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Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take symmetry into account in various ways, either by asking that the motion planners themselves admit compatible symmetries, or by exploiting the symmetry to motion plan between functionally equivalent configurations. We will survey the main definitions due to Colman-Grant, Lubawski-Marzantowicz, B\l{}aszczyk-Kaluba and Dranishnikov, and some related notions. We conclude with a short list of open problems., Comment: This is a preprint version of an invited survey article for a book project with the working title "Topology and AI", edited by Michael Farber and Jes\'us Gonz\'alez

The projective span of a smooth manifold is defined to be the maximal number of linearly independent tangent line fields. We initiate a study of projective span, highlighting its relationship with the span, a more classical invariant. We calculate the projective span for all Wall manifolds, which are certain mapping tori of Dold manifolds.

We compare three definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative group cohomologies, giving examples of strict inequality in all cases where it can occur. We prove and make use of Stallings--Swan type results which characterise the groups of equivariant cohomological dimension one. Some of our examples are relevant to Farber's problem which asks for an algebraic characterisation of the topological complexity of discrete groups. In particular, the topological complexity of a group is not in general given by a relative cohomological dimension of the product relative to the diagonal subgroup., Comment: 21 pages

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell--Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases., Comment: v2: Final version, to appear in Topol. Methods Nonlinear Anal. Removed Lemma 2.3, whose proof contained a gap, and replaced with a reference

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant Lusternik-Schnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with ${\sf cat}_G(\pi)={\sf cd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings-Swan type result for families of subgroups which do not contain all finite subgroups., Comment: v3: 23 pages. Added Remark 2.7 and strengthened Example 4.3

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.

The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center., Comment: v4: Final version, accepted for publication in Glasgow Mathematical Journal. 11 pages

We show that the directed topological complexity (as defined by E. Goubault) of the directed $n$-sphere is $2$, for all $n\ge1$., Comment: 6 pages, 1 figure. v2: To appear in the Journal of Applied and Computational Topology

In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\mathsf{TC}}^{\mathcal{D}}$ and the connections between ${\mathsf{TC}}^{\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\mathsf{TC}$-type invariant $\widetilde{\mathsf{TC}}$ that can be used to give an upper bound for $\mathsf{TC}$, $$\mathsf{TC}(X)\le {\mathsf{TC}}^{\mathcal{D}}(X) + \left\lceil \frac{2\dim X -k}{k+1}\right\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov., Comment: 20 pages

We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the Lusternik--Schnirelmann category of a symplectically aspherical manifold equals its dimension. Symplectically hyperbolic manifolds are symplectically atoroidal, as are symplectically aspherical manifolds whose fundamental group does not contain free abelian subgroups of rank two. Thus we obtain many new calculations of topological complexity, including iterated surface bundles and symplectically aspherical manifolds with hyperbolic fundamental groups. Our result also applies in the greater generality of cohomologically symplectic manifolds., Comment: 12 pages, final version, to appear in Math. Z

The Topological Period-Index Conjecture is an hypothesis which relates the period and index of elements of the cohomological Brauer group of a space. It was identified by Antieau and Williams as a topological analogue of the Period-Index Conjecture for function fields. In this paper we show that the Topological Period-Index Conjecture holds and is in general sharp for spin$^c$ 6-manifolds. We also show that it fails in general for 6-manifolds., Comment: v2, 14 pages. Significant improvements were made in the exposition, following comments from an anonymous referee. The main results and the essence of their proofs remain unchanged. v3: Corrected one reference

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$., Comment: This revision contains a few additional comments, among them is Corollary 3.5.4

We present upper and lower bounds for symmetrized topological complexity $TC^\Sigma(X)$ in the sense of Basabe-Gonz\'alez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square $SP^2(X)$. We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres., Comment: v2: 15 pages, 2 figures

A Poincar\'e-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 J\"{a}nich presented a Poincar\'{e}-Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalised setting. In this expository note we review the Poincar\'e-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions., Comment: 13 pages, 3 figures. v2: Final version, to appear in Journal of Geometry and Physics

Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1

We consider the topological complexity of subgroups of Artin's braid group consisting of braids whose associated permutations lie in some specified subgroup of the symmetric group. We give upper and lower bounds for the topological complexity of such mixed braid groups. In particular we show that the topological complexity of any subgroup of the n-strand braid group which fixes any two strands is 2n-3, extending a result of Farber and Yuzvinsky in the pure braid case. In addition, we generalise our results to the setting of higher topological complexity., Comment: v2: 12 pages, added Section 6 on higher topological complexity

It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod 2 homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient., Comment: 4 pages, no figures

