An important notion in homotopy theory is the Lusternik–Schnirelmann category (LS category) of a topological space. This notion is important not only as a purely mathematical object (see, e.g., the book [CLOT03]) but also in computer science as it is closely related to the topological complexity of motion planning; see, e.g, [Far03, Far04, FM20].
The LS category, , of a topological space is the smallest (if it exists) such that can be covered by open sets such that the inclusion of each of the open sets is nullhomotopic in . One difficulty when working with the LS category is that it is often difficult to determine. For example, determining whether is equivalent to contractibility of . This is known to be undecidable if is a simplicial complex of dimension at least ; see [VKF74, §10] and [Tan16, Appendix] while it is an open problem whether this is decidable for simplicial complexes of dimension .111Via tools in [HAMS93] (using the exercise on page 8) decidability of this problem is equivalent to determining whether a given balanced presentation of a group presents a trivial group. In this form, the problem is mentioned for example in [BMS02].
In order to bound the LS category from above we can use some closely related notions. One of them is the geometric category, which requires that the open sets covering are already contractible. (For more details see again [CLOT03].) If is a polyhedron, this is equivalent to finding the minimum number of subpolyhedra covering
each of which is contractible. This may make estimatingsometimes easier. However, determining whether is still equivalent to contractibility of .
Next step in this direction has been done by Borghini [Bor20] who introduced PL geometric category of a compact (connected) polyhedron . It is the minimum number of PL collapsible subpolyhedra of that cover . (See Section 2 for the precise definition of PL collapsibility.) In this case is equivalent to asking whether is PL collapsible. For -complexes, it is not hard to derive from known results that this is a polynomially checkable criterion (by performing the collapses greedily on an arbitrary triangulation).
Given a -dimensional triangulated polyhedron , it can be checked in polynomial time whether .
Borghini further proved [Bor20] that a connected -dimensional polyhedron has PL geometric category at most . For connected -polyhedra, this means that the only options are , , or . One of the main aims in [Bor20] is to provide a partial characterization of polyhedra with (which we do not reproduce here). All these positive results suggest that computing could be easy for -polyhedra.
However, this turns out to be false. We show that deciding whether is NP-hard. This also means that one should not expect much more than a partial characterization of polyhedra with , at least in algorithmic sense.
Given a -dimensional triangulated polyhedron , it is NP-hard to decide whether .
A useful step towards our proof is that we observe a relation between and shellability (of some triangulation) of . (Shellability will be discussed in detail in Section 2.)
If a -dimensional polyhedron admits a (pure) shellable triangulation, then .
It has been shown by Goaoc, Paták, Patáková, Tancer and Wanger [GPP19] that shellability is NP-hard already for -dimensional simplicial complexes. In addition, the reduction in [GPP19] is quite resistant with respect to subdivisions. Thus, we could hope to prove Theorem 2 in the following way: Consider a complex that appears in the reduction in [GPP19]. If is shellable, then by Proposition 3 ( stands for the polyhedron of ). If we were able to show the other implication: ‘if is not shellable, then ’, we would immediately get a proof of Theorem 2. Unfortunately, the other implication, stated this way, is not true: with some more effort (which we do not do here), it could be shown that every complex from the reduction in [GPP19] satisfies . However, this problem can be circumvented. We construct certain enriched complex (by attaching a torus in a suitable way to every triangle of —it may be slightly surprising that this indeed helps). It turns out that stays for shellable but it grows to for non-shellable (coming from [GPP19]). This will prove Theorem 2.
We point out that Proposition 3 as stated is not really necessary in the proof of Theorem 2. But we state it here as it provides the motivation for our approach as well as it can be seen as a complementary result to the results of Borghini [Bor20] providing some sufficient (or necessary) conditions for .
Simplicial complexes, polyhedra and subdivisions.
