Jesús González

BP-theoretic instabilities to the motion planning problem in 4-torsion lens spaces

The Brown-Peterson cohomology for skeleta of the classifyi ng space of the group Z4 Z4 is analyzed in order to describe obstructions to the motion p la ning problem for a particle moving in a 4-torsion lens space. We di scuss the relationship of this situation to the Euclidean immersion problem for 2e-torsion lens spaces, and the way this leads to an alternative approach to the classica l immersion problem for real projective spaces.

Symmetric topological complexity as the first obstruction in Goodwillie’s Euclidean embedding tower for real projective spaces

This paper explains why Goodwillie-Weiss calculus of embeddings can offer new information about the Euclidean embedding dimension of P m only for m � 15. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential—but critical—high-order obstructions in the corresponding Taylor towers. For m � 16, the relation TC S (P m ) � n is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding P m � R n . A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis’ BP-approach to the immersion problem of P m . A form of the Euler class viewpoint is applied to show TC S (P 3 ) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber’s work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber’s lead, this concept is connected to TC S (C(S)), the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P 3 .

Higher topological complexity of subcomplexes of products of spheres and related polyhedral product spaces

We construct ``higher motion planners for automated systemsnwhose spaces of states are homotopy equivalent to a polyhedralnproduct space

Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

Z(K,{(S^{k_i},star)})

Motion planning in real flag manifolds

, {e.g. robot arms withnrestrictions on the possible combinations of simultaneously movingnnodes.} Our construction is shown to be optimal by explicitncohomology calculations. The higher topological complexity ofnother {families of} polyhedral product spaces is also determined.

On the immersion problem for 2(Sup r)-torsion lens spaces

The clique number of a random graph in the Erdos-Renyi model G(n,p) yields a random variable which is known to be asymptotically (as n tends to infinity) almost surely within one of an explicit logarithmic (on n) function r(n,p). We extend this fact by showing that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint complete subgraphs with as many vertices as r(n,p). The result is motivated by and applied to the sequential motion planning problem on random right angled Artin groups. Indeed, we give an asymptotical description of all the higher topological complexities of Eilenberg-MacLane spaces associated to random graph groups.