# Geometric complexity of embeddings in

###### Abstract.

Given a simplicial complex , we consider several notions of geometric complexity of embeddings of in a Euclidean space : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any -complex with simplices which topologically embeds in , , can be PL embedded in with refinement complexity . Families of simplicial -complexes are constructed such that any embedding of into has an exponential lower bound on thickness and refinement complexity as a function of the number of simplices of . This contrasts embeddings in the stable range, , , where all known bounds on geometric complexity functions are polynomial. In addition, we give a geometric argument for a bound on distortion of expander graphs in Euclidean spaces. Several related open problems are discussed, including questions about the growth rate of complexity functions of embeddings, and about the crossing number and the ropelength of classical links.

## 1. Introduction.

We use the abbreviation -complex to mean an -dimensional simplicial complex. By general position any -complex embeds into the Euclidean space in the stable range, . There have been recent developments, cf. [7, 21], in quantifying the geometry of embeddings, and polynomial “complexity” has been established in the stable range. There are several possible notions of thickness and distortion of which we will mention four. We also consider refinement complexity, the minimal number of simplices needed for a PL embedding of a given complex into . We prove that for the refinement complexity and for the notion of thickness defined below the lower bound shifts from polynomial to at least exponential when the dimension passes from to . We do not presently know if there is a similar transition for the other notions of geometric complexity of embeddings. For the refinement complexity we also establish an upper bound: we prove that any -complex, , with simplices and with trivial van Kampen’s obstruction can be PL embedded in with refinement complexity . Our conclusions on thickness and refinement complexity are summarized in a table appearing near the end of section 4.

Section 2 considers the smallest codimensions . There are families of - and -complexes for which the refinement complexity with respect to embeddings into , , is a non-recursive function of the number of simplices. We conjecture that the thickness (defined below) of these complexes is also non-recursive, see section 2. This construction [16], stated as theorem 2.1, is based on Novikov’s theorem on the algorithmic unrecognizability of spheres , , and it is motivated by the work [16] on algorithmic undecidability of the embedding problem in codimensions .

In theorem 2.4 we show that in the relative setting there are also embedding problems in codimension : pairs where is a subcomplex of a -complex with a fixed embedding , , such that the refinement complexity (and conjecturally thickness) of any extension to an embedding is a non-recursive function. These examples are based on groups with undecidable word problem.

The rest of the paper addresses the question of what kinds of complexity functions can be realized in higher codimensions nearer the stable range. To formulate an upper bound for -complexes in and to fix the notation we state the following definition.

###### Definition 1.1.

Refinement complexity (rc). Given a complex and , is the minimal number of simplices in a PL embedding of into . We say if there is no PL embedding . There is also an extrinsic form: given a piecewise mooth embedding , is the minimum number of simplices of a PL embeddings of piecewise smoothly isotopic to .

###### Theorem 1.2.

Let be an -complex with simplices which topologically embeds in , . Then for any .

The proof of this theorem in section 3 is a quantitative version of the proof that the vanishing of the van Kampen cohomological obstruction for -complexes in is sufficient for embeddability for .

The main result of section 4 is the construction of -complexes in with an (at least) exponential lower bound on refinement complexity and thickness. To formulate this result, we first state two relevant notions of embedding thickness.

Consider embeddings of an -dimensional simplicial complex into the unit ball in . Following [7], we say that such an embedding has Gromov-Guth thickness at least if the distance between the images of any two non-adjacent simplices is at least . Assuming that local combinatorial complexity of is bounded, it is shown in [7] that if then is a sharp (up to an summand in the exponent) bound for the Gromov-Guth thickness of in , where is the number of vertices of . This is a generalization of earlier work of Kolmogorov and Barzdin [10] which, predating a formal definition of expander graphs, used their properties to establish the sharp bound for embedding thickness of graphs in -space. Next we define the notion of thickness used in this paper.

###### Definition 1.3.

Consider embeddings which are cell-wise smooth. (Explicitly this means that each closed -simplex is carried into by some embedding of a neighborhood .) We say that the thickness of such an embedding is at least if the distance between the images of any two non-adjacent simplices is at least and all simplices have embedded -normal bundles. (In the codimension zero case there is no normal bundle requirement on -simplices embedded in .)

This definition, like other notions of thickness and distortion, has two versions: thickness of an isotopy class of an embedding (where one takes the supremum over all embeddings isotopic to ), and intrinsic thickness of (the supremum over all embeddings into ).

