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We show that, up to small error, the analog category of a finite group records the size of its largest Sylow subgroup. 

We study probabilistic variants of the Lusternik–Schnirelmann category and topological complexity, which bound the classical invariants from below. We present a number of computations illustrating both wide agreement and wide disagreement with the classical notions. In the aspherical case, where our invariants are group invariants, we establish a counterpart of the Eilenberg– Ganea theorem in the torsion-free case, as well as a contrasting universal upper bound in the finite case. 

In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing the recent paper [2], we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles in Rd. This algorithm is optimal, it has minimal possible topological complexity for any d ≥ 3 odd. Besides, we describe a modification of this algorithm which is optimal for d ≥ 2 even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes. 

Parametrized motion planning algorithms \cite{CFW} have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles.Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.