key: cord-0689673-wnglmr7e authors: Tanaka, Kohei title: Parametrized topological complexity of poset-stratified spaces date: 2021-12-18 journal: J Appl Comput Topol DOI: 10.1007/s41468-021-00085-z sha: 7a8e45df72548225bf10fd3a6a4e9ee67f13da94 doc_id: 689673 cord_uid: wnglmr7e In this paper, parametrized motion planning algorithms for a fiberwise space [Formula: see text] over a poset P are studied. Such an algorithm assigns paths in a space X decomposed into subspaces with the index set P, that do not cross the boundaries of the separated regions. We compute the parametrized topological complexity of [Formula: see text] , which is one less than the minimal number of local parametrized motion planning algorithms used for designing non-cross-border robot motions in X. The robotic motion planning problem considers how robots move from an initial point to a final point. The central theme in the motion planning problem is to assign a path that connects x and y to each pair (x, y) of points in the space. Farber introduced a numerical invariant TC(X ) (Farber 2003) , called the topological complexity of a space X , which indicates the complexity of the design of motion planning algorithms in X . The equality TC(X ) = n implies that we need at least n + 1 local motion planning algorithms to move robots in X . The author would like to thank the reviewers for helpful comments and suggestions. This work was partially supported by JSPS KAKENHI Grant Number JP20K03607. B Kohei Tanaka tanaka@shinshu-u.ac.jp In contrast, various efficient motion planning algorithms such as symmetric motion (Farber and Grant 2007; Basabe et al. 2014) , monoidal (reserved) motion Sakai 2010, 2012) , equivariant motion (Colman and Grant 2012; Dranishnikov 2015) , and directed motion planning algorithms (Goubault et al. 2020; Borat and Grant 2020) have been developed. Recently, Cohen, Farber, and Weinberger introduced parametrized motion planning algorithms for fibrations to study collision-free motion planning (Cohen et al. 2021) , [CFW] . The original definition of parametrized topological complexity TC(π ) of a fibration π : E → B was defined as the sectional category of the associated fibration : E I B → E 2 B , (γ ) = (γ (0), γ (1)). Here, E 2 B = E × B E is the fiberwise product over B, and E I B consists of paths γ : I = [0, 1] → E such that π • γ is constant, i.e., γ maps into the fiber π −1 (b) for some b ∈ B. In other words, TC(π ) is defined as one less than the smallest number of open sets covering E 2 B with local homotopy sections of . A more general setting for fiberwise spaces (not necessarily fibrations) was considered by García-Calcines [Gar] . The parametrized topological complexity TC(π ) in his sense agrees with the one given by Cohen, Farber, and Weinberger when π is a fibration. In this study, we focus on parametrized motion planning algorithms for a fiberwise space over a poset regarded as a T 0 -Alexandroff space. Such a fiberwise space π : X → P is called a stratified space over P, and e p = π −1 ( p) is called a stratum of p ∈ P. Typical examples of poset-stratified spaces include simplicial complexes or, more generally, (normal) CW complexes with the face posets. A parametrized motion planning algorithm for a poset-stratified space π : X → P assigns a path I → X in a stratum e p to each pair (x, y) of points in e p . This algorithm effectively works for motion planning in a local area. For example, when we go on domestic travel in a country, a parametrized motion planning algorithm on the Earth (decomposed into countries) proposes a route in the country that does not cross the border, while a standard motion planning algorithm may suggest a route through a different country. In recent years, the spread of COVID-19 has imposed severe restrictions on cross-border travel. Parametrized motion planning algorithms on poset-stratified spaces can contribute to the design of intra-country routes for regional tourism. In this study, we compute several examples of TC for poset-stratified spaces. We show that TC(π ) = 0 for the stratified space π : X → P(X ) associated with a simpli-cial complex, or more generally, a regular CW complex X with the face poset P(X ). Furthermore, the parametrized topological complexity of a couple of fundamental stratifications on the cone and the suspension of a space is considered. As a result, TC(X → P(X )) = ∞ for some familiar CW complexes, such as; sphere S n , bouquet B k = ∨ k S 1 , torus T n = n S 1 , and real (complex) projective space RP n (CP n ) with the canonical (minimal) cell decomposition (Example 3.7). This is caused by the definition of TC(π : E → B) using open sets that cover the fiberwise product E 2 B . We can not construct a parametrized motion planning algorithm on an open neighborhood of