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

Abstract

Abstract. Let M be a simply connected closed manifold of dimension n. We study the rational homotopy type of the configuration space of 2 points in M , F (M, 2). When M is even dimensional, we prove that the rational homotopy type of F (M, 2) depends only on the rational homotopy type of M . When the dimension of M is odd, for every x ∈ H n−2 (M, Q), we construct a commutative differential graded algebra C(x). We prove that for some x ∈ H n−2 (M ; Q), C(x) encodes completely the rational homotopy type of F (M, 2). For some class of manifolds, we show that we can take x = 0.

The Lambrechts–Stanley model of configuration spaces

“…Totaro [Tot96] has shown that for a smooth complex compact projective variety, the spectral sequence only has one nonzero differential. When k = 2, then G A (2) was known to be a model of Conf 2 (M ) either when M is 2-connected [LS04] or when dim M is even [Cor15].…”

Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

“…We deduce the rational homotopy invariance of Conf(W, 2). The rational homotopy invariance of Conf (W, 2) when W is a closed 2-connected has been established in [8], and [2] gives partial results in the 1-connected case. When W is not simply-connected, [12] shows that there is no rational homotopy invariance.…”

“…Moreover a CDGA model of W \K (that is, an algebraic model in the sense of Sullivan of this rational homotopy type, see Section 2.1) can be explicitely constructed out of any CDGA model of Diagram (2).…”

“…5.4. A CDGA model forConf(W,2) when W is We apply the model constructed in Section 5.3 to disk bundles.Let ξ be a vector bundle of even rank, 2k, for some k ≥ 2, over some 2-connected closed manifold, M, of dimension m. Then the disk bundle Dξ is a compact manifold of dimension m + 2k with boundary the sphere bundle Sξ.…”