We assume that the reader is familiar with simplicial complexes (abstract or geometric). Because we also want to work with polyhedra, we will be using geometric simplicial complexes (with a single exception that the input for any computational problem we consider is the corresponding abstract simplicial complex). That is, a simplicial complex is for us a collection of (geometric) simplices embedded in some such that two simplices intersect in a face of both of them; and a face of any simplex in the complex belongs again to the complex. For more details on simplicial complexes, we refer to textbooks such as [RS82, Mat03].
We work with polyhedra as defined in [RS82]. When we say ‘polyhedron’ we always mean a compact polyhedron. Because every compact polyhedron can be triangulated, an equivalent definition is that a polyhedron is the underlying space of some finite simplicial complex (a.k.a. the polyhedron of ).
A simplicial complex is a subdivision of a complex if and every is a subset of some . Given a subcomplex of , then the subcomplex of corresponding to is the complex .
Collapsibility and PL collapsibility.
Given a simplicial complex , a face is free, if it is contained in a unique maximal face. A complex collapses to a subcomplex by an elementary collapse, if is obtained from by removing a pair of faces where , is free and is the maximal face containing .222Some authors allow more general elementary collapse removing a face and all faces containing it provided that is contained in a unique maximal face. This is only a cosmetic change in the resulting notion of collapsible complex because this more general elementary collapse can be emulated by a sequence of elementary collapses according to our definition. We also say in this case that collapses to through . A simplicial complex collapses to a subcomplex , if there is a sequence of elementary collapses starting collapsing gradually to . A complex is collapsible, if collapses to a point.
A polyhedron is PL collapsible if some triangulation of is a collapsible simplicial complex. Similarly, a simplicial complex is PL collapsible if is PL collapsible polyhedron. Here, we should point out a certain subtlety in the definition of PL collapsible simplicial complex: If is PL collapsible, then there is some triangulation of which is collapsible (in simplicial sense). This triangulation needn’t be a priori a subdivision of . However, by [Hud69, Theorem 2.4] we may assume that actually is a subdivision of . This also affects our earlier definition of . We get if and only if some subdivision of can be covered by collapsible subcomplexes while it does not matter with which triangulation of we start.
In general, collapsibility and PL collapsibility of a simplicial complex differ because PL collapsibility allows an arbitrarily fine subdivision before starting the collapses. In this paper, we need both and we carefully distinguish these two notions.
A simplicial complex is pure if all its maximal faces have the same dimension. A shelling of a pure complex is an ordering of all its maximal faces into a sequence such that for every the subcomplex of with the underlying space is pure and -dimensional. (Here we use the notation for geometric simplicial complexes, thus are actual geometric simplices.) A complex is shellable if it admits a shelling.
There are some similarities between collapsible and shellable simplicial complexes. However, in general, these two notions differ. For example, on the one hand a collapsible complex is always contractible as an elementary collapse keeps the homotopy type but shellable complexes need not be contractible. On the other hand, the union of two triangles meeting in a single vertex is a complex which is collapsible but not shellable. The following description of -complexes admitting a shellable subdivision has been given by Hachomori [Hac08].
Theorem 4 ([Hac08]).
Let be a -dimensional simplicial complex. Then the following statements are equivalent:
The complex has a shellable subdivision.
The second barycentric subdivision is shellable.
The link of each vertex of is connected and becomes collapsible after removing triangles where denotes the reduced Euler characteristic.
Hachimori’s theorem easily implies Proposition 3:
Proof of Proposition 3.
Let be a pure shellable triangulation of . By Theorem 4 there is a list of triangles such that the resulting complex is collapsible after removing these triangles. Now we build an auxiliary complex from by subdividing each of the triangles as in Figure 1. We also build a complex by removing the middle triangle from each subdivided in . The complex is a subcomplex of and it is not hard to see that collapses to . Hence is collapsible as well. Then is one of the two collapsible polyhedra covering . The second polyhedron is obtained by taking the union of and connecting them along the -skeleton of so that the resulting complex is collapsible. ∎
In our auxiliary computations, we will offer need homology groups, including the exact sequence for pairs, the Mayer-Vietoris exact sequence and the Lefsechtz duality. In general, we refer to the literutare such as [Hat02, Mun84] for details (in case of Lefschetz duality, we will recall its statement when used).
