Rational topological complexity

Abstract

We give a new upper bound for Farber's topological complexity for rational spaces in terms of Sullivan models. We use it to determine the topological complexity in some new cases, and to prove a Ganea-type formula in these and other cases. 55M30, 55P62

Computations in rational sectional category

“…If X is a space modelled by (ΛV, d) and ǫ : (ΛV, d) → Q is the augmentation then msc(ǫ) is the classical module category of X 0 and Hsc(ǫ) is the rational Toomer invariant of X. Also, if µ : (ΛV, d) ⊗ (ΛV, d) → (ΛV, d) is the multiplication, then sc(µ) = tc(X) and msc(µ) = mtc(X), as defined in [11].…”

“…We will denote X 0 the rationalisation of X and f 0 : X 0 → Y 0 the rationalisation of f . Following the scheme of Jessup-Murillo-Parent in [11], the following approximation to rational sectional category can easily be deduced:…”

A mapping theorem for topological complexity

“…[14,18]). Whilst we use minimal models in some of our examples and applications, we do not make use of an algebraic, or minimal model, version of TC n (−) as such.…”

Rational approximations of sectional category and Poincaré duality

“…Let X be a simply connected space of the homotopy type of a CW complex of finite type, and let (Λ(V ), d) be a Sullivan model of X. Inspired by the classical algebraic description of rational L.-S. category due to Félix and Halperin [3], Jessup, Murillo, and Parent [6] consider the multiplication µ : (ΛV ⊗ ΛV, d) → (ΛV, d), which is a model of the fibration ev 0,1 : X I → X × X, and define the invariants tc(X) and mtc(X) in terms of the projections…”