a 0-cell in the above case of non-regular CW complexes. In order to consider algorithms on more flexible regions, we compute the generalized version TC g (π ) of TC(π ) using arbitrary sets that separate E 2 B . For example, TC g (π : X → P(X )) becomes finite for any finite CW-complex X , unlike the case of the non-generalized version TC. The paper is organized as follows. Section 2 recalls the idea of parametrized topological complexity based on the papers (Cohen et al. 2021) [CFW, Gar] . Furthermore, we review a reconstruction method for stratified spaces from their combinatorial data (Tamaki 2018) to compute TC and TC g . In Sect. 3, we compute TC for poset-stratified spaces including simplicial complexes, regular CW complexes, cones, and suspensions. We show that TC(X → P(X )) = ∞ for some non-regular CW complexes X . In Sect. 4, we compute TC g for the poset-stratified spaces given in Sect. 3. This section briefly reviews the definitions and properties on parametrized topological complexity and stratified spaces. We deal only with path-connected spaces in this paper. First we review the definition and properties on parametrized topological complexity based on prior papers (Cohen et al. 2021) [CFW, Gar] . For a fiberwise space π : E → B, we consider the subspace where c is the constant path at a point in B. For the fiberwise product we have : E I B → E 2 B given by (γ ) = (γ (0), γ (1)). For a subspace U of E 2 B , a continuous (strict) local section U → E I B of is called a parametrized motion planning algorithm on U . The original idea of parametrized topological complexity was defined as the sectional category of the associated map for fibrations (Cohen et al. 2021) [CFW]. Definition 2.1 Let p : E → B be a fiberwise space. The sectional category secat( p) of p is the minimal number n such that B is covered by n +1 open subsets U 0 , · · · , U n , where each U i admits a homotopy local section of p. That is, we have s i : U i → E such that p • s i is homotopic to the inclusion U i → B. If no such number exists, we set secat( p) = ∞. If p : E → B is a (Hurewicz) fibration, the sectional category above agrees with one less than the minimal number of open sets covering B with strict local sections of p. Example 2.2 Several topological invariants are expressed as sectional categories. (1) For a space X with a base point x 0 , the based path space P X = {γ : I → X | γ (0) = x 0 } is equipped with a fibration ev 1 : P X → X given by ev 1 (γ ) = γ (1). The sectional category secat(ev 1 ) agrees with the LS(Lusternik-Schnirelmann) category cat(X ) originally defined as the minimal number n such that X is covered by n + 1 categorical open sets. Here, a subset A of X is categorical if the inclusion A → X is null homotopic. (2) For a space X , the free path space X I = {γ : I → X } is equipped with a fibration ev : X I → X 2 = X × X given by ev(γ ) = (γ (0), γ (1)). The topological complexity TC(X ) is defined as the sectional category secat(ev) (Farber 2003) . The parametrized topological complexity of a fibration π is defined as secat( ) in Cohen et al. (2021) . It should be noted that the associated map always becomes a fibration if π is a fibration. García-Calcines considered the topological complexity for general fiberwise spaces including non-fibrations [Gar]. Definition 2.3 Let π : E → B be a fiberwise space. The parametrized topological complexity TC(π ) is the minimal number n such that E 2 B is covered by n + 1 open subsets U 0 , · · · , U n , where each U i admits a parametrized motion planning algorithm. If no such number exists, we set TC(π ) = ∞. Our TC(π ) in Definition 2.3 agrees with the one given by García-Calcines [Gar] for fiberwise spaces which are not necessarily fibrations. When π is a fibration, our TC(π ) also agrees with the one given by Cohen et al. (2021) [CFW] . Moreover, when the base space B = * consists of a single point, the parametrized topological complexity TC(π ) agrees with the standard topological complexity TC(E) of the total space introduced in Farber (2003) . The fundamental properties of the parametrized topological complexity were compiled in Cohen et al. (2021) [Gar]. A special case of the above proposition is the following corollary. Corollary 2.6 Let π : E → B be a fiberwise space. For a subspace B ⊂ B and the restriction π = π |π −1 (B ) : π −1 (B ) → B , we have TC(π ) ≤ TC(π ). In particular, we have TC(π −1 (b)) ≤ TC(π ) for each b ∈ B. The next property is the homotopy invariance of TC(π ). We consider the following commutative diagram with a map f between fiberwise spaces: Even if π and π are not fibrations, the next proposition holds by the same argument in the proof of (Cohen et al. 2021 Proposition 5.2). Proposition 2.7 Let π : E → B and π : E → B be fiberwise spaces, and