Higher topological complexity of planar polygon spaces having small genetic codes

A motion planning algorithm in a path-connected topological space $X$ is defined as a section of the free path space fibration $\pi:X^{I}\to X\times X,$ where $\pi(\gamma)=(\gamma(0),\gamma(1)),$ and $X^{I}$ denotes the free path space of $X$, equipped with the compact open topology. To analyze the complexity of designing a motion planning algorithm for the configuration space $X$ of a mechanical system, Farber [11] introduced the notion of topological complexity. The topological complexity of a space $X$, denoted by $\mathrm{TC}(X)$, is defined as the smallest positive integer $r$ for which $X\times X$ can be covered by open sets $\{U_{1},\dots,U_{r}\}$, with each $U_{i}$ admitting a continuous local section of $\pi$. The number $\mathrm{TC}(X)$ represents the minimal number of continuous rules required to implement a motion planning algorithm in the space $X$. Farber [11, Theorem 3] showed that $\mathrm{TC}(X)$ is a numerical homotopy invariant of a space $X$.

In [16], Rudyak introduced the higher analogue of topological complexity. For a path-connected space $X$, consider the fibration $\pi_{k}:X^{I}\to X^{k}$ defined by

The higher topological complexity of $X$, denoted by $\mathrm{TC}_{k}(X)$, is the smallest positive integer $r$ for which $X^{k}$ can be covered by open sets $\{U_{1},\dots,U_{r}\}$, such that each $U_{i}$ admits a continuous local section of $\pi_{k}$ . Note that when $k=2$, the higher topological complexity $\mathrm{TC}_{k}(X)$ coincides with $\mathrm{TC}(X)$. The above definition makes also sense for $k=1$, but $\mathrm{TC}_{1}(X)$ is always equal to $1$ (see [16]). Similar to $\mathrm{TC}(X)$, the invariant $\mathrm{TC}_{k}(X)$ is associated with motion planning problems, where the input includes not only an initial and final point but also an additional $k-2$ intermediate points.

An older homotopy invariant of topological spaces, known as the Lusternik-Schnirelmann (LS) category, was introduced by Lusternik and Schnirelmann in [15]. The LS-category of a space $X$, denoted by $\mathrm{cat}(X)$, is the smallest positive integer $r$ such that $X$ can be covered by open subsets $V_{1},\dots,V_{r}$ such that each inclusion $V_{i}\hookrightarrow{}X$ is null-homotopic. The following inequalities were established in [1] shows how the LS category and higher topological complexity are related:

Determining the precise values of these invariants is usually a challenging task. Over the past two decades, several mathematicians have significantly contributed to approximating these invariants with bounds. To be more specific, Farber [11, Theorem 7] gave a cohomological lower bound on the topological complexity, and this concept was extended to the higher topological complexity by Rudyak in [16, Proposition 3.4]. Let $\Delta_{k}:X\to X^{k}$ be the diagonal map and $\mathrm{zcl}_{k}(X)$ denote the cup-length of $\mathrm{ker}(\Delta_{k}^{*})$, where cup-length is the length of the largest nontrivial product of cohomology classes. Rudyak in [16, Proposition 3.4] proved the following inequality, generalizing Farber’s cohomological lower bound on the topological complexity.

We refer to the non-negative integer $\mathrm{zcl}_{k}(X)$ as the higher zero-divisors-cup-length of $X$. On the other hand, if $\mathrm{cl}(X)$ denotes the cup-length of a cohomology ring of $X$. Then there is an inequality

(see [2, Proposition 1.5]). For a paracompact space, there is a usual dimensional upper bound on the higher topological complexity given as follows:

Moreover, an additional upper bound for $\mathrm{TC}_{k}(X)$ is formulated in terms of the homotopy dimension of the space (see [1, Theorem 3.9]).

In this paper, we are interested in studying the (higher) motion planning problem for planar polygon spaces. These spaces can be viewed as equivalence classes of oriented planar $n$-gons with consecutive side lengths $\alpha_{1},\dots,\alpha_{n}\in(0,\infty)$ for some $n\in\mathbb{N}$, identified under translation, rotation, and reflection. Practically, one can regard the sides of a polygon as the linked arms of a robot. Then the higher topological complexity of a polygon space describes the minimum number of motion-planning rules required to maneuver the robot from an initial configuration to a final configuration, ensuring the motion passes through a fixed sequence of intermediate configurations.

We now briefly recall what planar polygon spaces are and some basic information. A length vector is a tuple of positive real numbers. The planar polygon space associated with a length vector $\alpha=(\alpha_{1},\dots,\alpha_{n})$, denoted by $\mathrm{M}_{\alpha}$, is the collection of all closed piecewise linear paths in the plane with side lengths $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$, considered up to orientation-preserving isometries. Equivalently, we can describe $\mathrm{M}_{\alpha}$ as

where $S^{1}$ is the unit circle and the group of orientation-preserving isometries $\mathrm{SO}_{2}$ acts diagonally. The planar polygon space (associated with $\alpha$) viewed up to isometries is defined as

where $\mathrm{O}_{2}$ is the orthogonal group acting on $\mathbb{R}^{2}$, including all linear isometries.

The spaces $\mathrm{M}_{\alpha}$ and $\overline{\mathrm{M}}_{\alpha}$ are also called moduli spaces of planar polygons. It is clear that $\mathrm{M}_{\alpha}$ is a double cover of $\overline{\mathrm{M}}_{\alpha}$. A length vector $\alpha$ is called generic if $\sum_{i=1}^{n}\pm\alpha_{i}\neq 0$. For such a length vector $\alpha$, the moduli spaces $\mathrm{M}_{\alpha}$ and $\overline{\mathrm{M}}_{\alpha}$ are closed, smooth manifolds of dimension $m:=n-3$ (see [12]). In the rest of this paper, the length vectors are assumed to be generic unless stated otherwise.

The planar polygon spaces have been studied extensively. For example, Farber and Schütz [10] proved that the integral homology groups of $\mathrm{M}_{\alpha}$ are torsion-free. They also described the Betti numbers in terms of the combinatorial data associated with the length vector. The mod-$2$ cohomology ring of $\overline{\mathrm{M}}_{\alpha}$ was computed by Hausmann and Knutson in [13].

The program of studying the motion planning problem for planar polygon spaces was initiated by Donald Davis. In his several works [6, 7, 8, 9], he has shown that in most cases $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})$ is either $2m$ or $2m+1$. The estimates for $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})$, are primarily determined by the usual dimensional upper bound and cohomological lower bound, respectively, with only a few exceptions. Davis utilized the genetic code description to investigate these bounds, noting that the homeomorphism type of $\overline{\mathrm{M}}_{\alpha}$ is determined by its genetic code $\alpha$. In this paper, we initiate the study of the higher topological complexity of planar polygon spaces, focusing on spaces with small genetic codes. We establish higher analogues of Davis’s results [7] through a systematic investigation, leveraging a novel small genetic code description for the mod-$2$ cohomology ring of $\overline{\mathrm{M}}_{\alpha}$.

In this section, we provide the necessary background on planar polygon spaces, which is essential for stating and proving our upcoming results.

We denote the set $\{1,\dots,r\}$ by $[r]$, for $r\in\mathbb{N}$. There are two important combinatorial objects associated with the length vector $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$.

We may simply use "short" as a shorthand for $\alpha$-short when the context is clear. The collection of short subsets can be quite large. However, there is another combinatorial object associated with length vectors that further compacts the short subset data. It is important to note that the diffeomorphism type of a planar polygon space does not depend on the ordering of the side lengths. Therefore, we assume that the length vector satisfies $\alpha_{1}\leq\alpha_{2}\leq\dots\leq\alpha_{n}$.

For a length vector $\alpha$, consider the collection of subsets of $[n]$ :

along with the partial order $\leq$ defined as follows:

If $A_{1},A_{2},\dots,A_{k}$ are the maximal elements of $S_{n}(\alpha)$ with respect to $\leq$ then the genetic code of $\alpha$ is denoted by $\langle A_{1},\dots,A_{k}\rangle$. Each $A_{i}$’s is called a gene and a subset $A_{i}\setminus\{n\}$ is called gee for each $1\leq i\leq k$ (see [7] for more details).

Let $G$ be the genetic code of $\alpha$. Then it is easy to see that $G$ uniquely determines the collection $S_{n}(\alpha)$. Consequently, it follows from [14, Lemma 1.2] that if the genetic codes of length vectors $\alpha$ and $\beta$ are the same, then the corresponding planar polygon spaces are diffeomorphic.

We now recall the mod-$2$ cohomology algebra $H^{*}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$ of $\overline{\mathrm{M}}_{\alpha}$ in terms of the genetic code information. This result was originally established by Hausmann and Knutson in [13, Corollary 9.2], and later reinterpreted by Davis (see [7, Theorem 2.1]) as follows.

The following result is an immediate application of the previous theorem, which is crucial in proving some of the results in the next section.

We now set up some notations that will be used throughout the paper. Let $p_{j}:X^{k}\rightarrow X$ be the projection onto the $k$-th factor. There is an induced map on the cohomology ring

where $\mathbb{K}$ is a field. Now considering $X=\overline{\mathrm{M}}_{\alpha}$ and $\mathbb{K}=\mathbb{Z}_{2}$ we set the following notions

1. (1)

   $R_{j}:=p_{j}^{*}(R)$

2. (2)

   $w_{j}:=p_{j}^{*}(V_{1})$

3. (3)

   $\bar{R}_{j}:=R_{j}+R_{j-1}$

4. (4)

   $\bar{w}_{j}:=w_{j}+w_{j-1}$

Observe that $\bar{R}_{j}\in\mathrm{ker}(\Delta_{k}^{*})$ and $\bar{w}_{j}\in\mathrm{ker}(\Delta_{k}^{*})$, where $\Delta_{k}:X\to X^{k}$ is the diagonal map.

We now compute the LS category of planar polygon spaces and obtain bounds on the higher topological complexity of $\overline{\mathrm{M}}_{\alpha}$ whose dimension is $m=n-3$.

