Abstract. We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold.
We construct an invariant of t-structures on the derived category of a Noetherian ring. This invariant is complete when restricting to the category of quasi-coherent complexes, and also gives a classification of nullity classes with the same restriction. On the full derived category of Z we show that the class of distinct t-structures do not form a set.
Let M be a simply connected closed manifold and consider the (ordered) configuration space F.M; k/ of k points in M . In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F.M; k/. We prove that our model it is at least a † k -equivariant differential graded model.We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold.
55P62, 55R80
Let f : P ֒→ W be an embedding of a compact polyhedron in a closed oriented manifold W , let T be a regular neighborhood of P in W and let C := W T be its complement. Then W is the homotopy push-out of a diagram C ← ∂T → P . This homotopy push-out square is an example of what is called a Poincaré embedding.We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f . When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C ≃ W f (P ). Moreover we construct examples to show that our restriction on the codimension is sharp.Without restriction on the codimension we also give differentiable modules models of Poincaré embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement.
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