The notion of thickness introduced above, taking into account the size of the normal bundle, arises naturally when one considers isotopy classes of embeddings of manifolds. In particular, related measures of complexity have been investigated in knot theory, cf. [2, 12, 19].

In contrast to the known polynomial bounds on thickness, refinement complexity and distortion (discussed below) of embeddings in the stable range, we show the following.

###### Theorem 1.4.

For each there exist families of -complexes with simplices and bounded local combinatorial complexity which embed into , and the thickness of any such embedding is at most . Moreover,

Here the constants depend on .

Remarks. 1. The phrase “bounded local combinatorial complexity” means that in a family the maximum number of simplices incident to any fixed simplex is bounded independently of .

The proof of theorem 1.4 is given in section 4. At the end of that section we state an addendum extending this theorem to -complexes in for all . A table summarizing our results on refinement complexity and thickness of -complexes in for various values of is included at the end of section 4.

Other related families of -complexes in are discussed in section 5. This construction is a geometric implementation of nested commutators and other recursively defined elements in the free group. The key feature is that a word of length can be created using -cells. It seems likely that these examples also have exponentially small thickness as in theorem 1.4, however at present this is an open problem. Section 5 also formulates related questions about the thickness and the crossing number of links in .

Next we review some of the known results about distortion of embeddings into Euclidean spaces. Given a subset and two points , let denote the intrinsic metric: the infimum of the lengths of paths in connecting and . The distortion of is defined as

(1.1) |

The distortion of an isotopy class of an embedding is the infimum over all embeddings isotopic to . Measuring the distortion is often a rather subtle problem. A recent advance in the subject is due to J. Pardon [21] who used integral geometry of curves on surfaces to show that the distortion of torus knots is unbounded, , answering a 1983 question of M. Gromov [6]. A different construction of families of knots with unbounded geometry using the volume of branched covers is due to Gromov-Guth [7].

The intrinsic distortion of a complex with respect to embeddings in is defined as

(1.2) |

where the infimum is taken over all embeddings . It is established in [1] that any -point metric space can be embedded in the Euclidean space of dimension with distortion . The vertex sets of expander graphs (discussed in more detail in section 6) of bounded degree with the graph metric are examples of metric spaces for which this distortion bound is tight (even for embeddings into the Hilbert space ) [11].

However if one is interested in embeddings into the Euclidean space of a fixed dimension , the bounds on distortion are polynomial: [14] gives upper and lower bounds on distortion of an -point metric space into , (recall that means: asymptotically):

(1.3) |

In section 6 we analyze the distortion of embeddings of expander graphs in a Euclidean space of a fixed dimension, where the metric space is taken to be the entire graph (including its edges), not just the vertex set. We give a geometric argument, to some extent following the natural approach of [10, 7] of slicing the embedding by codimension hypersurfaces, to show that for families of expander graphs of bounded degree the distortion with respect to embeddings into has a lower bound , see theorem 6.1. It is an interesting question whether there is a transition to exponential distortion for embeddings of complexes below the stable range, similar to the behavior of thickness discussed in section 4; for intrinsic distortion we do not have a candidate family to propose.

## 2. Non-recursive complexity of embeddings in codimensions .

It is proved in [16] that the embedding problem for both - and - complexes in is undecidable, for . Their starting point is Novikov’s theorem that there can be no decision procedure for determining if a -complex , , which we may assume is a homology -sphere, is actually homeomorphic to . A second ingredient is Newman’s theorem [20] that any PL embedding (in the technical sense: a piecewise-linear embedding of a subdivision of the standard triangulation of) is topologically bicollared. Then the topological Schoenflies theorem implies that the closed complementary domains of any such embedding are homeomorphic to , the -ball. From this it follows directly that there can be no algorithm either for determining the embeddability of or in , where the first complex is a homology -ball obtained by deleting the interior of a single -simplex and the second complex is the -skeleton . Let and denote these undecidable families.

The following theorem, giving a refinement complexity (rc) - analogue of these statements, is given in [16, Corollary 1.2]. Modulo a technical conjecture 2.3 below, the statement also holds for thickness, see conjecture 2.2.

###### Theorem 2.1.

For each there exists a sequence of -complexes (-complexes) with simplices which after subdivision PL embed in , but there is no recursive function of which lower bounds .

