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

##### 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)
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