In all our computations, we work with homology with -coefficients. When working with simplicial complexes, we use simplicial homology. In particular, when we speak of -chains, then we can identify a -chain with a collection of -simplices (in its support). (Similarly, a -cycle is such a collection with trivial boundary.) In case of polyhedra, we use singular homology. However, we of course implicitly use that the simplicial and singular homology groups are (naturally) isomorphic (for a simplicial complex and its polyhedron).
3 PL collapsibility of 2-complexes
It is a folklore result going back at least to Lickorish (according to [HAMS93]) that simplicial -complexes can be collapsed greedily:
Let be a collapsible -complex. Assume that collapses to a subcomplex . Then is collapsible as well. In particular, it can be checked in polynomial time whether a simplicial -complex is collapsible.
For PL collapsibility we can essentially deduce the same conclusion as for collapsibility as soon as we observe that PL collapsibility of a -complex does not depend on the choice of the subdivision, which also might be a folklore.
Let be a simplicial complex of dimension at most and be a subdivision of . Then is collapsible if and only if is collapsible.
In the proof of the lemma we use the following observation.
Let be a triangle with vertices . Let be an arbitrary subdivision of . Then collapses to a subcomplex given by subdivisions of the edges and .
We greedily perform collapses through free edges of which are not in . Let be the resulting complex. We observe that contains no triangle. Indeed, every edge contained in some triangle of is either an edge of or it has to be contained in both neighboring triangles (otherwise we could continue with collapses). This means, because the dual graph of is connected, that once there is a single triangle of in , then contains all triangles of which is a contradiction.
Thus, contains no triangles and it has the same homotopy type as . That means that is a tree. Now we greedily perform collapses of edges not in through vertices of degree . By essentially the same argument as above, only the edges of remain (otherwise, we would find a cycle in ). ∎
Proof of Lemma 6.
First, we show that if is collapsible, then is collapsible by induction on the number of simplices of (the case of one vertex is trivial). Assume that arises from by the first elementary collapse in some collapsing of . First, assume that it removes an edge and a triangle . Perform the collapses from Observation 7 on obtaining a complex . Then is a subdivision of . Thus, it collapses by induction. The other option is that the first elementary collapse removes some vertex and some edge . Then we obtain a subdivision of by collapses on removing and the subdivided edge in direction from towards .
Now, we show that if is collapsible, then is collapsible again by induction on the number of simplices of . Assume that is collapsible. This implies that contains a free face (a vertex or an edge) which subdivides a face of which again has to be free. We perform a collapse on through obtaining . As in the previous paragraph, we also may collapse to a subdivision of . By Proposition 5 we get that is collapsible. Therefore, is collapsible by induction which also implies that is collapsible. ∎
4 NP hardness of PL geometric category 2
In this section we prove Theorem 2. As we have sketched in the introduction, in our construction we need to attach a torus to every triangle of a certain intermediate complex. We start with the details regarding this attachment.
4.1 Attaching tori
First, let us us consider the standard torus . An important curve in is the longitude where ‘’ stands for some fixed point in .
Definition 8 (Enriched complex ).
Given a simplicial complex , we define the enriched complex as follows. For each triangle we consider a copy of the standard torus with longitude triangulated as in Figure 2. We get as a result of gluing all tori to so that we identify with . In the sequel, we consider as well as all the tori as subcomplexes of .
If admits a covering by two collapsible subcomplexes such that both and contain the whole 1-skeleton of then can be also covered by two collapsible subcomplexes.
Split each to two annuli and as in Figure 2. (Both of them are subcomplexes of and they share on one of their boundaries.) Take as the union of and all annuli for . Then and cover . In addition, they are both collapsible because collapses to as each collapses to . ∎
We continue with the main technical lemma for our reduction.
