Lambrechts, Pascal; Stanley, Don Poincaré duality and commutative differential graded algebras. (English) Zbl 1172.13009 Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 497-511 (2008). Let \((A,d)\) be a commutative differential graded algebra over a field \(k\). It is assumed that \(A\) is non-negatively graded with \(A^0 = k\) and finitely generated as a \(k\)-algebra. We say that \(A\) is simply connected if \(A^1 = 0\), and that \(A\) is an oriented Poincaré duality algebra of dimension \(n\) if there is a linear map \(\epsilon: A \to k\) such that the induced bilinear forms \(A^i \otimes A^{n-i} \to k\) are non-degenerate.The main result in this paper is that any commutative differential graded algebra \((A,d)\) over \(k\) that has cohomology that is simply-connected and satisfies Poincaré duality of dimension \(n\) is weak equivalent to a commutative differential graded algebra \((A',d')\) that is simply-connected and satisfies Poincaré duality of dimension \(n\). The proof of the theorem is constructive. The result has applications to the study of configuration spaces and in string topology. Reviewer: Eivind Eriksen (Oslo) Cited in 3 ReviewsCited in 12 Documents MSC: 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 16E45 Differential graded algebras and applications (associative algebraic aspects) 55P62 Rational homotopy theory 18G55 Nonabelian homotopical algebra (MSC2010) Keywords:Poincaré duality; commutative differential graded algebra PDF BibTeX XML Cite \textit{P. Lambrechts} and \textit{D. Stanley}, Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 497--511 (2008; Zbl 1172.13009) Full Text: DOI arXiv Link OpenURL