Buchstaber Invariants of Universal Complexes

Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes,which now can be seen as "colorings".The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K,then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex.It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3.This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex.     

Quandle homology and complex volume

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.     

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ¹-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov's proportionality principle of simplicial volume.     

Every finite complex is the classifying space for proper bundles of a virtual Poincaré duality group

We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.     

Additional structures in Smith theory

This paper is devoted to the development of the apparatus of Smith theory. In it we introduce operations and and also Poincare duality, which have not been considered previously. New connections are established between specific objects of Smith theory and objects of ordinary homology theory. The account is given for the case of a simplicial action on a finite simplicial space, the proofs are given briefly, most intermediate calculations are omitted.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 17–23, 1982.     

Homotopy type of circle graph complexes motivated by extreme Khovanov homology

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper, we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant (up to suspension), and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and find it convincing to generalize it to every circle graph (intersection graph of chords in a circle). In particular, we prove it for the families of cactus, outerplanar, permutation and non-nested graphs. Conversely, we also give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres. We also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem shedding light on previous results by Csorba, Nagel and Reiner, Jonsson and Barmak. We study the implications of our results to knot theory; more precisely, we compute the real-extreme Khovanov homology of torus links T(3, q) and obtain examples of H-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.     

Discrete Differential Calculus on Simplicial Complexes and Constrained Homology*

Let V be a finite set. Let K be a simplicial complex with its vertices in V .In this paper, the author discusses some differential calculus on V . He constructs some constrained homology groups of K by using the differential calculus on V . Moreover, he defines an independence hypergraph to be the complement of a simplicial complex in the complete hypergraph on V . Let L be an independence hypergraph with its vertices in V .He constructs some constrained cohomology groups of L by using the differential calculus on V .     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

(Co)Homology Theories for Oriented Algebras

We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally, we also study the fundamental groups of the bound quivers involved in the pullbacks.     

Enriched homology and cohomology modules of simiplicial complexes

For a simplicial complex Δ on {1, 2,…, n} we define enriched homology and cohomology modules. They are graded modules over k[x ₁,…, x _n ] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein^* complexes Δ, and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c, d, n) and cyclic polytopes of even dimension. This paper is to a large extent a complete rewriting of a previous preprint, “Hierarchies of simplicial complexes via the BGG-correspondence”. Also Propositions 1.7 and 3.1 have been generalized to cell complexes in [11].     

A description of the fundamental group in terms of commutators and closure operators

A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of functors as coefficients. This makes it possible to calculate the fundamental groups corresponding to many interesting reflections arising, for instance, in the categories of groups, rings, compact groups and simplicial loops.     

Persistent Intersection Homology

The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection homology gives useful information about the relationship between an embedded stratified space and its singularities. We give an algorithm for the computation of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcomplexes, and we prove its correctness. We also derive, from Poincaré Duality, some structural results about persistent intersection homology.     

An exact homology sequences approach to the controllability of systems

Some results are given on the homology groups of the abstract simplicial complex associated to a multivariable control system, as an extension of the central result of J. Casti (J. Math. Anal. Appl.68 (1979), 347–370) to the multivariable case. The method used is “polyhedral dynamics” as termed by J. Casti, who introduced it. A given control system is mapped into a simplicial complex and then its homological structure is studied. These results, obtained in the multivariable case by the use of the Mayer-Vietoris homology sequence and of the relative homology sequence, present topological invariants for control systems thus creating the possibility of a new topological classification of multivariable systems. An example is given to clarify the approach.     

Higher-Order (F,α,β,ρ,d,E)-Convexity in Fractional Programming

In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.     

Cohen–Macaulayness and squarefree modules

In this paper we consider the category of squarefree modules over the polynomial ring and an exact duality functor, which is an extension of the Alexander dual of a simplicial complex. We give a relationship between the squarefree components of local cohomology groups of a squarefree module and the Tor groups of its dual. With this result it is shown that a squarefree module is sequentially Cohen–Macaulay if and only if the dual is componentwise linear. Received: 7 June 1999 / Revised version: 6 September 2000     

Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology.We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.     

Higher commutators in the loop space homology of K-products

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex K. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley-Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen-Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.     

一类带反凸约束的非线性比式和问题的全局优化算法

本文针对一类带有反凸约束的非线性比式和分式规划问题,提出一种求其全局最优解的单纯形分支和对偶定界算法.该算法利用Lagrange对偶理论将其中关键的定界问题转化为一系列易于求解的线性规划问题.收敛性分析和数值算例均表明提出的算法是可行的.     

Equivariant homology and duality

This note is concerned with stable G-equivariant homology and cohomology theories (G a compact Lie group). In important cases, when H-equivariant theories are defined naturally for all closed subgroups H of G, we show that the G-(co)homology groups of G x_H X are isomorphic with H-(co)homology groups of X. We introduce the concept of orientability of G-vector bundles and manifolds with respect to an equivariant cohomology theory and prove a duality theorem which implies an equivariant analogue of Poincaré-Lefschetz duality.

     

Classical Invariants for Global Actions and Groupoid Atlases

A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and groupoid atlases, such as the strong fundamental groupoid, the weak and strong nerves, classifying spaces and homology groups. We relate all these new invariants to classical constructions in topological spaces, simplicial complexes and simplicial sets. This way we obtain new combinatorial formulations of classical and non classical results in terms of groupoid atlases.