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20
The Bergman complex of a matroid and phylogenetic trees
 THE JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2005
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Operads and knot spaces
 J. Amer. Math. Soc
"... 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 ..."
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Cited by 40 (3 self)
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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
Configuration spaces with summable labels
 Proceedings of BCAT98
"... Let M be an nmanifold, and let A be a space with a partial sum behaving as an nfold 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 ..."
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Cited by 16 (1 self)
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Let M be an nmanifold, and let A be a space with a partial sum behaving as an nfold 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 nfold 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)).
A pairing between graphs and trees
, 2006
"... 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, ..."
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Cited by 9 (3 self)
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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 oddgraded 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
The Tree Representation of Σ_n+1
, 1996
"... We show that the space of fullygrown ntrees 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 nontrivial reduced homology group is ffl \Delta (Ind \Sigma n+1 \Sigma n Lie n \Gamma Lie n+1 ): Introduction ..."
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Cited by 7 (3 self)
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We show that the space of fullygrown ntrees 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 nontrivial 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 fullygrown ntrees, 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...
Higher homotopy operations
"... 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 Wconstruction of Boardman and Vogt, applied to the appropriate diagram categor ..."
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Cited by 7 (4 self)
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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 Wconstruction 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.
Complexes of trees and nested set complexes
"... 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 st ..."
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Cited by 7 (0 self)
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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 ktrees by Hanlon, and we propose yet another, in the context of nested set complexes more natural, generalization. 1.
Nested set complexes of Dowling lattices and complexes of Dowling trees
"... Abstract. Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Qn(G) (see [8]) and of its subposet of the Gsymmetric partitions Q0 n(G) which was recently introduced by Hultman in [20], together with the complex of Gs ..."
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Cited by 3 (0 self)
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Abstract. Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Qn(G) (see [8]) and of its subposet of the Gsymmetric partitions Q0 n(G) which was recently introduced by Hultman in [20], together with the complex of Gsymmetric phylogenetic trees T G n. Hultman shows that the complexes T G n and e∆(Q 0 n (G)) are homotopy equivalent and CohenMacaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov [12] shows that in fact T G n is subdivided by the order complex of Q0 n (G). We introduce the complex of Dowling trees Tn(G) and prove that it is subdivided by the order complex of Qn(G) and contains T G n as a subcomplex. We show that Tn(G) is obtained from T G n by successive coning over certain subcomplexes. It is well known that Qn(G) is shellable, and of the same dimension as T G n. We explicitly and independently calculate how many homology spheres are added in passing from T G n to Tn(G).
The preWDVV ring of physics and its topology
, 2005
"... We show how a simplicial complex arising from the WDVV (WittenDijkgraafVerlindeVerlinde) 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 ..."
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We show how a simplicial complex arising from the WDVV (WittenDijkgraafVerlindeVerlinde) 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 CohenMacaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated preWDVV ring. 1