##
**Effective topological complexity of spaces with symmetries.**
*(English)*
Zbl 1385.55003

Let \(PX\) denote the space of all paths in the topological space \(X\), and let \(\pi:PX\to X\times X\) denote the end-point fibration \(\pi(\gamma) = (\gamma(0),\gamma(1))\). A motion planner on an open subset \(U\subseteq X\times X\) is a local section \(s:U\to PX\) of \(\pi\). The topological complexity of \(X\), denoted \(\mathsf{TC}(X)\), is defined to be the minimum integer \(\ell\) such that \(X\times X\) can be covered by open sets \(U_1,\ldots , U_\ell\), each of which admits a motion planner. The invariant \(\mathsf{TC}(X)\) was introduced by M. Farber as part of his topological study of the motion planning problem in robotics; see the papers [M. Farber, Discrete Comput. Geom. 29, No. 2, 211–221 (2003; Zbl 1038.68130); Topology Appl. 140, No. 2–3, 245–266 (2004; Zbl 1106.68107)], where several basic properties (such as homotopy invariance) and calculations are given.

The invariant \(\mathsf{TC}(X)\) has attracted much attention from homotopy theorists, and so it should come as no surprise that several equivariant generalizations have been studied which take into account the action of a group \(G\) on \(X\). Among these are the “equivariant topological complexity” of H. Colman and M. Grant [Algebr. Geom. Topol. 12, No. 4, 2299–2316 (2012; Zbl 1260.55007)], the “invariant topological complexity” of W. Lubawski and W. Marzantowicz [Bull. Lond. Math. Soc. 47, No. 1, 101–117 (2015; Zbl 1311.55004)] and the “strongly equivariant topological complexity” of A. Dranishnikov [Topology Appl. 179, 74–80 (2015; Zbl 1304.55003)]. Each of these versions asks for motion planners which are equivariant with respect to various induced actions of \(G\) on \(X\times X\) and \(PX\), and as a result they can often be (much) larger than \(\mathsf{TC}(X)\).

The authors of the present paper take a different view, that symmetries of the space \(X\) should be used to help, rather than hinder, the task of motion planning. They therefore define a new \(G\)-homotopy invariant, denoted \(\mathsf{TC}^{G,\infty}\), which they call the “effective topological complexity”, which satisfies \(\mathsf{TC}^{G,\infty}(X)\leq\mathsf{TC}(X)\) for all \(G\)-spaces \(X\). The notation arises from the fact that \(\mathsf{TC}^{G,\infty}\) is the limit of a non-increasing sequence \((\mathsf{TC}^{G,k})_{k=1}^\infty\), with \(\mathsf{TC}^{G,1}=\mathsf{TC}\).

The definition of \(\mathsf{TC}^{G,k}\) is as follows. Given a \(G\)-space \(X\), write \[ \mathcal{P}_k(X) = \{ (\gamma_1,\ldots , \gamma_k)\in (PX)^k \mid G\gamma_i(1)=G\gamma_{i+1}(0)\text{ for }1\leq i\leq k-1\} \] for the space of paths with \(k-1\) ‘jumps’ by elements of \(G\). The map \(\pi_k:\mathcal{P}_k(X)\to X\times X\) defined by \(\pi_k(\gamma_1,\ldots , \gamma_k)=(\gamma_1(0),\gamma_k(1))\) turns out to be a fibration. Then \(\mathsf{TC}^{G,k}(X)\) is defined to be the minimum integer \(\ell\) such that \(X\times X\) can be covered by open sets \(U_1,\ldots , U_\ell\), each of which admits a local section of \(\pi_k\).

The rest of the paper gives several properties and calculations of this new invariant. These include: a cohomological lower bound in terms of the nilpotency of the kernel of the cup product map in \(H^*(X/G;\Bbbk)\), when \(G\) is finite and \(\Bbbk\) is a field of characteristic zero or prime to the order of \(G\); a product formula, bounding the effective topological complexity of a product in terms of the effective topological complexities of the factors; computations of the effective topological complexities of \(\mathbb{Z}/p\)-spheres. The paper concludes with several open questions.

A survey article [A. Ángel and H. Colman, Contemp. Math. 702, 1–15 (2018; Zbl 1387.55006)] has since been published, which contains a comparison of the four equivariant generalizations of topological complexity mentioned in this review.

The invariant \(\mathsf{TC}(X)\) has attracted much attention from homotopy theorists, and so it should come as no surprise that several equivariant generalizations have been studied which take into account the action of a group \(G\) on \(X\). Among these are the “equivariant topological complexity” of H. Colman and M. Grant [Algebr. Geom. Topol. 12, No. 4, 2299–2316 (2012; Zbl 1260.55007)], the “invariant topological complexity” of W. Lubawski and W. Marzantowicz [Bull. Lond. Math. Soc. 47, No. 1, 101–117 (2015; Zbl 1311.55004)] and the “strongly equivariant topological complexity” of A. Dranishnikov [Topology Appl. 179, 74–80 (2015; Zbl 1304.55003)]. Each of these versions asks for motion planners which are equivariant with respect to various induced actions of \(G\) on \(X\times X\) and \(PX\), and as a result they can often be (much) larger than \(\mathsf{TC}(X)\).

The authors of the present paper take a different view, that symmetries of the space \(X\) should be used to help, rather than hinder, the task of motion planning. They therefore define a new \(G\)-homotopy invariant, denoted \(\mathsf{TC}^{G,\infty}\), which they call the “effective topological complexity”, which satisfies \(\mathsf{TC}^{G,\infty}(X)\leq\mathsf{TC}(X)\) for all \(G\)-spaces \(X\). The notation arises from the fact that \(\mathsf{TC}^{G,\infty}\) is the limit of a non-increasing sequence \((\mathsf{TC}^{G,k})_{k=1}^\infty\), with \(\mathsf{TC}^{G,1}=\mathsf{TC}\).

The definition of \(\mathsf{TC}^{G,k}\) is as follows. Given a \(G\)-space \(X\), write \[ \mathcal{P}_k(X) = \{ (\gamma_1,\ldots , \gamma_k)\in (PX)^k \mid G\gamma_i(1)=G\gamma_{i+1}(0)\text{ for }1\leq i\leq k-1\} \] for the space of paths with \(k-1\) ‘jumps’ by elements of \(G\). The map \(\pi_k:\mathcal{P}_k(X)\to X\times X\) defined by \(\pi_k(\gamma_1,\ldots , \gamma_k)=(\gamma_1(0),\gamma_k(1))\) turns out to be a fibration. Then \(\mathsf{TC}^{G,k}(X)\) is defined to be the minimum integer \(\ell\) such that \(X\times X\) can be covered by open sets \(U_1,\ldots , U_\ell\), each of which admits a local section of \(\pi_k\).

The rest of the paper gives several properties and calculations of this new invariant. These include: a cohomological lower bound in terms of the nilpotency of the kernel of the cup product map in \(H^*(X/G;\Bbbk)\), when \(G\) is finite and \(\Bbbk\) is a field of characteristic zero or prime to the order of \(G\); a product formula, bounding the effective topological complexity of a product in terms of the effective topological complexities of the factors; computations of the effective topological complexities of \(\mathbb{Z}/p\)-spheres. The paper concludes with several open questions.

A survey article [A. Ángel and H. Colman, Contemp. Math. 702, 1–15 (2018; Zbl 1387.55006)] has since been published, which contains a comparison of the four equivariant generalizations of topological complexity mentioned in this review.

Reviewer: Mark Grant (Aberdeen)

### MSC:

55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |

68T40 | Artificial intelligence for robotics |