Abstract not found

Abstract not found

Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated

Let M be an n-manifold, and let A be a space with a partial sum behaving as an n-fold loop sum. We define the space C(M; A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I n, ∂I n; A) is an n-fold classifying space of C(I n; A), and for n = 1 it is homeomorphic to the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M; A) is homotopic to the mapping space Map(M, C(I n, ∂I n; A)).

We show that the space of fully-grown n-trees has the homotopy type of a bouquet of spheres of dimension n \Gamma 3 and that the character of the representation of \Sigma n+1 on its only non-trivial reduced homology group is ffl \Delta (Ind \Sigma n+1 \Sigma n Lie n \Gamma Lie n+1 ): Introduction We investigate a certain space Tn called the space of fully-grown n-trees, and we prove that it has the homotopy type of a bouquet of spheres of dimension n \Gamma 3. The symmetric group \Sigma n+1 acts in a natural way on Tn , so the reduced homology of Tn affords an integral representation of \Sigma n+1 , which we call the tree representation. It has dimension (n \Gamma 1)!. Its restriction to \Sigma n\Gamma1 is isomorphic (over the integers) to the regular representation. We show that the character of the tree representation is ffl \Delta (Ind \Sigma n+1 \Sigma n Lie n \Gamma Lie n+1 ) where ffl is the alternating character, and Lie n is the character of the representation of \Sigm...

Abstract. We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. 1.

In this paper we develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi and Arnold identities. Our first main result is that on these quotients the pairing is perfect, which makes it an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. Passing from the operads to free algebras over them, we get canonical models for cofree Lie coalgebras. The functionals on free Lie algebras which result are defined without reference to the embedding of free Lie algebras in tensor algebras. In the course of establishing our main results we reprove standard facts about the modules Lie(n). We apply the pairing to develop product, coproduct and (co)operad structures, defining notions such as a partition of forests which may be useful elsewhere. Remarkably, we find the cooperad which dual to the Poisson operad more manageable than the Poisson operad itself. This pairing arises as the pairing between canonical bases for homology and cohomology of configurations in Euclidean space. We elaborate on this topology in the expository paper [8]. A variant of this pairing first appears in work of Melancon and Reutenaur on odd-graded free Lie algebras [5]; see Section 4. The pairing was independently developed and applied by Tourtchine [12, 13]; see further commentary at the end of Section 1. We give a unified, explicit, and fully

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex ∆n is homotopy equivalent to a wedge of (n − 2)! spheres of dimension n − 4. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring. 1

Abstract. We exhibit an identity of abstract simplicial complexes between the wellstudied complex of trees Tn and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another, in the context of nested set complexes more natural, generalization. 1.

Abstract. Let Π (k) be the poset of partitions of {1,2,..., (n − 1)k + 1} with (n−1)k+1 block sizes congruent to 1 modulo k. We prove that the order complex ∆(Π (k) (n−1)k+1) is a subdivision of the complex of k-trees T k n, thereby answering a question posed by Feichtner [F, 5.2]. The result is obtained by an ad-hoc generalization of concepts from the theory of nested set complexes to non-lattices. The complex of k-trees T k n