Proof. The Tarski-Seidenberg theorem [25] on quantifier elimination implies that arbitrarily quantified sentences in first order semi-real algebraic geometry are decidable. Clearly the existence of a linear embedding (no refinement) of a fixed finite complex in can be reduced to quantified polynomial inequalities. For example the statement that there are disjoint line segments in may be expressed as:

Hence “linear embedding” is decidable. Let be the subsequence of either family of complexes (dimension or ) discussed prior to theorem 2.1 which actually do have PL embeddings into . Consider the function . If this function were upper bounded on by any recursive function we could write an algorithm for deciding the PL embeddability of the family . We would run the (in principle) Tarski-Seidenberg algorithm on each of the finitely many subdivisions of containing simplices for each . If one of these subroutines found a solution to the quantified embedding inequalities we would know that PL embed in . If by the time we have checked all subdivisions with simplices no solution has been found we could halt the search knowing further subdivision to be fruitless. serves as the in the statement. This contradiction proves the theorem. ∎

###### Conjecture 2.2.

For each there exists a sequence of -complexes (-complexes) with simplices which can be piecewise smoothly embedded in but the intrinsic thickness satisfies:

where is any positive recursive function.

To discuss the relation between theorem 2.1 and conjecture 2.2, note that our definition 1.3 of piecewise smooth embeddings is rather minimal. It has the virtue that because the closed simplices are smoothly embedded, the complex automatically satisfies the Whitney stratification conditions (A and B). However in other regards our definition is bare bones. It does not control how closely adjacent simplices may approach each other. For example our definition allows the -complex with two -simplices joined to form the letter V to be embedded in by:

where , is any smooth function ( for ). might satisfy , or even decay nonrecursively.

While the minimal nature of our definition is adequate for the main results in section 4, it makes it quite technical (but we think possible) to prove a lemma relevant to theorem 2.1 in the context of thickness. We choose to state this lemma as conjecture 2.3 which we recommend to anyone with technical interest in stratified sets:

###### Conjecture 2.3.

If admits a piecewise smooth embedding of thickness into (more specifically, into the unit ball ) then where is a recursively computable function of .

Next we establish non-recursive complexity in codimension 2 for a family of relative embedding problems.

###### Theorem 2.4.

For each there exists a sequence of -complex pairs with a fixed embedding such that the refinement complexity (and conjecturally the thickness) of any extension to an embedding is a non-recursive function of the number of simplices of .

Proof. Indeed, consider a -complex realizing a finite presentation of a group with undecidable word problem. Some thickening of to a -dimensional -handlebody , , embeds into the -sphere . There is a sequence of curves in , say those exhausting the kernel -skeleton of , bounding disks which embed in , so that the refinement complexity of such embeddings is necessarily non-recursive as a function of the length of the boundary curve. Otherwise an algorithm which searched by brute force for such disks up to a (recursively) computed upper bound on complexity would succeed in solving the word problem in .

Consider the complement , a codimension zero subcomplex of . For each the curve may be assumed to be embedded in the -skeleton of for a sufficiently fine triangulation. Define . Then is a -dimensional pair satisfying the conclusion of the theorem.

The dimension of the complex can be reduced to by considering -skeleton of , with the same fixed embedding into . Repeat the construction above with in place of , yielding pairs . Now the group in which we have to study the word problem is the original group free product with a finitely generated free group. The free summand does not affect the decision problem. ∎

The codimension relative examples with a fixed embedding of in theorem 2.4 do not admit an immediate generalization to the absolute setting . Indeed, fixing the fundamental group is a crucial ingredient in the proof, and if one considers arbitrary embeddings of a complex into the fundamental group of the complement is not controlled by Alexander duality. (The Stallings theorem gives a partial control of the fundamental group, modulo its lower central series for some codimension embeddings, see section 5 for more details.)

## 3. An upper bound for -complexes in .

By general position a PL embedding of an -complex in the stable range does not require any subdivisions. At the other extreme discussed in section 2, in codimensions (and in codimension in the relative setting), the refinement complexity (definition 1.1) is non-recursive. In this section we establish an upper bound at the edge of the stable range, for -complexes in :

Theorem 1.2. Let be an -complex with simplices which topologically embeds in , . Then for any .

To be more specific, the upper bound will be obtained in terms of max(number of -simplices, number of -simplices). Examples in section 4 complement this upper bound on refinement complexity with an exponential lower bound.