Let be a polyhedron which is a union of two subpolyhedra and . Assume that is the torus and assume that and intersect exactly in the longitude of . Assume that can be covered by two contractible subpolyhedra . Then and is nullhomologous in as well as in .
Let for . The lemma is implied by the following two claims where all the homology is considered with coefficients.
If , then .
If , then .
If , then , belongs to and is nullhomologous in .
If , then , belongs to and is nullhomologous in .
Indeed, the conjunction of the claims implies that only option is that and thus we can use the conclusions of Claim 10.2. Therefore, it remains to prove the claims. In each of the claims, we only prove the first item as the other one is symmetric.
Proof of Claim 10.1(i).
Let be the regular neighborhood of inside , which is homotopy equivalent to . Then is a surface with boundary. Thus we may apply the Lefschetz duality333Lefschetz duality (see e.g. Theorem 3.43 in [Hat02]) over : Let be an -dimensional compact manifold with boundary . Then for every . obtaining
where the second isomorphism follows from the fact that the homology and the cohomology groups are isomorphic over a field.
Now let be the closure of the complement of in , that is, . By the excision property of homology, and then by (1)
Finally, we consider the long exact sequence of the pair:
The map is induced by the inclusion . Because of (2), the map is surjective. The inclusion can be decomposed into inclusions and . (Note that as and cover .) By functoriality of homology, must be surjective as well. Therefore, . ∎
Proof of Claim 10.2(i).
Let . Consider the Mayer-Vietoris exact sequence:
As we assume that is contractible, we get . Therefore, is surjective (from exactness). As we also assume that
, there is a nonzero vector. We know as is surjective. In particular, . On the other hand, . Therefore, . This gives as we need. Using that is surjective again, we get . Because we actually get and . This gives that is nullhomologous in . ∎
4.2 Construction form [Gpp19]
As we sketched in the introduction, we use the construction from [GPP19] as an intermediate step. Given that this construction is somewhat elaborated, we prefer to state it as a blackbox only mentioning the properties that we need in our reduction.
The NP-hardness in [GPP19] is proved by a reduction form the classical -satisfiability problem. An input for the -satisfiability problem is a -CNF formula , that is, a boolean formula in conjunctive normal form where every clause contains exactly three literals.444A literal is some variable or its negation ; a clause with literals is a (sub)formula of form where are literals. A formula is in conjunctive normal form, if it can be written as where are clauses. The output is the answer whether the formula is satisfiable, that is, whether it is possible to assign the variables TRUE or FALSE so that the formula evaluates to TRUE in this assignment. It is well known that this problem is NP-hard.
Proposition 11 ([Gpp19]).
There is a polynomial time algorithm that produces from a given -CNF formula (with variables) a pure -dimensional complex with the following properties.
contains pairwise disjoint triangulated -spheres , one for each variable.
The second homology group, , is generated by the spheres . In particular, and no triangle outside the spheres is contained in a -cycle.
If is satisfiable, then there are triangles in for every such that becomes collapsible after removing these triangles. In addition, for every , there are at least two options how to pick in . (Such a choice can be done independently in each yielding at least collapsible subcomplexes.)
If an arbitrary subdivision of becomes collapsible after removing some triangles, then is satisfiable.
The proof of the proposition consists mostly of references to [GPP19]. However, a few items are not as explicitly stated in [GPP19] as we need them here, thus we explain in detail how all the items of the proposition can be deduced from the text in [GPP19].
The construction of is given in Section 4 of [GPP19]. The spheres of item (i) are the spheres introduced in §4.3 of [GPP19]. For checking the other items, we first point out that [GPP19, Proposition 12] states that the number of variables, , is equal to the reduced Euler characteristic .
It is stated in Remark 13 in [GPP19] that is homotopy equivalent to the wedge of -spheres; in particular, . Then item immediately follows as the disjoint spheres generate a subspace of dimension in . Unfortunately, Remark 13 is only a side remark in [GPP19] and it is not proved there. Therefore, we explain in Appendix A, Proposition 12, how does Remark 13 of [GPP19] follow from their tools.
Item (iii), using is the content of Proposition 8(ii) in [GPP19] with the addendum that it is also necessary to check the proof: In the beginning of Section 7 of [GPP19], it is specified that the triangles are removed in certain regions . By checking the construction of in §4.3 of [GPP19], these regions are in correct spheres ( in our notation; in the notation of [GPP19]) and in addition there are at least two choices of the removed triangle for every (actually exactly three choices).
Item (iv), using , is exactly the content of Proposition 8(iii). ∎
4.3 The final reduction
Proof of Theorem 2.
Given a 3-CNF formula and its corresponding complex we construct its enriched complex . (See Definition 8.) Theorem 2 is proved by showing that is satisfiable if only if as 3-satisfiability is an NP-hard problem.
is satisfiable can be covered by two collapsible subcomplexes.
If the formula is satisfiable then by Proposition 11(iii) is collapsible after removal of triangles, one from each sphere , and for each there are at least two options, say , how to pick such triangle. Therefore, the subcomplexes
are collapsible subcomplexes of and they cover it. Moreover and contain whole 1-skeleton of . Indeed, the complex is pure thus every edge of is contained in at least one triangle and in addition in at least two triangles if it is an edge in some of the spheres . In order to get or , at most one triangle is removed from each . Therefore, each edge of is still contained in at least one triangle of and in at least one triangle of . Then Observation 9 implies that can be covered by two collapsible subcomplexes.
A subdivision of can be covered by two collapsible subcomplexes is satisfiable.
First, we sketch the idea: Let and be the two collapsible subcomplexes of covering it. We want to verify the assumption in Proposition 11(iv) in order to deduce that is satisfiable. For this, we need a subdivision of such that removing triangles from this subdivsion yields a collapsible complex. In fact, our subdivision will be trivial, thus we need to find triangles in such that their removal yields a collapsible complex. We will take , say, and we will (essentially) deduce that in each there must be such that must miss at least one triangle in the subdivided . These triangles are the triangles we want to remove from . However, we need several intermediate claims to deduce that the resulting complex is indeed collapsible. (We will use the second complex only very sparingly in order to verify the assumptions of Lemma 10.)
Let be the subcomplex of corresponding to in this subdivision. (Let us recall that this means that is formed by simplices such that .) Let .
The complex is a collapsible subcomplex of .
Our aim is to show that collapses to . Then it follows from Proposition 5 that is collapsible.
We pick an arbitrary triangle of . Recall that is the torus attached to . (See Definition 8.) Let be the subcomplex of corresponding to . Note that (the subdivsion of) belongs to by Lemma 10. We also observe that is not a subcomplex otherwise would contain a nontrivial -cycle which is not possible if it is collapsible.
Now we proceed similarly as in the proof of Observation 7. We greedily perform collapses in on simplices of with the exception that we are not allowed to remove the simplices belonging to (the subdivision of) . (See Figure 3 for a realistic example of the intersection of and .) Let be the resulting complex. We first observe that contains no triangles of as at least one triangle is missing and the dual graph to our triangulation of is connected even after removing the dual edges crossing . Therefore, is a graph. Due to our restriction on collapses, subdivided is inside this graph. We observe that no other (graph theoretic) cycle may belong to this graph. Indeed, another cycle would contain an edge which is not in , thus not contained in any triangle of . Therefore, such a cycle could not be filled with a -chain, and thus it would be necessarily homologically nontrivial in which is a contradiction with the fact that is contractible (obtained by collapses from a collapsible complex). Thus, we may conclude that is the subdivided with a collection of pendant trees. However, these pendant trees have to be actually trivial as they get collapsed during the greedy collapses.
Altogether we have collapsed to a complex which agrees with on while we have removed all simplices of except those that belong to . Now we pick another triangle of and we remove (via collapses) the simplices of except those belonging to by an analogous approach. After passing through every triangle of , we get exactly as required. ∎
Now, for any triangle let be the subcomplex of corresponding to this triangle.
For every triangle , is nullhomologous in .