let f : E → E be a homotopy equivalence over B (satisfying π • f = π ). If we have a map g : E → E of f over B with a fiberwise homotopy g • f B id E , then TC(π ) ≤ TC(π ). Corollary 2.8 If fiberwise spaces π : E → B and π : E → B are fiberwise homotopy equivalent, then TC(π ) = TC(π ). The topological complexity TC(X ) = 0 if and only if X is contractible. A similar property of the parametrized topological complexity was studied in Cohen et al. (2021) [Gar] for fibrations or fiberwise pointed spaces. Unfortunately, a poset-stratified space is neither a fibration nor a fiberwise pointed space in general. For a general fiberwise space π : E → B, a condition equivalent to TC(π ) = 0 can be described as follows: where H (x, y, t) = (H 1 (x, y, t), H 2 (x, y, t)). Hence, TC(π ) = 0. This subsection reviews the definition and properties on poset-stratified spaces. A poset-stratified space is roughly a space decomposed into subspaces (called strata) with the index poset P such that the inclusion relation on the closures of strata corresponds to the partial order on P. Detailed observations on decompositions and poset-stratified spaces can be found in Tamaki and Tanaka (2019) , Yokura (2020) . A poset P can be regarded as a T 0 -Alexandroff space whose open sets are closed under infinite intersection. Open sets of P are filters (upper sets) of P, that is, subsets closed under the upper order. Conversely, a T 0 -Alexandroff space X can be regarded as a poset with the partial order x ≤ y defined by x ∈ O y , where O y is the minimal open neighborhood of y (the intersection of all open sets including y). From this perspective, we identify T 0 -Alexandroff spaces with posets. We focus on fiberwise spaces π : X → P over posets P. The following definition of poset-stratified spaces is essentially based on Tamaki and Tanaka (2019). A stratified space over a poset P is an open surjective continuous map π : X → P such that each stratum e p = π −1 ( p) is connected and locally closed. Our stratified space π : X → P is required to be an open map because of the compatibility of the orders. Let π : X → P be a fiberwise space over a poset P. The map π is further an open map if and only if it satisfies the following condition: e p ⊂ e q if and only if p ≤ q for any p, q ∈ P (Remark 2.2 Tamaki 2018). A CW complex X has a natural map π : X → P(X ) to the face poset P(X ) given by π(x) = e if x ∈ e. Here, the face poset P(X ) consists of (open) cells of X with the relation e ≤ e if, and only if, e ⊂ e . This map π is not always continuous; however, the normality (the axiom of the frontier) makes π continuous. Recall that a CW complex is normal if each pair of cells e p , e q satisfying e p ∩ e q = ∅ implies e p ⊂ e q . It should be noted that the above term "normal" is a different concept from a space satisfying Axiom T4. Proposition 2.14 (Corollary 3.7 of Tamaki and Tanaka (2019)) If X is a normal CW complex, then the canonical map X → P(X ) to the face poset is a stratified space. When we deal with stratified spaces with infinite strata, the CW condition is a useful property in homotopy theory, as is the case with cell complexes. A stratified space π : X → P is CW if it satisfies the following two conditions: (1) The boundary ∂e p of a stratum e p is covered by a finite number of strata. (2) The space X has the weak topology with respect to the closures of strata {e p | p ∈ P}. A stratified space π : X → P is called locally finite if every point x ∈ X has an open neighborhood U intersecting with a finite number of strata. Lemma 2.16 (Proposition 2.21 of Tamaki (2018)) Any locally finite stratified space π : X → P is CW. We present an overview of the paper (Tamaki 2018 ) about a reconstruction method of stellar stratified spaces by the face categories. This reconstruction method plays a central role in computing the parametrized topological complexity of poset-stratified spaces in this paper. A stellar stratified space is a generalized idea of CW complex introduced in Tamaki (2018), Tamaki and Tanaka (2019) . A CW complex is constructed by gluing disks along the boundaries. On the other hand, a stellar stratified space is constructed by attaching star-shaped cells. Let S be a space. ]. An element x ∈ C X is denoted by (1 − t)y + tv for some y ∈ S and 0 ≤ t ≤ 1. Let S be a space. A subset D ⊂ C S is an aster if for any x ∈ D, the line segment between v and x is contained in D. That is, if x is described as Definition 2.18 Let π : X → P be a stratified space. A characteristic map of a stratum e p is a continuous map ϕ p : D p → e p from an aster D p ⊂ C S p for some space S p that satisfies the following conditions: A stratum e p is called thin if the domain of the characteristic map D p → e p is a thin aster. A stellar stratified space X is a