Proof of theorem 1.2. We start by briefly recalling the definition of the van Kampen obstruction and refer the reader to [4] for more details. Predating a formal definition of cohomology, Van Kampen’s 1933 paper [26] gave a rough description of a cohomological embedding obstruction for -complexes in . Any general position map gives rise to a -equivariant -dimensional simplicial cochain on , where is the union of all products of simplices that have at least one vertex in common. Given -dimensional , is defined as the algebraic intersection number in . The cohomology class of is independent of the map : a general position homotopy between two maps has finitely many non-generic times when an -cell intersects an -cell, and it is not difficult to see that , differ by the coboundary of the sum over of the products -cell-cell [4]. (Note that the action of on the cohomology coefficients is trivial [22, 17] and not as stated in [4].) Moreover, for any cochain representative of there is a map such that [4, Section 2.4, Lemma 2].

The vanishing of the van Kampen obstruction is clearly a necessary condition for embeddability of in for any . For it is also a sufficient condition, due to the validity of the Whitney trick, see [22], [4, Theorem 3] and the discussion below. (For the obstruction is incomplete [4].)

Suppose an -complex with simplices embeds in . First map it linearly into , in general position. That is, consider a generic map of the vertices of to and extend linearly to get a map . The rank of the -th cellular cochain group of is , and the possible values of the van Kampen cochain on each -cell are , so . Here and in other estimates below we will work with the -norms on the cochain groups , where the - (respectively -) dimensional product cells are taken as orthonormal generators.

There are two steps in changing this map to a PL embedding. Since embeds in , the van Kampen obstruction vanishes and is a coboundary. Therefore, as discussed above, there are “finger move” homotopies pushing -simplices of across -simplices, so that the result is a PL map with , in other words for any two non-adjacent -simplices of . Each finger move homotopy amounts to taking a connected sum of an -simplex with a small oriented PL -sphere linking an -simplex of (cf. figure 2.1 in [4]). The connected sum is taken along a straight line segment in general position with . This move requires a constant (depending on the dimension ) number of subdivisions of the simplex .

The second step consists of applications of the Whitney move to remove the intersections among the images of the -simplices under . In general there are three types of Whitney moves: corresponding to

(1) a pair of intersections of two non-adjacent simplices ,

(2) an intersection point of two distinct adjacent -simplices, and

(3) a self-intersection point of an -simplex.

To begin with, the map is linear on each simplex and so there are no intersections of types (2), (3). The finger moves implementing a homotopy to may introduce intersections of type (2), however since the finger move always involves non-adjacent simplices , self-intersections will not be introduced and so the move of type (3) is not needed.

Correspondingly, there are two types of Whitney circles : the usual Whitney circle consisting of two arcs pairing up the intersection points in two non-adjacent simplices , and a slightly different type connecting an intersection between two adjacent simplices through a common vertex. In either case, the Whitney circle consists of a bounded number (to be specific, at most ) line segments. For the Whitney disk in each case take the cone on the Whitney circle . A generic choice of the cone point ensures that the interior of is disjoint from . Therefore each Whitney disk consists of at most 6 -simplices.

Due to the bounded combinatorial complexity of the Whitney disk discussed above, the classical Whitney move of type (1), changing one of the simplices , by an ambient PL isotopy, requires an a priori bounded number of subdivisions. The Whitney move of type (2) is described in lemma 5 in [4]. It eliminates an intersection point between two adjacent simplices while also introducing a bounded number of subdivisions.

Since each finger move gives rise to at most new intersection points, the number of Whitney moves is bounded by times the number of finger moves. The remaining problem is to get an estimate on the number of finger moves needed to get from to . This problem is equivalent to getting an upper bound on the -norm of a -cochain which maps onto the van Kampen -cocycle . We will work with the -norms below, however the required estimate on the -norm will follow since (recall that the ranks of are ) .

Denote by the matrix representing the coboundary map with respect to the chosen cell basis. The minimal norm of a cochain such that is upper bounded by , where

(3.1) |

Therefore we need to estimate the smallest absolute value of a non-zero eigenvalue of the symmetric matrix . The coboundary map from - to -cochains of the original complex has entries, the rest are zeros. The matrix has ’s, the rest of the matrix entries are zeros. Then the sum of the absolute values of the entries of is bounded in absolute value by .

Let be the characteristic polynomial of with variable . Unless the original linear map happened to be embedded and no subdivision was necessary, must have a non-zero eigenvalue, otherwise the non-zero obstruction element could not be a coboundary. It follows that must have at least two non-zero terms; let the term of lowest degree read: . To estimate how small (in absolute value) a root of could possibly be, we take as small in absolute value as possible, and all the coefficients of higher terms as large as possible. Estimating this bound by computing determinants we find the maximum is . We get a (smallest case) equation: .

Summing the geometric series obtain . We find a lower bound on : can be no smaller than . Then the minimal norm of a cochain with is ∎

Remarks. 1. The upper bound on in theorem 1.2 is established in terms of the weakest possible isoperimetric constant of the coboundary map of the deleted product that works for any . If one is interested in a specific -complex with a larger isoperimetric constant, the proof above would give a more optimal bound on the refinement complexity of .

2. A more careful linear algebra estimate may improve the power on in the exponent in the statement of the theorem. It does not seem likely however that a minor improvement along these lines would close the gap with the lower bound proved in the next section.

3. It seems reasonable to believe that a version of theorem 1.2 also holds for thickness. Such a statement would follow if one established an estimate on thickness in terms of the number of simplices in a PL embedding. For example, it is plausible that thickness .

## 4. Exponentially thin -complexes in .

This section gives the proof of theorem 1.4. Note that most -dimensional complexes do not embed into , and for those complexes that do embed a given generic map into may not in general be approximable by embeddings. The idea of the proof is to construct a sequence of -complexes which “barely” embed into : any embedding necessarily involves an exponential amount of linking, and as a consequence an exponential bound on thickness and recursive complexity.

Let denote the -skeleton of the -simplex, , with a single -simplex removed. Denote the boundary -sphere of the missing simplex by , and let be the -sphere spanned in by the vertices which are not in .

###### Proposition 4.1.

The -complex embeds into . Moreover, for any embedding the mod- linking number is non-zero.

This proposition essentially follows from the work of van Kampen [26]. We give an outline of the argument below and refer the reader to lemma 6 in [4] for more details. ([4] states the proof for -complexes in ; the argument for all is directly analogous.)

The complex for is the complete graph on vertices (well-known from the Kuratowski planarity criterion), which can be drawn on the plane with a single intersection point between two non-adjacent edges. Van Kampen [26] (and independently Flores [3]) observed that this is also true for : can be mapped into with a single double point between two non-adjacent -simplices. For example, for this can be seen concretely as follows: consider the -skeleton () of . To get the -skeleton of add another vertex inside the -sphere in , and in respectively. Connect by an edge intersecting in a point outside of . It is not difficult to see that all -cells embed disjointly, except for the -cell with vertices whose boundary circle links the -sphere . Adding this -cell introduces a single double point with one of the four faces forming the -sphere.

Van Kampen proved that the embedding obstruction (discussed in the proof of theorem 1.2 in section 3) is non-trivial for the complex by starting with the map with a single double point discussed above and showing that any homotopy preserves (mod 2) the sum of the cochain over all -cells. Now suppose the linking number is zero in the statement of the proposition for some embedding . Attach an -cell to and map it into , thus extending to a map . By assumption the intersection number equals zero (mod 2). Then the sum of the values of the cochain over all -cells is zero, a contradiction. This concludes an outline of the proof of proposition 4.1. ∎

We are now in a position to construct the complexes used in the proof of theorem 1.4. For convenience of the reader we restate it here:

Theorem 1.4. For each there exist families of -complexes with simplices and bounded local combinatorial complexity which embed into and the thickness of any such embedding is at most . Moreover,

(4.1) |

Here the constants depend on .

Proof. Fix and for each define the -complex to be the -fold mapping telescope

(4.2) |

where the last arrow denotes the degree map from to the -sphere from the statement of proposition 4.1. Sometimes the dimension of this sphere will be dropped and it will be denoted .

Concretely each stage of the mapping telescope is a copy of triangulated so that receives an induced triangulation as , and is triangulated as the -fold branched cover of along , for some . The map between each product segment of the telescope is the aforementioned branched cover. Therefore has -simplices and the local combinatorial complexity of does not increase with .

Next observe that the embedding discussed in proposition 4.1 extends to an embedding . Recall that the -cell bounded by can be mapped into creating a single double point with a non-adjacent -cell. In particular, a collar on the boundary -sphere of embeds into disjointly from the rest of the complex . A disk normal bundle over in , restricted to the collar, which we will parametrize as , also embeds into disjointly from the rest of . The sphere is identified with in this normal bundle.

