Generating functions and topological complexity

Abstract

We examine the rationality conjecture raised in [1] which states that (a) the formal power series r≥1 TCr+1(X) · x r represents a rational function of x with a single pole of order 2 at x = 1 and (b) the leading coefficient of the pole equals cat(X). Here X is a finite CWcomplex and for r ≥ 2 the symbol TCr(X) denotes its r-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

Right-angled Artin groups, polyhedral products and the TC-generating function

“…Indeed, a remarkable 3-cell complex X is constructed in [31] with the property that cat(X ×X) = cat(X) = 2. In [14], the higher topological complexities of X were shown to be TC r (X) = r = cat(X r ) (r ≥ 2). In view of Theorem 1.1, this yields P X (1) = 1 < cat(X).…”

“…It should be noted that, in this example, P X (1) = cup(X), the cup-length of X, a situation that, to the best of our knowledge, holds in all cases where all TC r (X) are known. It is tempting to insert this clause in the modified form of the rationality conjecture at the end of [14].…”

On the growth of topological complexity

“…As is observed in [3], Question 1.1 is asking whether or not TC r+1 (X) = TC r (X) + cat(X) for all r large enough. Farber, Kishimoto and Stanley [3] proved that if TC r (X) = zcl r (X; k) and cat(X) = cup(X; k) for some field k and all r large enough, then F X (x) satisfies the condition in Question 1.1, where zcl r (X; k) and cup(X; k) denote the r-th zero-divisors cup-length and the cup-length of X over k, respectively. They also showed that F X (x) does not always satisfy the condition in Question 1.1.…”

“…By applying Theorem 1.4, we give a rational space X, to which the criterion of Farber, Kishimoto and Stanley [3] mentioned in Section 1 does not apply, but F X (x) satisfies the condition of Question 1.1.…”

Higher topological complexity of hyperbolic groups

“…Here cat denotes the Lusternik-Schnirelmann category. While a counter-example to the rationality conjecture for finite CW-complexes was found in [FKS20], the rationality conjecture for groups of type F remains open. It is known to hold, e.g.…”