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Fibrewise stable rational homotopy
Author(s) -
Félix Yves,
Murillo Aniceto,
Tanré Daniel
Publication year - 2010
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtq023
Subject(s) - mathematics , homotopy , pure mathematics , nilpotent , commutative property , differential (mechanical device) , space (punctuation) , type (biology) , algebraic number , algebra over a field , discrete mathematics , mathematical analysis , computer science , biology , aerospace engineering , operating system , ecology , engineering
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 .