By general position a degree map can be perturbed to an embedding , . Then the mapping cylinder embeds level-wise into , where the target sphere is identified with . Proceeding by induction and embedding the spheres in , , the cells forming the mapping telescope (4.2) embed in , showing that embeds into .

To establish the bound on embedding thickness of the complexes claimed in theorem 1.4, consider any simplex-wise smooth embedding into the unit ball in , . Denote by the left-most -sphere in (4.2). According to proposition 4.1, the mod- linking number is non-zero. Switching to the integer-valued linking number, it follows that

(4.3) |

Recall from the introduction that the thickness of an embedding is defined as the supremum of such that the distance between the images of any two non-adjacent simplices is at least and all simplices have embedded -normal bundles. Next we will use the Gauss linking integral to get an upper bound on .

For a -component link in whose components are parametrized by , the classical Gauss integral in 3-space computes the linking number in terms of the triple scalar product of , and :

Denoting by parametrizations of the spheres in , consider the higher-dimensional analogue (cf. [23] for a more detailed discussion) of the Gauss integral computing the degree of the corresponding map , equal (up to a sign) to the linking number:

(4.4) |

Since is an embedding into the unit ball and the embedding thickness is , it follows that . Then the integral (4.4) has an upper bound in terms of the -volume , the -volume and the thickness , leading to the inequality

(4.5) |

Recall that that the spheres are considered with standard triangulations. Then

(4.6) |

where the prime denotes the largest volume among the simplices forming the two linking spheres.

We will next obtain a lower bound on the -volume of the -normal bundles over the relevant simplices of in terms of their -, respectively -volumes . There is a polynomial expression, the classical Weyl tube formula [27], for the volume of an -regular neighborhood of a submanifold of a Euclidean space in terms of the volume of the submanifold. For example, the Weyl formula for closed surfaces in states:

cf. [5, section 1.2]. Note that in general this formula does not immediately give a lower bound for the volume in terms of the area since higher order terms may be negative (and we don’t have an apriori estimate on how small is). Instead we will use a more direct argument to get a rough estimate relating the volumes of the submanifold and of its normal bundle.

For brevity of notation let denote the relevant - (respectively -) simplex in the sphere , respectively in , and let denote its dimension. Let be the distance from to its nearest focal point and set . Consider the restriction of the exponential map (defined on the tangent bundle of ) to the -disk normal bundle over . Denote the -regular neighborhood of in by .

We will derive a rough bound

(4.7) |

The bound on the differential is particularly easy to see in the case of a hypersurface, pictured in figure 1: due to the choice of where is the distance to the nearest focal point, the tangent vectors to do not contract by more than a factor when mapped to any point within . (In some directions the tangent vectors expand - however we are looking for just a very rough estimate so we take the “worst case” bound in every direction.) This gives a lower bound of on the eigenvalues of the differential restricted to vectors tangent to , and the vectors tangent to the normal disk are mapped isometrically by .

Proving (4.7) in general (for submanifolds of higher codimension) requires a bit more care, and we will refer to [5] for details of the statements below. Fix and let be a unit vector normal to at . Denote by the boundary of the tube of radius around . Then the principal curvatures of at the point satisfy the differential equation ([5, Corollary 3.5] states this in a more general setting where is replaced with an arbitrary Riemannian manifold.)

The first of these principal curvatures of may be thought of as corresponding to the directions tangent to , and they are given by , . The rest correspond to the directions normal to , , (they are equal to the corresponding principal curvatures of the -sphere normal bundle, since the exponential map is an isometry in these directions). Then as expected the determinant of the differential of the exponential map may be expressed in terms of the first principal curvatures:

(4.8) |

see [5, (3.27)]. The principal curvatures satisfy the inequality , where as above denotes the distance to the nearest focal point [5, section 8.1, Lemma 8.9]. We assumed that , so According to (4.8) then , concluding the proof of the estimate (4.7).