stratified space X → P with a family of characteristic maps {ϕ p } p∈P such that the boundary ∂e p = e p − e p of each stratum e p is covered by the strata indexed by P < p = {q ∈ P | q < p}. A stellar stratified space is called a stellar complex if all of the strata are thin. Let π : X → P be a stellar stratified space. A stratum e p is regular if the characteristic map ϕ p : D p → e p is a homeomorphism. When all of the strata are regular, π is called regular. For a stratified space π : X → P and a subposet Q ⊂ P, we consider the stratified subspace π Q = π |π −1 (Q) : π −1 (Q) → Q. Even if π admits a stellar structure, the restriction may not present a stellar structure on π Q . This is because the restriction does not preserve quotient maps in general (see Tamaki 2018 Section 6). However, the restriction preserves regular stellar structures because the restriction of a homeomorphism is again a homeomorphism onto its image (and a quotient map). Lemma 2.20 Let π : X → P be a regular stellar stratified space, and let Q ⊂ P be a subposet. The restriction π Q : π −1 (Q) → Q is again a regular stellar stratified space. The assumption ensures that the characteristic map ϕ p : q is again an aster because it is obtained by removing a part of the boundary ∂ D p from D p . We have a homeomorphism for each q ∈ Q. It provides a regular stellar structure on π Q . A typical example of stellar stratified space is a cell complex. A cell complex X is a special case of stellar complex. An n-cell e is equipped with a characteristic map ϕ : D n → e, and an n-disk D n can be regarded as a thin aster D n = S n−1 {0} with the boundary ∂ D n = S n−1 . For a poset P, the nerve semi-simplicial set N P consists of totally ordered subsets in P: with the face maps deleting elements. The geometric realization of N P is denoted by B P, and is called the classifying space or order complex of P. This is a special case of the classifying space of a loop-free top-enriched category in Definition 2.28. Any point in B P is uniquely expressed as a pair of a ∈ Int( n ) and a totally ordered subset p 0 < p 1 < · · · < p n in P for some n ≥ 0. The classifying space B P is equipped with a natural continuous map τ : B P → P defined by τ (a, p 0 < · · · < p n ) = p n . We can naturally consider B P as a stratified space over P by τ . Lemma 2.23 If P is a locally finite poset, then τ : B P → P is a locally finite stratified space. Proof Any point x ∈ B P belongs to a unique open simplex indexed by a totally ordered subset p 0 < · · · < p n in P. The open neighborhood x ∈ U = τ −1 (P ≥ p n ) consists of a finite number of strata. Thus, τ is locally finite. The classifying space B P has a natural stellar structure. More generally, stellar structures on the classifying spaces of loop-free top-enriched categories have been considered in Tamaki and Tanaka (2019) . (2019)) The stratified space τ : B P → P over a locally finite poset P admits a stellar structure as follows: A stratum e p consists of open simplices indexed by a totally ordered subset with the maximal element p. The classifying space B(P ≤ p ) can be expressed as B(P 0(< 0)} denotes the k-dimensional upper(lower) hemisphere. The face poset R = P(S n ) consists of 2n + 2 points, and the fiberwise product (S n ) 2 R consists of pairs of points lying in the same cell. According to Farber's computation of TC(S n ) (Farber 2003) , the shortest arc provides a motion planning algorithm s on U = S n × S n − {(x, −x)}. The restriction of s to (S n ) 2 R ⊂ U maps into (S n ) I R . Thus, TC(S n → R) = 0 for any n ≥ 1, while the usual topological complexity TC(S n ) = 1 for n odd, 2 for n even. Theorem 3.4 For a locally finite regular CW complex X , we have TC(X → P(X )) = 0. Proof The product X 2 is a regular CW complex because of the local finiteness. Similarly to the discussions in Theorem 3.2, the fiberwise product X 2 P is a regular CW stellar stratified space over the face poset P = P(X ), and the diagonal : X ∼ = B P → X 2 P is a fiberwise deformation retract of X 2 P . Our desired result follows from Proposition 2.11. The cone C X and the suspension X of a space X admit stratifications C X = {v} ∪ (C X − {v}) and X = {v} ∪ ( X − {v}), where v is the top vertex. The stratifications are quite simple, however, we will show that TC becomes infinite when X is not contractible. As a result, TC(X → P(X )) also becomes infinite for some non-regular CW complex X (see Example 3.7). Let J denote the poset {0 < 1}. A fiberwise space X → J corresponds to choosing an open set (or closed set) in X . We consider the stratified space π J : C X → J on the cone C X = X {v}, where π J (v) = 0 for the top vertex v and π J (Ĉ X) = 1 for C X = C X − {v}. Theorem 3.5 For π J : C X → J , we have Proof First, we assume that X is contractible. A