Coreduction Homology Algorithm

This paper presents a new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes. The algorithm—particularly strong for low-dimensional sets embedded in high dimensions—runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. Both authors are partially supported by Polish MNSzW, Grant N201 037 31/3151.     

Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index

We introduce a new approach to the algorithmic computation of the Conley index for continuous maps. We use the technique of splitting an index pair into two layers which is inspired by the work of Mrozek, Reineck and Srzednicki [M. Mrozek, J.F. Reineck, R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000) 4171–4194]. The main advantage of our construction over the approach based directly on the one introduced by Mischaikow, Mrozek and Pilarczyk [K. Mischaikow, M. Mrozek, P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005) 199–229] is that our cubical sets have the excision property. Moreover, our solution has some advantages in comparison to the approach recently proposed by Mrozek [M. Mrozek, Index pair algorithms, Found. Comput. Math. 6 (2006) 457–493].     

From Leibniz Homology to Cyclic Homology

For an algebra R over a commutative ring k, a natural homomorphism _*: HL_*+1(R) HH_* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map _* is surjective when R = gl(A), A an algebra over a characteristic zero field. If f: A B is an algebra homomorophism, the relative groups HL_* (gl(f)) are studied, where gl(f): gl(A) gl(B) is the induced map on matrices. Letting HC_* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL_*+1(gl(f)) HC_* (f) in the characteristic zero setting.     

Strong Homology Groups of Continuous Maps

In this paper, we define coherent morphisms of chain maps and homology groups of morphisms of this type. We construct strong homology groups of continuous maps of compact metric spaces and prove that for each continuous map f : X?→?Y , there exists a long exact homological sequence. Moreover, we show that for each inclusion i : A?→?X of compact metric spaces, there exists an isomorphism $ {{\bar{H}}_n}(i)\approx {{\bar{H}}_n}\left( {X,A} \right) $ .     

Homology cobordism and classical knot invariants

In this paper, we define and investigate -homology cobordism invariants of -homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the -homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given -homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot. Received: May 7, 2001     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

Phantom Maps from a Strong Homology Viewpoint

Let X and Y be connected nilpotent CW-complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.     

Hochschild Homology of Twisted Tensor Products

We compute the Hochschild homology of some twisted tensor products of algebras, which are a natural generalization of the Ore extensions. We apply our result to the ring D _Q,P(X,/X) of differential operators of the multiparametric affine space, the ring of coordinates of the quantum symplectic 2v-dimensional space and the ring of coordinates of the quantum 2v-dimensional Euclidean space.     

图映射拓扑序列熵的可交换性

主要研究图上连续自映射拓扑序列熵的可交换性,证明了对任意无界的正整数递增序列A=(ai)∞i=1和任意的连续图映射f,g都有hA(fog):hA(g of).解决了Balibrea F等人在相关文献中提出的一个猜想.     

d n = 0: Generalized Homology

We study the generalized homology associated with a nilpotent endomorphism d satisfying d ^N = 0,For simplicial modules, we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalization of the notion of graded differential algebra in this context.     

F连续线性映象空间

本文采用（１）中给出的Ｆｕｚｚｙ拓扑线性空间的定义，并利用Ｆｕｚｚｙ拓扑线性空间的有关理论，对Ｆｕｚｚｙ连续线性映象空间作了较深入的研究。     

一个Cluster-Tilted代数的Hochschild同调与循环同调

基于Furuya构造的一个Cluster-Tilted代数的极小投射双模分解,用组合的方法计算了Cluster-Tilted代数的Hochschild同调空间的维数与基.当基础域的特征为零时,也计算了代数的循环同调群的维数.     

Measuring and computing natural generators for homology groups

We develop a method for measuring homology classes. This involves two problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(βn³log²n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. Finally, we prove the stability of our result. The algorithm can be adapted to measure any given class.     

具有零拓扑熵的图映射的攀援集的测度

本文研究了具有零拓扑熵的图映射f的性质,证明了在任意有限f-不变的Borel 测度μ下,其攀援集的外μ-测度都是零.     

Transitivity and Dense Periodicity for Graph Maps

In 1984, Blokh proved [A. M. Blockh, On transitive mappings of one-dimensional branched manifolds, Differential-Difference Equations and Problems of Mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 131, pp. 3–9, 1984] that any topologically transitive continuous map from a graph into itself which has periodic points has a dense set of periodic points and has positive topological entropy (in this proof a crucial role is played by the specification property, which implies these two statements). Also, he characterized the topologically transitive continuous graph maps without periodic points. Unfortunately, this clever paper is only available in Russian (except for a translation to English of the statements of the theorems without proofs—see [A. M. Blockh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat. Nauk, 42(5(257)) (1987), pp. 209–210]).     

一个自入射Koszul代数的Hochschild同调与循环同调

代数的Hochschild同调群与其对应的Gabriel箭图的循环圈有着紧密的联系.本文基于Furuya构造的一个四点自入射Koszul代数的极小投射双模分解,用组合的方法计算了该代数的Hochschild同调空间的维数,并用循环圈的语言给出该代数的Hochschild同调空间的一组k-基.进一步,当基础域k的特征为零时,我们也得到了该代数的循环同调群的维数.     

Homology of Centralizers

Given a group Π, we study the group homology of centralizers Π _g , g ? Π, and of their central quotients Π _g /〈 g〉. This study is motivated by the structure of the Hochschild and the cyclic homology of group algebras, and is based on Quillen's approach to the cyclic homology of algebras via algebra extensions. A method of computing the de Rham complex of a group algebra by means of a Gruenberg resolution is also developed.     

Leibniz Homology of Extended Lie Algebras

Leibniz homology is a noncommutative homology theory for Lie algebras. In this paper, we compute low-dimensional Leibniz homology of extended Lie algebras.     

On minimal subgroups of finite groups

In this paper, the minimal subgroups of a finite group G are studied. Using the normalizes of Sylow subgroups of G, we obtain some sufficient conditions for G to be p-milpotent and supersolvable, which ae extensions of the related results of Burnside, Ito, Buckley and Asaad.