A model for configuration spaces of points

Abstract

The configuration space of points on a D-dimensional smooth framed manifold may be compactified so as to admit a right action over the framed little D-disks operad. We construct a real combinatorial model for these modules, for compact smooth manifolds without boundary. Contents 1. Introduction 1 2. Compactified configuration spaces 5 3. The Cattaneo-Felder-Mnev graph complex and operad 7 4. Twisting Gra M and the co-module * Graphs M 10 5. Cohomology of the CFM (co)operad 14 6. The non-parallelizable case 21 7. A simplification of * Graphs M and relations to the literature 23 8. The real homotopy type of M and FM M 29 9. The framed case in dimension D = 2 33 Appendix A. Remark: Comparison to the Lambrechts-Stanley model through cyclic C ∞ algebras 36 Appendix B. Example computation: The partition function of the 2-sphere 38 Appendix C. Pushforward of PA forms 41 References 43 * Graphs M ⊂ * Graphs M that still comes with a map of dg Hopf collections * Graphs M → Ω PA (FM M ).Our first main result is the following.1 A (dg) Hopf collection C for us is a sequence C(r) of dg commutative algebras, with actions of the symmetric groups S r . A (dg) Hopf cooperad is a cooperad in dg commutative algebras.

Stable cohomology of graph complexes

“…In this section we give a combinatorial definition of the graph complexes considered in this paper. These complexes have appeared at other places in the literature, for example [7,12]. We say that a (directed) graph with n vertices and k edges is an ordered set of k pairs (i, j) of numbers i, j ∈ {1, .…”

“…Auxiliary graph complexes Graphs (g) (n). In [7] Campos and the third author define dg graphical cocommutative coalgebra models for framed configuration spaces of n points on surfaces (i.e., for W g if m = 1 in our notation). As auxiliary objects in some proofs below, we will need to use closely analogous graph complexes defined for general m, which we denote by Graphs (g) (n).…”

A model for framed configuration spaces of points

“…The ordered configuration space of k points on M is given by latter object carries a natural action of a Fulton-MacPherson-Axelrod-Singer-version of the framed E n -operad FM fr n . Some of the authors described in [CW16;Idr16] a graphical real model Graphs M (k) for Conf k (M ). Elements of Graphs M (k) are linear combinations of undirected diagrams with k numbered "external" vertices, some further "internal" vertices, and zero, one, or more decorations in H(M ) at each vertex:…”

“…Our second main result is to upgrade the dg commutative algebra model Graphs M of FM M so as to capture also the action of FM M n on FM M : Theorem 2 (See Theorem 18). There is a graph complex Graphs M n , which is a model for FM M n , and which acts on the model Graphs M of FM M from [CW16], so that the map (2) respects the (homotopy) comodule structures on both sides. Remark 3.…”

“…Remark 3. In this paper we will deviate notationally from [CW16] and denote the graphical commutative coalgebra model of the configuration space by Graphs M instead of * Graphs M . We hope that no confusion arises.…”