contraction on X implies that C X and I are fiberwise homotopy equivalent over J , where we regard I as a fiberwise space π I : I → J given by π I (1) = 0 and π I [0, 1) = 1. Corollary 2.8 ensures the equality TC(π J ) = TC(π I ) = 0 because I has the linear motion planning algorithm that is parametrized with respect to π I . (2) TC(B k → P(B k )) = ∞ because Corollary 2.6 shows (3) TC(T n → P(T n )) = ∞ because Corollary 2.6 shows (4) TC(RP n → P(RP n )) = ∞ because Corollary 2.6 shows TC(RP n → P(RP n )) ≥ TC(S 1 → {e (0) < e (1) }) = ∞. (5) TC(CP n → P(CP n )) = ∞ because Corollary 2.6 shows The CW complexes X given in the above example have infinite TC(X → P(X )). This is because TC uses open sets that cover the fiberwise product. If we use arbitrary subspaces instead of open sets with parametrized motion planning algorithms, we can consider the generalized version TC g of TC, and obtain a different result from Example 3.7 (see Example 4.20). Cohen et al. (2021) may not hold for non-fibrations by Theorems 3.5 and 3.6. (1) For a fibration π : E → B, the inequality TC(π ) ≤ cat(E 2 B ) holds, as mentioned in (Section 7 Cohen et al. 2021) . However, it is not true for general fiberwise spaces. For a non-contractible space X and the stratified space π J : C X → J , the fiberwise product (C X) 2 J is contractible to (v, v) by the contraction induced from the natural linear contraction on C X to v. Thus, cat((C X) 2 J ) = 0, whereas TC(π J ) = ∞. (2) Proposition 2.9 is not true for general fiberwise spaces. For a circle S 1 with the minimal cell decomposition S 1 → J , the fiberwise product is a torus with two open 1-cells removed, and is homotopy equivalent to S 1 ∨S 1 ∨S 1 . Each stratum is a contractible open cell; however, TC(S 1 → J ) = ∞. With another stratification on the cone C X over J , we have π E : C X → J , defined by π E (X ) = 0 for X = X × {0} and π E (C X + ) = 1 for C X + = C X − X . In this subsection, we will show that TC(π E ) equals to TC(X ) or TC(X ) + 1. We have TC(X ) ≤ TC(π E ) ≤ TC(X ) + 1. Proof It is obvious that TC(X ) = TC(π −1 E (0)) ≤ TC(π E ) by Corollary 2.6. We show the other inequality TC(π E ) ≤ TC(X ) + 1. Let U be an open subset of X 2 with a motion planning algorithm s : U → X I . We consider an open set and a parametrized motion planning algorithms : U → (C X) I J , given bỹ In contrast, the open set C X 2 + = C X + × C X + ⊂ (C X) 2 J admits a motion planning algorithm given by a contraction on C X + . This is a parametrized motion planning algorithm C X 2 + → (C X) I J because it only works in C X + . If TC(X ) = m with open sets U 0 , U 1 , · · · , U m covering X 2 , where each U i admits a motion planning algorithm, then U 0 , · · · , U m , and C X 2 + constitute an open cover of (C X) 2 J with parametrized motion planning algorithms. Hence, TC(π E ) ≤ m + 1 = TC(X ) + 1. A natural question to ask at this point is whether TC(π E ) = TC(X ) or TC(π E ) = TC(X ) + 1. We have not completely solved the problem, but some cases show TC(π E ) = TC(X ). Proposition 3.10 If X is contractible, then TC(π E ) = TC(X ) = 0. Proof The cone C X is fiberwise homotopy equivalent to C{0} = I over J . The interval I admits the linear motion planning algorithm, which is a parametrized motion planning algorithm. Hence, TC(π E ) = TC(I → J ) = 0. For example, in the case of non-contractible space X = S n shows TC(π E ) = TC(X ). In this case, the cone C S n = D n+1 is convex, and we can extend motion planning algorithms in S n to parametrized ones in D n+1 using linear combinations. admits a parametrized motion planning algorithms : U 0 → (C X) I J in the proof of Lemma 3.9. Furthermore, the contractible space C X also has a motion planning algorithm (not necessarily parametrized) h : (C X) 2 → (C X) I . We can construct the following motion planning algorithm: f (a)h(a) . The restriction of γ to W ∪ C X 2 + is a parametrized motion planning algorithm. The open sets U 1 , · · · , U m , and W ∪ C X 2 + cover (C X) 2 J with parametrized motion planning algorithms. Hence, TC(π E ) ≤ m = TC(X ). The next computation immediately follows from Proposition 3.11. Example 3.12 Consider the stratified space π E : D n → J for an n-disk D n = C S n−1 (n ≥ 2). Then we have TC(π E ) = TC(S n−1 ) = 1 for n even, 2 for n odd. For the general case, we leave it as a conjecture. Conjecture 3.13 TC(π E : C X → J ) = TC(X ) for any space X . In contrast, the suspension X is separated into three strata: the upper open cone C X + , equator X , and lower open cone C X − . This is a stratified space π E : X → E over the poset E = {−1 > 0 < 1} defined by π E (C X + ) = 1, π E (X ) = 0, and π E (C X − ) = −1. Note that the previous stratified space π E : C X → J is a stratified subspace of π E : X → E. Proposition 3.14 TC(π E : X → E) = TC(π E : C X → J ) for any space X . Proof It is obvious that We will show the converse inequality. Let U be an open set in (C X) 2 J with a parametrized motion planning algorithm s : U → (C X) I E . For a point a = [x, t] ∈ X , let −a ∈ X denote the vertically symmetrical point [x, −t]. We consider the open sets and U = U ∪ (−U ). The parametrized motion planning algorithm s on U can be extended tos on U as follows:s(a, b)(t) = −s(−a, −b)(t) for (a, b) ∈ −U , and t ∈ I . Hence, TC(π E : X → E) ≤ TC(π E : C X → J ). Consider the stratified space π E : S n → E for an n-sphere S n = S n−1 (n ≥ 2). Then we have that TC(π E ) = TC(S n−1 ) = 1 for n even, 2 for n odd, by Proposition 3.14 and Example 3.12. Example 3.7 suggests that it is impossible to construct parametrized motion planning algorithms on open sets covering the fiberwise product of some CW complexes. However, we can separate the fiberwise product into a finite number of subspaces (not necessarily open sets) with parametrized motion planning algorithms for finite CW complexes. From this perspective, we can consider another version of parametrized topological complexity. We briefly review the generalized version of parametrized topological complexity. This concept was considered for fibrations in Cohen et al. (2021) . Definition 4.1 For a fiberwise space π : E → B, the generalized parametrized topological complexity TC g (π ) is defined as the minimal number n such that the fiberwise product E 2 B admits a partition into n + 1 subsets where each U i admits a parametrized motion planning algorithm. In particular, when B = * consists of a single point, TC g (π ) = TC g (E) is called the generalized topological complexity of E. Clearly, the inequality TC g (π ) ≤ TC(π ) always holds for any fiberwise space π . The converse inequality also holds for nice fiberwise spaces. Theorem 4.2 (Proposition 4.7 in Cohen et al. (2021) , Corollary 2.8 in García-Calcines (2019)) For a locally trivial fibration π : E → B between metrizable separable ANR spaces E and B, we have TC g (π ) = TC(π ). In particular, TC g (X ) = TC(X ) for a space X having the homotopy type of a CW complex. Unfortunately, a poset-stratified space X → P is not a fibration, and the base poset P is not an ANR space in general. The above equality fails for X → P(X ), which is associated with the CW complexes X with the face poset P(X ) given in Examples 3.7 and 4.20. For a fiberwise space π : E → B, TC(π ) = 0 indicates that there exists a global section of : E I B → E 2 B . Hence, TC(π ) = 0 if and only if TC g (π ) = 0. A similar equality to Theorem 4.2 holds for LS category and its generalization. The generalized LS category cat g (X ) is defined as the minimal number n such that X is separated into n + 1 categorical subspaces. Theorem 4.4 (Corollary 2.10 in García-Calcines (2019)) cat g (X ) = cat(X ) for a space X having the homotopy type of a CW complex. As seen in Section 2.1, some fundamental properties of TC(π ) also hold for TC g (π ) because they do not depend on open sets. Proposition 4.5 Let π : E → B be a fiberwise space. (1) TC g ( f * π) ≤ TC g (π ) for the pull-back f * π : E × B X → X for a map f : X → B. In particular, TC g (π | π −1 (A) ) ≤ TC(π ) for A ⊂ B. (2) TC g (π ) ≤ TC g (π ) for a fiberwise space π : E → B with fiberwise maps f : E → E and g : E → E satisfying g • f B id E . In particular, TC g (π ) = TC g (π ) if π and π are fiberwise homotopy equivalent. We will deal with the computation of TC g (π ) for poset-stratified spaces π : X → P given in Sect. 3. The inequality TC g (π ) ≤ TC(π ) implies that TC g (X → P(X )) = 0 for a locally finite regular CW complex X with the face poset P(X ). Furthermore, TC g (τ : B P → P) = 0 for a locally finite poset P. Let us recall the stratifications π J : C X → J and π J : X → J respectively on the cone C X and the suspension X given in Sect. 3.2. Theorem 4.6 For π J : C X → J , we have TC g (π J ) = TC g (X ). Proof It is obvious that TC g (π J ) ≥ TC g (π −1 J (1)) = TC g (Ĉ X) = TC g (X ). We will show the converse inequality. Let TC g (X ) = m with subspaces U 0 , · · · , U m separating X 2 with motion planning algorithms s i : U i → X I . Recall the proof of Lemma 3.9. We extend U i to a subspace U i in (Ĉ X) 2 and s i to a motion planning with parametrized motion planning algorithms, and TC g (π J ) ≤ m = TC g (X ). If P is a finite poset, then X is separated into finite strata and X 2 P = p (e p ) 2 . The next lemma follows immediately from this fact. Lemma 4.7 For a stratified space π : X → P over a finite poset P, we have Theorem 4.8 For π J : X → J , we have Proof The contractible case can be shown by the same argument in the proof of Theorem 3.6. For a non-contractible space X , assume that TC g (π J ) = 0. Remark 4.3 