Representation theory for the Križ model

Abstract

The natural action of the symmetric group on the configuration spaces F (X, n) induces an action on the Križ model E(X, n). The representation theory of this DGA is studied and a big acyclic subcomplex which is Sn-invariant is described. Recently Lambrechts and Stanley [LaSt] constructed a (quasi)-model for the configuration space of a topological space with Poincaré duality cohomology; if such a space is formal, the model of Lambrechts-Stanley is reduced to the Križ model and this is the case of Kähler manifolds, see [DGMS]. Therefore all the results of this paper could be applied to (simply connected) formal closed manifolds (with few changes for the odd-dimensional manifolds).Let us remind the construction of Križ. We denote by p * i : H * (X) → H * (X n ) and p * ij : H * (X 2 ) → H * (X n ) (for i = j) the pullbacks of the obvious projections and by m the complex dimension of X (for cohomology groups we use rational or complex coefficients). The model E(X, n) is defined as follows: as an algebra E(X, n) is isomorphic to the exterior algebra with generators G ij , 1 ≤ i, j ≤ n (of degree 2m − 1) and coefficients in H * (X) ⊗n modulo the relations G

On the geometry and topology of partial configuration spaces of Riemann surfaces

“…More generally, let Σ be an arbitrary complex projective manifold of dimension m ě 1. The full configuration space FpΣ, K n q has a remarkable cdga model, [3] for details and references related to these models). As a graded algebra, E .…”

Cohomology of 3-points configuration spaces of complex projective spaces

“…More generally, the right hand side of the trapezoid E * * is acyclic, see [2]. The coordinates of the vector W in the next lemma correspond to the frame…”

“…In [9] Feichtner and Ziegler described the cohomology algebra of F (CP 1 , n) with integer coefficients. The symmetric structure of H * (F (CP 1 , n)) was considered in [2].…”

“…All the cohomology classes in H * are of even degree, hence E k q = 0 if k + q ≡ 1 (mod 2). For large values of k we use the S 3 -equivariant Poincaré duality, see [2].…”

“…(1) The elements of the canonical basis E (2m−1)+2k 1 correspond to the marked graphs (Γ (2,1) , [2] for a marked version of the graphs introduced in [17]. The possible marks (h , h ) are the pairs (0, k), (1, k − 1), .…”