Cech Complex Research Articles

Čech (Co-) Complexes as Koszul Complexes and Applications

In the main results of the paper it is shown that the Cech (co-) homology might be considered as an appropriate Koszul (co-) homology. Let $\check {C}_{\underline {x}}$ denote the Cech complex with respect to a system of elements $\underline {x} = x_{1},\ldots ,x_{r}$ of a commutative ring R. We construct a bounded complex ${\mathscr{L}}_{\underline {x}}$ of free R-modules and a quasi-isomorphism ${\mathscr{L}}_{\underline {x}} \overset {\sim }{\longrightarrow } \check {C}_{\underline {x}}$ and isomorphisms ${\mathscr{L}}_{\underline {x}} \otimes _{R} X \cong K^{\bullet }(\underline {x}-\underline {U}; X[\underline {U}^{-1}])$ and $\text {Hom}_{R}({\mathscr{L}}_{\underline {x}},X) \cong K_{\bullet }(\underline {x}-\underline {U};X[[\underline {U}]])$ for an R-complex X. Here $\underline {x} - \underline {U}$ denotes the sequence of elements x1 − U1,…,xr − Ur in the polynomial ring $R[\underline {U}] = R[U_{1},\ldots ,U_{r}]$ in the variables $\underline {U}= U_{1},\ldots ,U_{r}$ over R. Moreover $X[[\underline {U}]]$ denotes the formal power series complex of X in $\underline {U}$ and $X[\underline {U}^{-1}]$ denotes the complex of inverse polynomials of X in $\underline {U}$ . Furthermore $K_{\bullet }(\underline {x}-\underline {U};X[[\underline {U}]])$ resp. $K^{\bullet }(\underline {x}-\underline {U}; X[\underline {U}^{-1}])$ denotes the corresponding Koszul complex resp. the corresponding Koszul co-complex. In particular, there is a bounded R-free resolution of $\check {C}_{\underline {x}}$ by a certain Koszul complex. This has various consequences e.g. in the case when $\underline {x}$ is a weakly proregular sequence. Under this additional assumption it follows that the local cohomology $H^{i}_{\underline {x} R}(X)$ and the left derived functors of the completion ${\Lambda }_{i}^{\underline {x} R}(X), i \in \mathbb {Z},$ are a certain Koszul cohomology and Koszul homology resp. This provides new approaches to the right derived functor of torsion and the left derived functor of completion with various applications.

The Vietoris–Rips complexes of a circle

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.

Homology-Based Distributed Coverage Hole Detection in Wireless Sensor Networks

Homology theory provides new and powerful solutions to address the coverage problems in wireless sensor networks (WSNs). They are based on algebraic objects, such as Cech complex and Rips complex. Cech complex gives accurate information about coverage quality but requires a precise knowledge of the relative locations of nodes. This assumption is rather strong and hard to implement in practical deployments. Rips complex provides an approximation of Cech complex. It is easier to build and does not require any knowledge of nodes location. This simplicity is at the expense of accuracy. Rips complex can not always detect all coverage holes. It is then necessary to evaluate its accuracy. This work proposes to use the proportion of the area of undiscovered coverage holes as performance criteria. Investigations show that it depends on the ratio between communication and sensing radii of a sensor. Closed-form expressions for lower and upper bounds of the accuracy are also derived. For those coverage holes which can be discovered by Rips complex, a homology-based distributed algorithm is proposed to detect them. Simulation results are consistent with the proposed analytical lower bound, with a maximum difference of 0.5%. Upper bound performance depends on the ratio of communication and sensing radii. Simulations also show that the algorithm can localize about 99% coverage holes in about 99% cases.

Geometry-driven Collapses for Converting a Čech Complex into a Triangulation of a Nicely Triangulable Shape

Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing's house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point-cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Cech complex by a sequence of collapses. We proceed in two phases. Starting from the Cech complex, we first produce a sequence of collapses that arrives to the Cech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some robust covering of the shape.

Relative Beilinson monad and direct image for families of coherent sheaves

The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel’fand-Gel’fand, we give an alternate description in finite terms. Using this description we can characterize the generic complex, over the variety of finite free complexes of a given shape, as the direct image of an easily-described vector bundle. We can also give explicit descriptions of the loci in the base spaces of flat families of sheaves in which some cohomological conditions are satisfied—for example, the loci where vector bundles on projective space split in a certain way, or the loci where a projective morphism has higher dimensional fibers. Our approach is so explicit that it yields an algorithm suited for computer algebra systems.

A new proof of the Gerritzen-Grauert theorem

The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

Applications of Atiyah–Hirzebruch spectral sequences for motivic cobordism

In this paper, we study applications of Atiyah?Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel. For example, we compute the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky. The cobordism ring $MGL / 2^{*,*} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz. Moreover, we study algebraic cobordism of the classifying spaces $BG$ for algebraic groups over $\mathbb{C}$. For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.

The global topological meaning of cocycles in a Gauge group

This Letter shows explicitly that the descent sequence of cocycles in a gauge group realizes the Cech-de Rham double complex. Meanwhile the Cech complex corresponds to finite gauge transformations. This Letter also shows how the indices of each order of cocycles characterizes the obstructions on overlapping spheres in different dimensions and how these indices are equal to a common one.

Characteristic Pontryagin cycles in a Cech complex

Characteristic Pontryagin cycles in a Cech complex