implies TC(π J ) = 0; however, this is a contradiction by Theorem 3.6. Therefore, TC g (π J ) > 0. Moreover, X is separated into two contractible strata {v} andˆ X = X − {v}; therefore, TC g (π J ) ≤ 1 by Lemma 4.7. Hence, TC g (π J ) = 1. Next, we consider TC g for the stratifications π E : C X → J and π E : X → E given in Sec. 3.3. For π E : C X → J , we have TC g (X ) ≤ TC g (π E ) ≤ TC g (X ) + 1. The essential argument here is the same given in Lemma 3.9; however, the proof is simpler because we do not need open sets. We have Therefore, we will show the inequality TC g (π E ) ≤ TC g (X ) + 1. Let TC g (X ) = m and let X 2 be separated into m + 1 subspaces U 0 , · · · , U m with motion planning algorithms. The open cone C X + is contractible and TC g (C X + ) = 0. Lemma 4.7 provides We have the following conjecture similar to Conjecture 3.13. Conjecture 4.10 TC g (π E : C X → J ) = TC g (X ) for any space X . If Conjecture 3.13 is true, the above conjecture is also true for spaces having the homotopy type of a CW complex as the following result asserts. Proposition 4.11 Let X be a space having the homotopy type of a CW complex. If TC(π E : C X → J ) = TC(X ), then we have that TC g (π E : C X → J ) = TC g (X ). Proof Theorem 4.2 provides the following inequalities: Thus, TC(π E ) = TC(X ) implies TC g (π E ) = TC g (X ). The following equality follows from the argument in the proof of Proposition 3.14. Proposition 4.12 TC g (π E : X → E) = TC g (π E : C X → J ) for any space X . In this subsection, we will compute TC g (X → P(X )) for the non-regular CW complexes X given in Example 3.7. While TC(X → P(X )) = ∞, the calculation shows that TC g (X → P(X )) = cat(X ) in this case. A CW complex X is separated into contractible open cells (strata). Lemma 4.7 implies the following proposition. For a finite CW complex X with the face poset P(X ), we have TC g (X → P(X )) ≤ P(X ) − 1, where P(X ) stands for the cardinal of P(X ). Proposition 4.14 For a finite-dimensional CW complex X with the face poset P(X ), we have TC g (X → P(X )) ≤ dim X. Proof We consider the subset Each e is contractible; hence, we have a section U n → X I P(X) of . Thus, we have X 2 P(X) = U 0 U 1 · · · U dim X and TC g (X → P(X )) ≤ dim X . Using the results given above we are able to compute TC g of some CW complexes given in Example 3.7. The minimal cell decomposition on a sphere S n consists of two cells. Proposition 4.13 provides TC g (S n → P(S n )) ≤ 1. Moreover, TC(S n → P(S n )) = ∞ in Example 3.7 implies that there is no global section of . Hence, TC g (S n → P(S n )) = 1. For a bouquet B k = ∨ k S 1 with the cell decomposition given in Example 3.7, Proposition 4.14 provides TC g (B k → P(B k )) ≤ 1. Moreover, TC(B k → P(B k )) = ∞ in Example 3.7 implies that there is no global section of . Hence, TC g (B k → P(B k )) = 1. We now focus on lower bounds of TC g . For a fiberwise pointed space π : E → B with a section s : B → E, García-Calcines presented a fiberwise LS lower bound of TC(π ). We can similarly show the generalized version by the same argument in the proof of Proposition 13 in [Gar] . A subspace U of E is called fiberwise categorical over B if the inclusion U → E is fiberwise homotopic to s • p |U . The fiberwise LS category cat B (E) is the smallest number n such that E is covered by n + 1 fiberwise categorical open subsets over B. Similarly, the generalized fiberwise LS category cat B g (E) is the smallest number n such that E is separated into n + 1 fiberwise categorical subspaces over B. Proposition 4.17 provides a lower bound of TC and TC g respectively for fiberwise pointed space. However, a poset-stratified space is not a fiberwise pointed space, that is, it does not admit a section in general. Let P be a finite connected poset and let X be a T 1 space. Any continuous map P → X must be constant. Proof Let f : P → X be a continuous map. For a comparable pair p < q in P, However, the minimal open neighborhood P ≥ p of p includes q. Hence, V must include q. This contradiction implies that f ( p) = f (q) for any comparable pair p, q. If P is a finite connected poset, then any two points p, q in P are connected by comparable pairs: p = p 0 , p 1 , p 2 , · · · , p n = q, such that p i ≤ p i+1 or p i ≥ p i+1 for each i. Thus, f ( p) = f (q) for any p, q, and f is constant. The above lemma suggests that the fiberwise space X → P(X ) associated with a connected normal finite CW complex X is not pointed, except when X is a single point. It is difficult to construct a section of a continuous map X → P for a poset P and a Hausdorff space X . However, it may be possible for the classifying space B P instead of P. Theorem 4.19 Let X be a cylindrically normal CW complex with face category C = C(X ). If the canonical functor ρ : C → P to the