Poincaré duality and commutative differential graded algebras. (English) Zbl 1172.13009

Lambrechts, Pascal; Stanley, Don

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)

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