A remarkable DGmodule model for configuration spaces

Abstract

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

Rational model of the configuration space of two points in a simply connected closed manifold

“…The following result is an evident reformulation of the results of [6, Theorem 10.1] and [4]. Let (A, d, ǫ) be a Poincaré duality CDGA of formal dimension n. Let {a i } N i=1 be a homogeneous basis of A and denote by {a * i } N i=1 its Poincaré dual basis.…”

“…equipped with the semi-trivial structure (see Section 2.1) is a natural candidate to be a CDGA model of F (M, 2). The main result of [4] proves that it is when the manifold is 2-connected. In the following section we show that it also is when the manifold is 1-connected and of even dimension.…”

“…In [4] it is shown that A ⊗ A/(∆), where (∆) is the ideal generated by ∆ in A ⊗ A, is a CDGA model of F (M, 2) when M is at least 2-connected. This results implies that the rational homotopy type of F (M, 2) depends only on the rational homotopy type of M for a 2-connected closed manifold.…”

“…We will say that a manifold M is untwisted if C(0) is a CDGA model of F (M, 2) (see Definition 6.1). In [4] it is shown that all 2-connected closed manifolds are untwisted and in Section 4 we will show that every simply connected closed manifold of even dimension is untwisted. We prove in Section 6 that the product of two untisted manifolds is untwisted.…”

Poincaré duality and commutative differential graded algebras

“…In case A has differential 0, as it always does in Kriz's situation, this model agrees with the model of Kriz. In [9] out of a differential Poincaré duality algebra model A for M we construct a model of F (M, k) in some module category. Our model is also a CDGA and agrees with the Kriz model in case d(A) = 0.…”