Monotonicity of the Schwarz genus

Abstract

Schwarz genus g(ξ) of a fibration ξ : E → B is defined as the minimal integer n, such that there exists a cover of B by n open sets that admit partial section to ξ. Many important concepts, including Lusternik-Schnirelmann category, Farber's topological complexity and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain type of morphisms between fibrations. Our main result is the following: if there exist a fibrewise map f : E → E ′ between fibrations ξ : E → B and ξ ′ : E ′ → B which induces an n-equivalence between respective fibres for a sufficiently big n, then g(ξ) = g(ξ ′). From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complexes has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension.

Topological complexity of real Grassmannians

“…In [11] we proved the following result: Theorem 5.1. ( [11, theorem 3.6]) Let X be a connected CW-complex X, and let A be a subcomplex of X containing X d .…”

“…In order to derive analogous results for topological complexity we rely on the following result from [11].…”

Manifolds with small topological complexity

“…They managed to compute topological complexity of many two-cell complexes, but the technical details are quite formidable, and the full analysis of three-cell complexes is probably very hard. Nevertheless, we were able to combine some of their computations with our results from [22] that relate topological complexity of a space with topological complexity of its skeleta, to show that some sphere bundles over spheres have topological complexity at least 4. We will work in the so-called meta-stable range and assume that 2k < l < 3k − 1.…”

“…Topological complexity of S k ∪ α e l was bounded from below in [12,Theorem 5.6]: TC(S k ∪ α e l ) ≥ 4. On the other hand, [22,Theorem 3.6] implies that cat(M ) ≥ cat(S k ∪ α e l ) = 3, therefore TC(M ) ≥ 3. Then we may apply [22,Theorem 3.1], which states that if…”

Topological complexity of real Grassmannians

“…For k = 4 our approach works only under some additional assumptions. In order to derive analogous results for topological complexity we rely on the following result from [12].…”