Poincaré duality and commutative differential graded algebras.

*(English)*Zbl 1172.13009Let \((A,d)\) 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.

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)