We proceed with the proof of theorem 1.4. The complex is embedded in the unit ball and the -normal bundles to the simplices are assumed to be embedded. Since is the distance to a closest focal point, . If then . If then the inequality (4.7) holds for in place of . In either case, as a consequence of (4.7) one has

(4.9) |

where is the constant in the expression for the volume of the ball . The effect of this constant is minor on the final estimate (4.13) on thickness below, so it is omitted in the last term in (4.9). To get a very rough upper bound on note that the -regular neighborhood of each simplex is embedded in the ball of radius , so its -volume is less than . Combine this with the estimate (4.9) where is largest volume simplex in as in (4.6), so , respectively :

(4.10) |

It follows that

(4.12) |

To a good approximation this implies

(4.13) |

This is the sought exponential thinness in terms of .

Proof of the bound (4.1) on refinement complexity. Instead of computing linking(-sphere, -sphere) by an integral formula, we can instead use intersection numbers in a generic projection to . Since for two linearly embedded simplices of dual dimensions the possible intersection numbers are: , or , there must be exponentially many pairs of simplices to account for the exponential linking number. Denote by the number of simplices in a PL embedding. Then . ∎

Addendum. The statement of theorem 1.4 (both for thickness and rc) applies to -complexes in , for any .

Proof of addendum. Consider the -complexes and their embeddings in the -sphere constructed in the proof of theorem 1.4. Applying (iterated) suspension, one gets embeddings . Recall the exponential bound (4.3) on the linking number of the -sphere and the -sphere for any embedding . Then for any embedding one has the same exponential bound on the linking number of and . By induction assume exponential linking is established between and in . Now look at any embedding of intersect a small -radius sphere about one of the embedded suspension points and apply the inductive hypothesis. This linking number in must of course agree with in . The rest of the proof is identical to that of theorem 1.4. ∎

Our results on thickness and refinement complexity are summarized in Table 1.

nd | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1P | 1P | 1P | 1P | 1P | 1P | |||||||||

2 | * | 1P | rr | EE | 1P | 1P | 1P | ||||||||

3 | * | rr | EE | EE | E | 1P | 1P | 1P | |||||||

4 | * | EE | Nn | EE | EE | E | 1P | 1P | |||||||

5 | * | Nn | Nn | EE | EE | EE | E | 1P | 1P | ||||||

6 | * | * | Nn | Nn | EE | EE | EE | EE | E | 1P | 1P | ||||

7 | * | Nn | Nn | EE | EE | EE | EE | EE | E | 1P |

Refinement complexity: 1=no refinement, E=(at least) exponential refinement, =exponential lower bound and upper bound for -complexes in , N=non-recursive refinement, r=we conjecture to be recursive.

Thickness: P=polynomial, E=(at least) exponential, n=non-recursive (modulo conjecture 2.3), r=we conjecture to be recursive.

cannot embed since .

Remarks. We conjecture decidable (recursive) behavior in in line with numerous recent results in -manifold topology. Cells (3,4) and (4,4) are filled in “EE” since one may trivially add an additional - or -simplex to the examples in (2,4). Similarly all “Nn” are certainly at least “NE”.

The interested reader may find it instructive to compare this table with table 1 on page 4 in [16] which lists the algorithmic complexity of the embedding problem for simplicial -complexes into . In particular, it is interesting to note that the algorithmic complexity of the embedding problem for in , is polynomial (this amounts to checking whether van Kampen’s cohomological obstruction vanishes [16]), while we have shown that the geometric complexity (thickness and also refinement complexity) of embeddings in these dimensions is at least exponential.

## 5. Further examples of -complexes in , and links in .

Section 4 introduced a family of -complexes which “barely” embed into : any embedding is necessarily exponentially thin. Here we examine the case in more detail. The main purpose of this section is to discuss and formulate a number of open questions about related families of -complexes in and also about classical links in -space. Their definition is based on the lower central series and other more general iterated word constructions.

This discussion of -complexes is motivated by two problems. First, the bound on thickness in section 4 is based on the integral formula for the linking number. It is an interesting question whether similar bounds on thickness may be obtained when the linking numbers vanish but a (suitably interpreted) higher-order Massey product is non-trivial. There are integral formulas that may be used to compute Massey products, but they do not seem to be directly suitable for getting a thickness estimate as above. At the end of the section we also pose a related question for links in -space where Massey products correspond to Milnor’s -invariants [18].

Another motivation is suggested by Milnor’s representation [18] of a link which for convenience of the reader is reproduced in figure 2. This is a Brunnian link, in particular the sublink formed by the components is the -component unlink, so the fundamental group of its complement is the free group generated by meridians to these components. The remaining curve represents a -fold commutator in this free group.