We present a class of subposets of the partition lattice n with the following property: The order complex is homotopy equivalent to the order complex of n – 1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn – 1-action on the homology of n – 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet Pnn
– 1 of the lattice of set partitions n, obtained by removing all elements with a unique nontrivial block. More generally, for 2 kn – 1, let Qnk
denote the subposet of the partition lattice n obtained by removing all elements with a unique nontrivial block of size equal to k, and let Pnk
=
i = 2kQni
. We show that Pnk
is Cohen-Macaulay, and that Pnk
and Qnk
are both homotopy equivalent to a wedge of spheres of dimension (n – 4), with Betti number
. The posets Qnk
are neither shellable nor Cohen-Macaulay. We show that the Sn-module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property. 相似文献
Based on regular boundary element method and a kind of linearity invariance under homotopy, a kind of numerical scheme of 2D steady-state Navier-Stokes equation in streamfunction-vorticity formulation is described. The flow inside a square cavity is used to illustrate this numerical scheme. 相似文献
Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGLnnp, n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGLnnp. In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGLn3n+2 and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGLn3n+2. How many homotopy equivalence classes of objects (L(W), δ) in DGLn3n+2 such that H*(W, d′)?H*(V, d) are there? Note that DGLn3n+2 is a subcategory of DGLnnp when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl. 相似文献
It is well known that the Hochschild cohomology of an associative algebra admits a G-algebra structure. In this paper we show that the dialgebra cohomology of an associative dialgebra has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex .
We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C*r(, )-bundle and connection, which can be used to construct a C*r(, )-index that lies in the Grothendieck group of C*r(, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem. 相似文献
Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.
A standing conjecture in -cohomology says that every finite -complex is of -determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, is -acyclic, we also show that the -determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for -Betti numbers.
Consider the multi-homogeneous homotopy continuation method for solving a system of polynomial equations. For any partition of variables, the multi-homogeneous Bézout number bounds the number of isolated solution curves one has to follow in the method. This paper presents a local search method for finding a partition of variables with minimal multi-homogeneous Bézout number. As with any other local search method, it may give a local minimum rather than the minimum over all possible homogenizations. Numerical examples show the efficiency of this local search method.