Lower bounds for topological complexity

Abstract

We introduce fibrewise Whitehead- and fibrewise Ganea definitions of monoidal topological complexity. We then define several lower bounds for the topological complexity, which improve on the standard lower bound in terms of nilpotency of the cohomology ring. The relationships between these lower bounds are studied.Comment: 19 pages, 1 figur

Spaces with high topological complexity

“…As it often happens in the fibrewise context however, the standard notation for the various fibrewise constructions becomes excessively complicated and difficult to read. As an attempt to avoid this inconvenience we use a more intuitive notation (introduced in [10]), based on the analogy between fibrewise constructions and semi-direct products. Indeed, whenever we perform a pointed construction (e.g a wedge or a smash-product) on some fibrewise space, the fibres of the resulting space depend on the choice of base-points, and we view this effect as an action of the base on the fibres.…”

“…The Whitehead-type characterization of the monoidal topological complexity (cf. [10,Theorem 3], see also [14,Section 6]) is: TC M (X) is the least integer n such that the map 1 ⋉ ∆ n : X ⋉ X → X ⋉ Π n X can be compressed into X ⋉ W n X by a fibrewise pointed homotopy.…”

“…[10,Corollary 4]) is: TC M (X) is the least integer n such that the fibrewise pointed map 1 ⋉ p n : X ⋉ G n X → X ⋉ X admits a section. Note that the fibres of these constructions are respectively the spaces X, Π n X, W n X, P X and G n X (the nth Ganea space).…”

“…In fact, Iwase and Sakai [14] found a useful characterization of the monoidal topological complexity as the fibrewise pointed LS-category (see Section 2 for details), which makes the above analogy even clearer. In [10] we exploited this new approach and introduced several lower bounds for TC M (X) that refine previously known estimates. Nevertheless, these bounds need not be precise, and in fact one can always construct spaces for which the difference between the estimate and the actual value of TC M (X) is arbitrarily large.…”

“…By a formula of Farber [7, Theorem 5.2] the topological complexity TC(X) of a (p − 1)-connected, m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. There are also various lower estimates for TC(X) such as the nilpotency of the ring H * (X × X, ∆(X)), and the weak and stable topological compexity wTC(X) and σTC(X) (see [10]). In general the difference between these upper and lower bounds can be arbitrarily large.…”