Rational approximations of sectional category and Poincaré duality (Q2789887)

The main result of this short paper is a proof of the following Theorem:NEWLINENEWLINE{Theorem 1.1}: Let \(p: E \to X\) be a fibration such that \(H^*(X;\mathbb{Q})\) is a Poincaré duality algebra. Then \(\mathrm{Mcat(p)}=\mathrm{Hsecat(p)}\).NEWLINENEWLINE The model sectional category of a fibration \(p: E \to X\), denoted \(\mathrm{Mcat}(p)\), is the least integer \(n\) such that the monomorphism of \(A_{PL}(X)\)-modules \(A_{PL}(j^np): A_{PL}(x)\to A_{PL}(*^n_XE)\) admits a homotopy retraction while the cohomology sectional category of \(p\), \(\mathrm{Hsecat(p)}\), is defined by \(\mathrm{Hsecat(p)}\leq n \Leftrightarrow H(A_{PL}(j^n(p)))\) is injective. Theorem 1.1 is an analogue of a result by Félix, Halperin, and Lemaire which says that the rational module category and rational Toomer invariant coincide on Poincaré duality complexes. The first three sections of the paper are devoted to the introduction, basic definitions, and proof of Theorem 1.1 while the last section applies Theorem 1.1 in the context of topological complexity. The authors show that if \(H^*(X)\) is a Poincaré duality algebra, then the corresponding invariants for topological complexity \(\mathrm{MTC}(X)\) and \(\mathrm{HTC}(X)\) coincide. Finally, it is noted that if \(H^*(X)\) is not a Poincaré duality algebra, then \(\mathrm{MTC}(X)\) and \(\mathrm{HTC}(X)\) need not coincide.