In most cases, to establish lower bounds for $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})$, Davis utilized two key group homomorphisms, $\phi$ and $\psi$. Specifically, $\phi:H^{m}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})\rightarrow\mathbb{Z}_{2}$ is the Poincaré duality isomorphism, and by choosing an uniform homomorphism $\psi:H^{m-1}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})\rightarrow\mathbb{Z}_{2}$ such that $(\phi\otimes\psi)(z)\neq 0$, where $z\in H^{2m-1}(\overline{\mathrm{M}}_{\alpha}\times\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$ a product of $(2m-1)$-many cohomology classes from the $\mathrm{ker}(\Delta^{*})$. Thus, one can conclude that $\mathrm{zcl}(X)\geq 2m-1$.

For example, if the genetic code of a length vector $\alpha$ is $\langle\{a,a+b,n\}\rangle$, then the uniform homomorphism in the sense of Davis is a homomorphism $\psi:H^{m-1}(\overline{\mathrm{M}}_{\alpha})\to\mathbb{Z}_{2}$ satisfying

• •

  $\psi(V_{i})=\psi(V_{j})$ if $i,j\leq a$ or if $a<i,j\leq a+b$, and

• •

  $\psi(V_{i}V_{j})=\psi(V_{i}V_{k})$ if $j,k\leq a$ or if $a<j,k\leq a+b$.

(Above, we have omitted writing powers of $R$ accompanying $V$’s as done by Davis in [7]).

Our main strategy is to determine lower bounds for $\mathrm{TC}_{k}(\overline{\mathrm{M}}_{\alpha})$ as follows. We will construct classes $\xi\in H^{2m-1}(\overline{\mathrm{M}}_{\alpha}\times\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})^{\otimes l}$ when $k=2l$ and $\xi^{\prime}\in H^{2m-1}(\overline{\mathrm{M}}_{\alpha}\times\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})^{\otimes l}\otimes H^{m}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$ when $k=2l+1$ such that $(\phi\otimes\psi)^{\otimes l}(\xi)\neq 0$ and $((\phi\otimes\psi)^{\otimes l}\otimes\phi)(\xi^{\prime})\neq 0$. Here the map $(\phi\otimes\psi)^{\otimes l}$ is given by the usual tensor product of maps as

defined by $(\phi\otimes\psi)^{\otimes l}(w_{1}\otimes\cdots\otimes w_{l})=(\phi\otimes\psi)(w_{1})\cdots(\phi\otimes\psi)(w_{l})$ for $w_{1},\dots,w_{l}\in H^{m}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})\otimes H^{m-1}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$, and $(\phi\otimes\psi)(w^{\prime}\otimes w^{\prime\prime})=\phi(w^{\prime})\psi(w^{\prime\prime})$ for $w^{\prime}\in H^{m}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$, $w^{\prime\prime}\in H^{m-1}(\overline{\mathrm{M}}_{\alpha};\mathbb{Z}_{2})$. Also, we will use the following homomorphisms

where the $2j-1$ th and $2j$ th entries are the maps $\phi$ and $\psi$ respectively, for $1\leq j\leq l$.

In [7], Davis has explicitly shown that $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})\in\{2m,2m+1\}$, for the genetic code $\alpha$ having gene sizes from the table below. For genes of size up to $4$, the following table depicts the number of possible genetic codes for the first few values of $n$.

A genetic code $\langle S,S^{\prime}\rangle$ is said to be of Type 1 if $1\in S\cap S^{\prime}$. For the case of $n=7$ and onward, we need the notion of Type 2. These are the genetic codes of specified size and which are not of Type 1 and their gees are those of a genetic code with $n=7$. Note that there are $27$ genetic codes of Type $2$ as highlighted in Table˜1. We refer the reader to [7, Table 2] for the list of all possible gees of the genetic codes of Type $2$ cases. For example, $\langle\{1,2,4,n\},\{3,4,n\},\{2,5,n\},\{1,6,n\}\rangle$ is the genetic code of Type $2$ having genes of size $4$, $3,3,3$ (see [7] for more details).

In this section, we obtain sharp lower bounds on $\mathrm{TC}_{k}(\overline{\mathrm{M}}_{\alpha})$ when the genetic code of $\alpha$ has a gene of size $2$.

Since $\overline{\mathrm{M}}_{\alpha}$ is a connected smooth manifold of dimension $m=n-3$, we know that $\mathrm{TC}_{k}(\overline{\mathrm{M}}_{\alpha})\leq km+1$. In [7, Theorem 2.2], Davis has shown that the $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})\in\{2m,2m+1\}$ if the genetic code of $\alpha$ is $\langle\{a,n\}\rangle$. We will now generalize this result in the higher setting.

For genetic codes having a gene of size two of the length vector $\alpha$, Davis in [7, Theorem 2.3] has shown that $\mathrm{TC}(\overline{\mathrm{M}}_{\alpha})\in\{2m,2m+1\}$. We will now prove a higher analogue of this result. In particular, we have: