Schedule for: 22w5182 - Topological Complexity and Motion Planning

We develop theory of sequential parametrized motion planning generalising the approach of parametrized motion planning, which was introduced recently. A sequential parametrized motion planning algorithm produces a motion of the system which is required to visit a prescribed sequence of states, in certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixed in advance but rather constitute part of the input of the algorithm. We give a detailed analysis of the sequential parametrized topological complexity of the Fadell - Neuwirth fibration. In the language of robotics, sections of the Fadell - Neuwirth fibration are algorithms for moving multiple robots avoiding collisions with other robots and with obstacles in Euclidean space. Besides, we introduce the new notion of TC-generating function of a fibration, examine examples and raise some interesting general questions about its analytic properties.

Configuration spaces represent the space that can occupy n particles moving along a certain topological space, for example a graph. A very useful tool in understanding these spaces is discrete Morse theory. In this talk I shall give bounds for the higher topological complexity of configuration spaces of graphs without cycles.

In 2005, Farber proved that the topological complexity of the configuration space of $k$ ordered points on a tree is as large as possible, at least for $k$ sufficiently large. He conjectured that the same should hold true for an arbitrary graph. Subsequently, Farber's argument was adapted to more general contexts by various authors. I will discuss the most general such adaptation to date, which treats higher TC and all planar graphs. I will then argue that no further adaptation is possible, articulating a precise sense in which Farber's argument fails fundamentally in the non-planar setting.

The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fibrewise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally we give sufficient conditions so that both notions agree.

There are several possible definitions of the cohomological dimension of a group $G$ relative to a given subgroup $H$. In this talk (which is based on joint work in progress with Ehud Meir and Irakli Patchkoria) I will recall the definitions of the Takasu, Adamson and Bredon relative cohomological dimensions, and discuss comparisons between them. When $K$ is a group acted on by $H$, we obtain competing definitions of the equivariant cohomological dimension $cd_H(K)$ by considering the cohomological dimension of the semi-direct product relative to $H$. Letting $K$ be a free group, we can give examples where the Adamson dimension is strictly less than the Bredon dimension, and where the Bredon dimension is strictly less than the Takasu dimension. I'll also speculate as to what such results could tell us about the topological complexity of groups.

In this talk we will define a simpler notion of symmetric TC more ad hoc to the motion planning problem which was the original motivation for the definition of TC. This is a homotopy invariant that we call bidirectional TC which serves as a first approximation of the symmetrized TC of a space. We will discuss properties of this invariant and show specific instances for which the symmetrized TC can be relaxed to the bidirectional setting. We will show how this approach allows us to define a sequence of TC invariants that approximate the symmetrized TC of a space.

For zoom participants this will be the group photo. Remember turn your camera on and your mic off.

The notion of topological complexity was introduced by M. Farber to estimate the complexity of a motion planning algorithm of a mechanical system. When there is a symmetry on the mechanical system or its configuration space, we may consider developing an equivariant version of topological complexity. In this talk, we study different situations of having symmetry and we propose a motion planning algorithm for symmetric navigator/walker robots using the concept of transversal.

For a topological space $X$, a free involution $\tau\colon X \to X$ and a Hausdorff space $Y$, we exhibit an unexpected connection between the sectional category of the double covers $q \colon X \to X/\tau$ and $q^Y \colon F(Y,2) → B(Y,2)$ from the ordered configuration space $F(Y, 2)$ to its unordered quotient $B(Y, 2) = F (Y, 2)/\Sigma_2$, and the Borsuk-Ulam property (BUP) for the triple $((X, \tau); Y )$. Explicitly, we demonstrate that the triple $((X, \tau ); Y )$ satisfies the BUP if the sectional category of $q$ is larger than the sectional category of $q^Y$ . This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. As application of our results, we present a new lower bound for the index in terms of sectional category. The author would like to thank grant #2022/03270-8, Sao Paulo Research Foundation (FAPESP) for financial support. This is joint work with Daciberg L. Goncalves.