face poset P = P(X ) has a section and B P is contractible, then (1) cat(X ) ≤ TC(π ), (2) cat(X ) = cat g (X ) ≤ TC g (π ), for the stratified space π : X → P. Proof (1) Our aim is to show the following inequalities: TC(π ) ≥ TC(τ * (π )) ≥ cat B P (X × P B P) ≥ cat(X × P B P) ≥ cat(X ). Let us focus on each of the inequalities. (i) First we consider the pullback τ * (π ) : X × P B P → B P of π along the natural map τ : B P → P. Proposition 2.5 ensures TC(τ * (π )) ≤ TC(π ). (ii) Let s : P → C denote a section of ρ : C → P and ι : BC → X denote the natural homeomorphism in Theorem 2.29. We notice that τ * (π ) is a fiberwise pointed space over B P because we have a section B P → X × P B P of τ * (π ) sending a to (ι (Bs(a) ), a). Hence, cat B P (X × P B P) ≤ TC(τ * (π )) by Proposition 4.17. (iii) Fiberwise categorical subsets in a space over the contractible classifying space B P are categorical subsets in the standard sense. Thus, we have cat(X × P B P) ≤ cat B P (X × P B P). (iv) The first projection X × P B P → X has a section X → X × P B P sending x to (x, Bρ(ι −1 (x))). This implies cat(X ) ≤ cat(X × P B P). (2) Similarly, we can show cat g (X ) ≤ TC g (π ). Theorem 4.4 shows the equality cat g (X ) = cat(X ). Thus, cat(X ) ≤ TC g (π ). Now, using all the machinery we have developed above, we are able to compute TC g for the CW complexes given in Example 3.7. Example 4.20 Let us recall the cylindrically normal CW complexes X and their face categories C(X ) given in Example 2.31. The canonical functor ρ : C(X ) → P(X ) admits a section and B P is contractible in each example. Hence, Theorem 4.19 and Propositions 4.13, 4.14 provide cat(X ) ≤ TC g (π ) ≤ min{P(X ) − 1, dim X }. These inequalities determine TC g of the following CW complexes: (1) The canonical functor C(S n ) → P(S n ) admits a section by choosing a point in S n−1 , and B(P(S n )) ∼ = I is contractible. Hence, TC g (S n → P(S n )) = 1, as shown in Example 4.15. (2) The canonical functor C(B k ) → P(B k ) admits a section by choosing a point S 0 , and B(P(B k )) ∼ = ∨ k I is contractible. Hence, TC g (B k → P(B k )) = 1, as shown in Example 4.16. (3) The canonical functor C(T n ) → P(T n ) admits a section as the product of sections of C(S 1 ) → P(S 1 ), and B(P(T n )) ∼ = I n is contractible. Hence, TC g (T n → P(T n )) = n. (4) The canonical functor C(RP n ) → P(RP n ) has a section given by the unit element in Z 2 , and B(P(RP n )) ∼ = n is contractible. Hence, TC g (RP n → P(RP n )) = n. (5) The canonical functor C(CP n ) → P(CP n ) has a section given by the unit element in U (1), and B(P(CP n )) ∼ = n is contractible. Hence, TC g (CP n → P(CP n )) = n. We have computed the (generalized) parametrized topological complexity of various poset-stratified spaces compiled in the following Table 1 . τ : B P Regular CW X X )) = 0 for any locally finite regular CW complex X (Theorem 3.4). Hence, TC(X → P(X )) and TC g (X → P(X )) strongly depend on the cell decomposition on X . A natural question to ask is whether there is a non-regular CW complex X satisfying TC(X → P(X )) = 0 or not. If TC(X → P(X )) = 0 is equivalent to the regularity of X , we can say that the (generalized) parametrized topological complexity for CW complexes measure the difference from regularity. Furthermore, in order to better understand the characteristics of TC and TC g for nonregular CW complexes with the face posets, we need more computational examples. As seen in Example 3.7, TC(X → P(X )) = ∞ for various non-regular CW complexes. Moreover, all examples in Example 4.20 show that TC g (X → P(X )) = cat(X Directed topological complexity of spheres Higher topological complexity and its symmetrization Topology of parametrised motion planning algorithms Parametrized topological complexity of collision-free motion planning in the plane Equivariant topological complexity On topological complexity of twisted products Symmetric motion planning. Topology and robotics Totally normal cellular stratified spaces and applications to the configuration space of graphs A note on covers defining relative and sectional categories Formal aspects on parametrized topological complexity and its pointed version Directed topological complexity Topological complexity is a fibrewise L-S category Erratum to Topological complexity is a fibrewise L-S category The topology of CW complexes Classifying spaces and spectral sequences On the Lusternik-Schnirelmann category of Peano continua The Lusternik-Schnirelmann category of metric spaces Combinatorial and toric homotopy Stellar stratifications on classifying spaces. Algebraic topology and related topics Decomposition spaces and poset-stratified spaces Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations The author states that there is no conflict of interest.