Fibrewise stable rational homotopy

Abstract

In this paper, for a given space B, we establish a correspondence between differential graded modules over C*(B; ℚ) and fibrewise rational stable spaces over B. This correspondence opens the door for topological translations of algebraic constructions made with modules over a commutative differential graded algebra. More precisely, given the fibrations E→B and E′→B, the set of stable rational homotopy classes of maps over B is isomorphic to Ext^*_C*(B;ℚ) (C*(E′; ℚ), C*(E; ℚ)). In particular, a nilpotent finite‐type CW‐complex X is a rational Poincaré complex if there exist non‐trivial stable maps over X_ℚ from (X × S^q)_ℚ to (X ∨ S^q+N)_ℚ for exactly one N.