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.     

Categorification of Persistent Homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category and show that if the target category is abelian, so is this category of interleavings.     

Embeddings of Kleinian Groups with Torsion

Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions. Received September, 16, 1999, Accepted September 14, 2000     

Homology Groups of Cubical Sets with Connections

Applied Categorical Structures - Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of “connection” as a new kind of degeneracy. In this paper,...     

On the Homology of Completion and Torsion

Let A be a commutative ring, and ${\mathfrak{a}}$ a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically ${\mathfrak{a}}$ -adically complete complexes and the category of cohomologically ${\mathfrak{a}}$ -torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso–Jeremias–Lipman, Schenzel and Dwyer–Greenlees.     

Homology Groups of Conormal Varieties

Let G be a real Lie group acting on real analytic manifold with finitely many orbits. We prove that the characteristic cycle map is a surjective homomorphism from the K-group of G-equivariant sheaves on X to the top homology group of the conormal variety of the G-action on X. We also show that the top homology group of the G-action on X is a free -module of rank equal to the number of G-orbits. This work was completed with the support of the Ministry of Science, Education and Sport of Croatia, and Infodesign d.o.o., Varaždin.     

Homology Groups of Simplicial Complements

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.     

Computational Tools in Weighted Persistent Homology

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p2 weighted persistent homology provided with some information on the mod p weighted persistent ...     

Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

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) $ .     

On Generalized Homology of Artin Groups

The Morava K-theory and the generalized BrownPeterson homology of groups related to the classical braid groups are calculated. Examples of such groups are the Artin groups and braid groups in handlebodies. Bibliography: 13 titles.     

Stable Groups and Algebraic Groups

A certain property of some type-definable subgroups of superstablegroups with finite U-rank is closely related to the Mordell–Langconjecture. This property is discussed in the context of algebraicgroups.     

Torsion Pairs in Stable Categories

Let 𝒞 be a triangulated category. When ω is a functorially finite subcategory of 𝒞, Jøtrgensen showed that the stable category 𝒞/ω is a pretriangulated category. A pair (𝒳, 𝒴) of subcategories of 𝒞 with ω ? 𝒳 ∩ 𝒴 gives rise to a pair (𝒳/ω, 𝒴/ω) of subcategories of 𝒞/ω. In this article, we find conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in terms of properties of the pair (𝒳, 𝒴). In particular, we obtain necessary and sufficient conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in the stable category 𝒞/ω when τω = ω, where τ is the Auslander–Reiten translation.     

Homology Stability for Unitary Groups II

In this paper, the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied. We obtain a much better range of homology stability compared to the results existing in the literature. An application of our results is given.     

二元广义外代数的Hochschild同调群

设Aq=k／(x2,xy+qyx,y2)是含有两个变量的广义外代数,基于Buch- weitz等人构造的极小投射双模解,广义外代数的各阶Hochschild同调群的维数被清晰地计算．