This is joint work with John Guaschi (Normandie Univ, UNICAEN, CNRS, LMNO, France) and Vinicius Casteluber Laass (Federal University of Bahia, Brazil).
By 1930 Ulam posed the question: Given a continuous map $f: S^n \to R^n$, does there exist a point $x\in S^n$ such that $f(x)=f(-x)$? The classical Borsuk-Ulam theorem asserts that, for any continuous map $f: S^n \to R^n$, there exists a point $x\in S^n$ such that $f(x)=f(-x)$ (see [1]). This result has been generalised in many directions and it continues to be an attractive and current topic. More generally, one may consider the situation where the space $S^n$ is replaced by a topological space endowed with a free involution of a finite group $G$, and possibly the target replaced by a space $Y$. In this talk we first present a short survey of the results which were obtained for the case where $X$ is a closed surface and $Y$ is either $R^2$ or a closed surface. We stress the rule of configuration spaces in our approach to study such generalisations of the Borsuk-Ulam property. We then present some new results when $X$ is a closed surface, $G=Z_n$ is the cyclic group of order $n$ and $Y=R^2$. The main result is that for $M$ a compact surface without boundary, and $\tau \colon Z_n \times M \to M$ a free action, the quadruple $(M,Z_n,\tau;R^2)$ has the Borsuk-Ulam property if and only if the following conditions are satisfied:
(1) $n \equiv 2 \ mod \ 4$.
(2) $M_\tau$ is non-orientable, and $(\theta_\tau)_{Ab} (\delta)$ is non trivial.
The main tool used is the configuration space of $R^2$, as well as the intermediate configuration space which is the quotient of the ordered configuration space by the group $Z_n$. The approach suggests a way to study the Borsuk-Ulam property for other finite groups $G$, as well as for maps into a surface.
Here is a non-exhaustive list of references relevant for the presentation.
[1] K. Borsuk, Drei Satze uer die $n$-dimensionale Euklidische Sphare, Fund. Math. 20 (1933), 177–190. <\br> [2] D. L. Goncalves, The Borsuk-Ulam theorem for surfaces, Quaest. Math. 29 (2006), 117–123.
[3] D. L. Goncalves, J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, Top. Appl. 157 (2010), 1742–1759.
[4] D. L. Goncalves, J. Guaschi, The homotopy fibre of the inclusion $F_n(M) \hookrightarrow \Pi_1^n M$ for $M$ either $S^2$ or $RP^2$ and orbit configuration spaces, arXiv:1710.11544, October (2017).
[5] D. L. Goncalves, J. Guaschi, Orbit configuration spaces and the homotopy groups of the pair $(\Pi_1^n M,F_n(M))$ for $M$ either $S^2$ or $RP^2$. Accepted for publication in Israel Journal of Mathematics, (2022).
[6] D. L. Goncalves, J. Guaschi, V. C. Laass, The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero, J. Fixed Point Theory Appl. (2019) 21:65.
[7] V. L. Hansen, Braids and coverings: selected topics, London Mathematical Society Student Texts 18, Cambridge University Press (1989).
[8] K. Murasugi, B. I. Kurpita, A study of braids, Mathematics and its Applications 484, Kluwer Academic Publishers, (1999).

(This is joint work with Isaac Carcacía-Campos and David Mosquera-Lois).
Tanaka ([4]) introduced the notion of categorical LS-category $\mathrm{ccat}\mathcal{C}$ of a small category $\mathcal{C}$. Among other properties, he proved an analog of Varadarajan's theorem for fibrations, relating the LS-categories of the total space, the base and the fiber.
In this talk, we recall the notion of homotopic distance $\mathrm{D}(F,G)$ between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, introduced by us in [2], which has $\mathrm{ccat} \mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},*)$ as a particular case. We consider another particular case, the distance $\mathrm{D}(p_1,p_2)$ between the two projections $p_1,p_2\colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$, which we call the categorical complexity of the small category $\mathcal{C}$.
We prove the main properties of those invariants. As a final result we prove a Varadarajan's theorem for the homotopic distance for Grothendieck bi-fibrations between small categories.
All these notions are inspired by the homotopic distance between continuous maps introduced by us in [3].
[1] Carcacía-Campos, I.; Macías-Virgós, E.; Mosquera-Lois, D. Homotopy invariants in small categories. preprint (2022).
[2] Macías-Virgós, E.; Mosquera-Lois, D. Homotopic distance between functors. J. Homotopy Relat. Struct. 15, No. 3-4, 537-555 (2020).
[3] Macías-Virgós, E.; Mosquera-Lois, D. Homotopic distance between maps. Proc. Camb. Philos. Soc. 172, No. 1, 73-93 (2022).
[4] Tanaka, K. Lusternik-Schnirelmann category for categories and classifying spaces. Topology Appl. 239, 65-80 (2018).

By Farber's famous result, the zero-divisor cup-length gives a lower bound for the topological complexity. In this talk we discuss the problem of finding mod 2 zero-divisor cup-length, and its higher analogs, of real Grassmannians and some related manifolds.

Let $f\colon X\to Y$ be a map. If $Y$ fibres over some base space $B$, then we can view $f$ as a map over $B$ and consider the fibrewise topological complexity of $f$ relative to $B$. We will examine several variants of this concept with respect to various definitions of topological complexity of a map and show how it can be used to model certain problems in robot manipulation planning.