We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied in the study of Farber's topological complexity for 2-cell complexes, as well as in the construction of a counter-example to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category., Comment: v4: 39 pages, incorporating comments from a referee. v3: 38 pages. Improved exposition and strengthened our results on TC of 2-cell complexes. v2: 36 pages. Completely re-written, with a new co-author and new title. We now calculate TC of many 2-cell complexes in the metastable range, and exhibit a counter-example to the TC analogue of Ganea's conjecture

We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and more generally for the rational sectional category of a map, in terms of the rational category of a certain auxiliary space. We use our results to deduce consequences for the global (rational) homotopy structure of simply connected, hyperbolic finite complexes., Comment: 22 pages

We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\times B$, whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect trivially. For instance, this assumption is satisfied whenever $A$ and $B$ are complementary subgroups of $\pi_1(X)$. This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zero-divisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group., Comment: 15 pages, 4 figures

In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system consisting of non-colliding autonomous entities performing tasks in space whilst avoiding collisions with several moving obstacles. The Isotopy Extension Theorem from manifold topology implies, somewhat surprisingly, that the complexity of this problem coincides with the complexity of the corresponding problem in which the obstacles are stationary., Comment: 10 pages; Final version, to appear in Proc. Amer. Math. Soc

We show that for any k>1, stratified sets of finite complexity are insufficient to realize all homology classes of codimension k in all smooth manifolds. We also prove a similar result concerning smooth generic maps whose double-point sets are co-oriented., Comment: 5 pages; v2: Very minor corrections, to appear in Topol. Methods Nonlinear Anal

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TC_n(X) is as low as possible, namely n-1., Comment: 9 pages

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension not exceeding 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through generalized concepts of axial maps and, alternatively, non-singular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's., Comment: 16 pages

We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik-Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the non-equivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the non-equivariant topological complexity., Comment: v1: 19 pages; v2: 14 pages. Final version, to appear in Algebraic & Geometric Topology

We prove that the Hatcher-Quinn and Wall invariants of a self-transverse immersion $f\co N^n\imm M^{2n}$ coincide. That is, we construct an isomorphism between their target groups which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher-Quinn invariant of an immersion $f\co N^n\imm M^{2n-1}$., Comment: 14 pages. v2: Final version, to appear in Glasgow Math. J

We show that in every codimension greater than one there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) which cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word `immersion' is replaced by `map with some restricted set of multi-singularities'., Comment: 13 pages; Final version, to appear in Bull. Lond. Math. Soc

We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups., Comment: Final version, to appear in Topology and its Applications

This article has been withdrawn due to an error in a proof of the main result., Comment: Paper withdrawn

The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.

We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity $\TC(X)$ (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one., Comment: 11 pages; minor revisions and 1 added reference; to appear in the Proceedings of the M. M. Postnikov Memorial Conference

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik - Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result., Comment: 10 pages; Revised version (changes only in section 5)

In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions of topological complexity of such algorithms and compare them with each other and with the usual (non-symmetric) concept of topological complexity. Using equivariant cohomology and the theory of Schwarz genus we obtain cohomological lower bounds for symmetric topological complexity. One of our main results states that in the case of aspherical manifolds the complexity of symmetric motion planning algorithms with fixed midpoint map exceeds twice the cup-length. We introduce a new concept, the sectional category weight of a cohomology class, which generalises the notion of category weight developed earlier by E. Fadell and S. Husseini. We apply this notion to study the symmetric topological complexity of aspherical manifolds.

We study motion planning algorithms for collision free control of multiple objects in the presence of moving obstacles. We compute the topological complexity of algorithms solving this problem. We apply topological tools and use information about cohomology algebras of configuration spaces. The results of the paper may potentially be used in systems of automatic traffic control.

We present a formula describing the action of a generalised Steenrod operation of $\Z_2$-type on the cohomology class represented by a proper self-transverse immersion $f\co M\imm X$, in terms of the equivariant double points of $f$ and the characteristic classes of its normal bundle. This generalises a classical result of R.\ Thom: If $\alpha\in H^k(X;\Z_2)$ is the ordinary cohomology class represented by $f\co M\imm X$, then $\mathrm{Sq}^i(\alpha)=f_* w_i(\nu_f)$., Comment: 10 pages; v3 final version, to appear in Acta Math. Hungarica

We prove a geometrical version of Herbert's theorem by considering the self-intersection immersions of a self-transverse immersion up to bordism. This generalises Herbert's theorem to additional cohomology theories and gives a commutative diagram in the homotopy of Thom complexes. The proof uses Koschorke and Sanderson's operations and the fact that bordism of immersions gives a functor on the category of smooth manifolds and immersions., Comment: 16 pages