Aspherical completions and rationally inert elements

Abstract

Let X be a connected space. An elementWe extend the results of [16] and prove in particular that if X ∪ f D n+1 is a Poincaré duality complex and the algebra H(X) requires at least two generators then [f ] ∈ π n (X) is rationally inert. On the other hand, if X is rationally a wedge of at least two spheres and f is rationally non trivial, then f is rationally inert. Finally if f is rationally inert then the rational homotopy of the homotopy fibre of the injection X → X ∪ f D n+1 is the completion of a free Lie algebra.

The Rational Homotopy Type of Homotopy Fibrations Over Connected Sums

“…is implies that, rationally, Ω has a right homotopy inverse. e following theorem was first proved in [5, eorem 5.1], though we prefer the statement found in [2]. Proof.…”

“…Lemma 2.2. Take the setup of Diagram (2). If the maps Ω and Ω have right homotopy inverses, then so does Ωℎ.…”

The rational homotopy type of homotopy fibrations over connected sums

“…This implies that, rationally, Ωi has a right homotopy inverse. The following theorem was first proved in [6, Theorem 5.1], though we prefer the statement found in [3].…”

“…By Theorem 4.1, the attaching maps for the top cells of C and L are rationally inert. We have the homotopy pullback below, which is the right-hand square of Diagram (3) Rationalising spaces and maps in this pullback square, we see that Proposition 3.3 would apply if the map Ωp has a rational right homotopy inverse, as the attaching map for the top cell of L is rationally inert. Thus, we would have a rational homotopy equivalence ΩM Ω(X#L).…”