Let . Let be the polyhedron of and all tori of except . Let and . Then , , and satisfy the assumptions of Lemma 10. Then we deduce that is nullhomologous in . Assume that is such that is the first torus to be removed in the proof of Claim 11.1, we can choose so. Then is exactly the polyhedron of in the proof of Claim 11.1. In particular, collapses to . As collapses provide a homotopy equivalence, we deduce that is nullhomologous in as well. ∎
If is a triangle which does not belong to any of the spheres , then is a subcomplex of .
For every , all triangles in except exactly one satisfy that is a subcomplex of .
Let . Due to Claim 11.2, it has to be possible to fill the subdivision of by some -chain in .
If does not belong to any of the spheres , then the only option for is to consist of simplices of . Indeed, if there is another such , then considering as a -chain, we get a nontrivial -cycle with support at least partially outside the spheres which contradicts Proposition 11(ii). Therefore, must be a subcomplex of which concludes (i).
Now for (ii), take . Then has to miss at least one triangle in otherwise subdivided forms a non-trivial -cycle in which is a contraction with Claim 11.1. Assume that in was chosen so that this missing triangle belongs to . Then splits (subdivided) to two hemispheres; one of them is formed by and another is formed by the union of subcomplexes taken over all triangles in different from . By using Proposition 11(ii) again, the only options are that consists of the simplices of one or the other (subdivided) hemispheres. But the hemisphere of is ruled out as misses a triangle of . Thus has to be filled by the other hemisphere. Then we conclude (ii) for all simplices in except exactly as required. ∎
In the light of Claim 11.3(ii), let be the unique triangle of such that is not a subcomplex of . Let be the subcomplex of obtained by removing all triangles and let be the subcomplex of corresponding to . Note that Claim 11.3 implies that is a subcomplex of . See Figure 4 for comparison of , and after using Claim 11.3.
collapses to .
The complexes and differ only so that may contain some simplices of for some (except those that subdivide ) which are not in .
Now, we continue analogously as in the proof of Observation 7 or Claim 11.1. We greedily collapse all simplices of in except those that subdivide . We first deduce that the resulting complex contains no triangles of as at least one triangle was missing in the beginning. Then we deduce that there is no graph-theoretic cycle among simplices of except the one corresponding to by the same argument as in the proof of Claim 11.1 (using that is collapsible). Then, we deduce that among the simplices of only the simplices subdividing remain in the complex. After repeating this approach for every we obtain . ∎
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Appendix A Appendix
Our aim in the appendix is to verify Remark 13 in [GPP19] which is stated but not proved in [GPP19]. The exact statement we need is given by the following proposition. We will provide all the necessary detail in order to verify correctness of Remark 13 of [GPP19]. On the other hand, we warn the reader that our proof is not self-contained but it relies on the construction of and partially the notation in [GPP19]; thus it is necessary to consult the contents of [GPP19].
The complex from [GPP19] is homotopy equivalent to the wedge of -spheres (where is the number of variables).
In the proof, we need the following simple lemma.
Let be simplicial complexes. Assume that and are contractible, then and are homotopy equivalent.
It is well known that contracting a contractible subcomplex is a homotopy equivalence [Mat03, Proposition 4.1.5]. Therefore, we get
as required. ∎
Proof of Proposition 12.
We follow essentially in verbatim the proof of Proposition 12 in [GPP19]. The only difference is that we use Lemma 13 instead of the weaker statement in [GPP19]: If and are contractible, then where stands for the reduced Euler characteristic.
As described in the proof of Proposition 12 in [GPP19], the complex can be transformed into certain complex by a series of steps when we decompose some intermediate complex as where and are contractible, and then we replace the intermediate complex with . Therefore, using Lemma 13 we get that the resulting complex , after performing all these steps is homotopy equivalent to .
By a further homotopy equivalence [GPP19] obtain another complex which is already (obviously) homotopy equivalent to the wedge of -spheres. Therefore, is homotopy equivalent to the wedge of -spheres. ∎