tau$ cannot be extended to a simplex of $F$ (disjoint with $\sigma$). Since $F$ contains complete $(\ell+1)$-skeleton, $\sigma$ can be extended (to $(s+1)$-simplex of $F$ disjoint with $\tau$) by both vertices of $F$ not contained in $\sigma\cup\tau$. The numbers [math] and $2$ have the same parity as required.

• •

Case $s>\ell$. In this case $\ell<t<k$.

Subcase when neither $\sigma$ nor $\tau$ contains $p$. Since $p$ is contained only in simplices of dimension at most $\ell+1$, neither $\sigma$ not $\tau$ can be extended by $p$ to a simplex of $F$. On the other hand, since $s,t<k$, the remaining vertex of $F$ can serve for extension of both $\sigma$ or $\tau$. The numbers $1$ and $1$ have the same parity as required.

Subcase when $\sigma$ or $\tau$ contains $p$. Since $t>\ell$ and $p$ can be only contained in a simplex of dimension at most $\ell+1$, we can without loss of generality assume that $p\in\sigma$ and $s=\ell+1$. Since $p$ does not belong to any $(\ell+2)$-simplex, it follows that $\sigma$ cannot be extended. On the other hand, $\tau$ can be extended in two ways to both vertices of $F$ not contained in $\sigma\cup\tau$. The numbers [math] and $2$ have the same parity as required.

∎

Proof of Lemma 5.

Extend the map $f$ to a general position PL map $g\colon|F|\to\mathbb{R}^{k+\ell+1}$. Since $\ell+2k<2(k+\ell+1)$, by general position to every point $x$ from the definition of $v(g)$ there corresponds a unique unordered pair of simplices of $F$, the sum of whose dimensions is $d$ and the intersection of whose $g$-images contains $x$. Since $f$ is an almost-embedding, for every such point $x$ there is a unique $i\in[3]$ such that $x\in g(\sigma_{i})\cap g(S_{i})$. This and Lemma 7 imply that $\sum_{i=1}^{3}|g(\sigma_{i})\cap g(S_{i})|\underset{2}{\equiv}v(g)=1\in\mathbb{Z}_{2}$. Hence one of the three summands is odd as required. ∎