The fibre of a cell attachment

Abstract

In view of understanding the Hopf algebra structure of the loop space homology H t (Q{E (J/ e " + ')) in terms of H t (tlE) and the map /, we consider the homotopy fibre F of the inclusion map (o: E c» £ {J f e°+ 1 . In [15], the case when H t (Clo]) is surjective (the "inert" case) was studied, and in [11] a weaker condition, called "lazy", was considered. Here we give several new characterizations of inert and lazy cell attachments in terms of properties of F. We also show how these results extend to the case of the mapping cone £ \J f CW of an arbitrary map / : W-*E.

Aspherical completions and rationally inert elements

“…Then d 0 d 1 w ∈ (∧ 2 W ) k+2 and so d 0 d 1 w ∈ R ∧ ∧W ⊕ Qa 2 . Thus ϕ(d 0 d 1 w) = 0 and again by (17)…”

The Homotopy Type of a Poincaré Duality Complex After Looping

“…Determining the conditions under which θ is a Hopf Algebra isomorphism is called the cell attachment problem. This has a long history, having been studied by Anick [1], Bubenik [3], Félix and Thomas [5], and Halperin, Hess, and Lemaire [11,7,8,9]. Lemaire [11], for one, found that θ is a Hopf algebra isomorphism whenever the morphism of graded R-vector spaces induced by the canonical surjection H * (ΩX; R) π → H * (ΩX; R) I is bijective, and R is a vector space of characteristic p.…”

“…This has a long history, having been studied by Anick [1], Bubenik [3], Félix and Thomas [5], and Halperin et al . [7][8][9]11]. Lemaire [11], for one, found that θ is a Hopf algebra isomorphism whenever the morphism of graded R-vector spaces Tor π p :…”