Joins of DGA modules and sectional category

TL;DR

This paper develops an explicit algebraic model for fiber joins of fibrations, introduces a new lower bound for sectional category that improves upon classical bounds, and applies it to compute or estimate topological complexity.

Contribution

It provides a new explicit semifree model for fiber joins and a more accurate lower bound for sectional category and topological complexity.

Findings

The new lower bound can be computed from any Sullivan model.

The difference between the new and classical bounds can be arbitrarily large.

Applied to the evaluation fibration, it yields a computable lower bound for TC(X).

Abstract

We construct an explicit semifree model for the fiber join of two fibrations p: E --> B and p': E' --> B from semifree models of p and p'. Using this model, we introduce a lower bound of the sectional category of a fibration p which can be calculated from any Sullivan model of p and which is closer to the sectional category of p than the classical cohomological lower bound given by the nilpotency of the kernel of p^*: H^*(B;Q) --> H^*(E;Q). In the special case of the evaluation fibration X^I --> X x X we obtain a computable lower bound of Farber's topological complexity TC(X). We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.