Related papers: On Hawaiian homology groups
By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\displaystyle{\varprojlim (Y_i^{(n)},y_i^*)}$, where $(Y_i^{(n)},y_i^*)=\bigvee_{j\leq i}(X_j^{(n)},x_j^*)$ is the wedge of $X_j^{(n)}$'s in which $X_j^{(n)}$'s are…
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.
Can we do a topological study of various classes of normal subgroups endowed with a hull-kernel-type topology? In this paper, we have provided an answer to this question. We have introduced as well a new class of normal subgroups called…
This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…
In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of…
There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation…
In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology…
This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as…
Homological stability has shown itself to be a powerful tool for the computation of homology of families of groups such as general linear groups, mapping class groups or automorphisms of free groups. We survey here tools and techniques for…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for various mapping class groups of a non--orientable surface $N$ with punctures and/or boundary.
Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as a generalization of the mapping class group. Using torsion invariants, we show that the abelianization of…
We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…
For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
Endowed with quotient topology inherited from the space of based loops, the fundamental group of the Hawaiian earring fails to be metrizable. The fundamental group of any space which retracts to the Hawaiian earring is also nonmetrizable.
Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient…
We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair…