DOCSLIB.ORG

Approximating Bounded, Non-Orientable Surfaces from Points

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximating Bounded, Non-Orientable Surfaces from Points Approximating Bounded, Non-orientable Surfaces from Points Anders Adamson Marc Alexa Department of Computer Science, Darmstadt University of Technology Fraunhoferstr. 5, 64283 Darmstadt [email protected] Abstract All approaches seem to assume that the represented sur- face is a solid and, thus, unbounded and globally orientable. We present an approach to surface approximation from This might stem from the fact that point clouds are mostly points that allows reconstructing surfaces with boundaries, acquired by scanning real-world (solid) objects, so that ar- including globally non-orientable surfaces. The surface is eas lacking points are considered inadequately sampled and defined implicitly using directions of weighted co-variances to be fixed. However, with the rising interest in unstruc- and weighted averages of the points. Specifically, a point tured point sets as a general surface representation, we feel belongs to the surface, if its direction to the weighted av- it is important to consider surfaces with boundary explic- erage has no component into the direction of smallest co- itly. Furthermore, enhancing reconstruction to faithfully variance. For bounded surfaces, we require in addition that deal with bounded, possibly non-orientable surfaces, in- any point on the surface is close to the weighted average of creases the robustness against sampling errors and makes the input points. We compare this definition to alternatives the algorithm more versatile. and discuss the details and parameter choices. Points on We extend our framework for approximating surfaces the surface can be determined by intersection computations. from scattered points [1] to handle smooth boundaries and We show that the computation is local and, therefore, no show that the surface could be (globally) non-orientable. globally consistent orientation of normals is needed. Conti- Specifically, we make the following contributions: nuity of the surfaces is not affected by the particular choice of local orientation. We demonstrate our approach by ren- Boundary definition: The surface is defined implicitly us- dering several bounded (and non-orientable) surfaces using ing compactly supported functions. We show how to ray casting. compute and define a smooth and natural boundary us- Keywords: point-sampled geometry, ray-surface inter- ing only the quantities we compute during surface ap- section, manifold, boundary, non-orientable proximation. Boundary behavior and parameters: We prove several properties about the behavior of the computations in- 1 Introduction volved in the estimation of the boundary. We use these approximations to solve the problem of parameter es- Point sets without additional topological information are timation and provide the user with an interval of rea- a popular representation of surfaces (see [3, 16, 32]). Real sonable parameters to choose from. world shapes are sampled using scanners resulting in an Orientation: The computation procedure for points re- initial surface representation consisting of points [21, 27]. quires only the intersection of a curve/ray with a plane. Points are also a suitable primitive for the display of sur- As the orientation of the plane’s normal is irrelevant, a faces, especially as the complexity of the shape results in global orientation of normals is not required. We show triangles being rendered into less than a pixel for the final that the continuity of the surface is not affected. display [26, 28, 18, 11, 10, 33]. For many modeling tasks, however, a (local) approxima- Despite these features, the point set could still define a tion of the surface based on the points is required. A variety solid and our implicit definition of the surface would allow of methods for the reconstruction of surfaces from points defining inside and outside. Note that we cannot handle exist. Some of these are approaches are global in nature, non-manifold surfaces, e.g., surfaces with self intersections. others allow locally computing surface parts but are based In the following, we briefly review some of the related work, on locally approximating a global implicit function. define the surface and explain how to compute it efficiently, Proceedings of the Shape Modeling International 2004 (SMI’04) 0-7695-2075-8/04 $20.00 © 2004 IEEE before we explain in detail the definition of the boundary are, e.g., positive. Our approach could, indeed, be inter- and how to choose the necessary parameters. preted in this way. 2 Related Work 3 Surface definition A variety of works tackle the (ill-posed) problem of re- We first give a brief definition of the surface as defined constructing a surface from an unstructured set of points. in [1] and establish several notations for later use. P = {p ∈ R3,i ∈{1,...,N}} Hoppe et al. [17] define a signed distance field from the Given points i on or S S points. For each point a normal direction is estimated and close to a surface . The point set surface P is defined as the normal is oriented. The signed distance to the surface the zero-set of the implicit function is defined as the normal component of the distance to the f(x)=n(x) · (x − a(x)), (1) closest point. x In a sense the surface definition of Hoppe et al. is related which describes the distance of to a plane with normal n(x) a(x) to Voronoi-based reconstruction techniques [8, 13, 4, 15, 6, through the weighted average of points .Forthe 5] as the surface is defined by a closest point relationship. surface definition to be reasonable (i.e. the surface is con- These techniques define the surface as a subset of the De- tinuously differentiable), we require the mappings defin- n : R3 → S2 launay triangulation of the point set. As such, they are all ing normal direction and weighted average 0 a : R3 → R3 global methods and generate a C surface. to be smooth in their input at least in a suf- Ω P Surfaces with higher order continuity from points are ficient neighborhood around . We will give particu- typically implicit. For the specific purpose of consolidat- lar choices for the mappings and the neighborhood below. ing registered range images Curless et al. [12] used a dis- These choices are useful for efficient and local computation crete representation (see also [23] for a comparable ap- of the surface, and they lead to natural definition of bound- proach). Radial basis functions with global [29, 9, 31] or aries. θ : R → R local [22, 19, 25] support are a more general technique for Let be a monotone decreasing function with r approximating functions from scattered constraints. Note local support θ, which is strictly positive inside the support, θ(χ)=0⇔ χ>r θ that all of these techniques are inherently global in nature i.e. θ. We will use as a weight function p x – some of them explicitly consider the case of incomplete determining the influence of a point i on a point in space . Ω point sampling and explain how, despite the missing points, The neighborhood is consequently defined as the region in space that is affected by the points: a globally unbounded and oriented surface would be recon- structed. Hierarchical methods based on blending individual local Ω= x| θ(||pi − x||) > 0 (2) implicit approximations such as [24] as well as moving least i squares type surface approximation [20, 3, 14] are local rep- Alternatively, we can define the distance of a point x to the resentations. While the MLS-based approaches explicitly point set as the minimum distance of x to any point in P: require the surface to be sampled sufficiently dense from a dP (x)=min||x − pi||. (3) closed manifold, hierarchical methods are rather concerned i with the problem of adequately filling un-sampled regions. Using this definition, Ω could be defined as the set of points Schaufler and Jensen [30] define the intersection of a ray with distance dP less than rθ, which also describes the and the point set as follows: In each point a disk is con- union of rθ-balls around the points in P. Note that all these structed using the point normal. A cylinder around the ray definitions of Ω are equivalent because we assume θ to be is intersected with the disks. The intersection is computed strictly positive in its support. as a weighted average of disks whose centers are inside the The weighted average of points a is defined (for points cylinder. Note that the reconstructed surface could have in Ω only) as boundary, because not every cylinder contains points. On the other hand, the boundary of the surface would depend piθ(||pi − x||) a(x)= i , x ∈ Ω. (4) on the particular rays chosen. i θ(||pi − x||) Our approach is also based on intersecting rays with the Normal directions are defined as directions of smallest surface [1], however, it could be understood as a robust weighted co-variance, using the same weights as before. computation of a particular implicit surface (which is iden- This could be understand as a least squares fit of a plane tical to the MLS definition [20]). Implicit surfaces typi- with unit normal n through x, i.e. the minimizer of cally represent solids. According to Bloomenthal et al. [7], 2 boundaries are generally represented by defining additional n, pi − x θ(||pi − x||) min i . (5) functions on the space and requiring that these functions ||n||=1 i θ(||pi − x||) 2 Proceedings of the Shape Modeling International 2004 (SMI’04) 0-7695-2075-8/04 $20.00 © 2004 IEEE It is well known that this constrained minimization prob- lem can be solved using the eigenvectors of the matrix W (x)={wjk} of weighted co-variances in the directions 3 of an orthonormal basis of R .Letei,i∈{0, 1, 2} be such a basis, then the coefficients of the matrix are given as wjk = ej, pi − xek, pi − xθ(||pi − x||).
Load more

Recommended publications
  • Vector Bundles on Projective Space
    Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X.
    [Show full text]
  • LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
    LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction).
    [Show full text]
  • Math 704: Part 1: Principal Bundles and Connections
    MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R).
    [Show full text]
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
    [Show full text]
  • Vector Bundles and Connections
    VECTOR BUNDLES AND CONNECTIONS WERNER BALLMANN The exposition of vector bundles and connections below is taken from my lecture notes on differential geometry at the University of Bonn. I included more material than I usually cover in my lectures. On the other hand, I completely deleted the discussion of “concrete examples”, so that a pinch of salt has to be added by the customer. Standard references for vector bundles and connections are [GHV] and [KN], where the interested reader finds a rather comprehensive discussion of the subject. I would like to thank Andreas Balser for pointing out some misprints. The exposition is still in a preliminary state. Suggestions are very welcome. Contents 1. Vector Bundles 2 1.1. Sections 4 1.2. Frames 5 1.3. Constructions 7 1.4. Pull Back 9 1.5. The Fundamental Lemma on Morphisms 10 2. Connections 12 2.1. Local Data 13 2.2. Induced Connections 15 2.3. Pull Back 16 3. Curvature 18 3.1. Parallel Translation and Curvature 21 4. Miscellanea 26 4.1. Metrics 26 4.2. Cocycles and Bundles 27 References 28 Date: December 1999. Last corrections March 2002. 1 2 WERNERBALLMANN 1. Vector Bundles A bundle is a triple (π,E,M), where π : E → M is a map. In other words, a bundle is nothing else but a map. The term bundle is used −1 when the emphasis is on the preimages Ep := π (p) of the points p ∈ M; we call Ep the fiber of π over p and p the base point of Ep.
    [Show full text]
  • Foundations of Algebraic Geometry Classes 35 and 36
    FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 35 AND 36 RAVI VAKIL CONTENTS 1. Introduction 1 2. Definitions and proofs of key properties 5 3. Cohomology of line bundles on projective space 9 In these two lectures, we will define Cech cohomology and discuss its most important properties, although not in that order. 1. INTRODUCTION As Γ(X; ·) is a left-exact functor, if 0 ! F ! G ! H ! 0 is a short exact sequence of sheaves on X, then 0 ! F(X) ! G(X) ! H(X) is exact. We dream that this sequence continues off to the right, giving a long exact se- quence. More explicitly, there should be some covariant functors Hi (i ≥ 0) from qua- sicoherent sheaves on X to groups such that H0 = Γ, and so that there is a “long exact sequence in cohomology”. (1) 0 / H0(X; F) / H0(X; G) / H0(X; H) / H1(X; F) / H1(X; G) / H1(X; H) / · · · (In general, whenever we see a left-exact or right-exact functor, we should hope for this, and in good cases our dreams will come true. The machinery behind this is sometimes called derived functor cohomology, which we will discuss shortly.) Before defining cohomology groups of quasicoherent sheaves explicitly, we first de- scribe their important properties. Indeed these fundamental properties are in some ways more important than the formal definition. The boxed properties will be the important ones. Date: Friday, February 22 and Monday, February 25, 2008. 1 Suppose X is a separated and quasicompact A-scheme. (The separated and quasicom- pact hypotheses will be necessary in our construction.) For each quasicoherent sheaf F on X, we will define A-modules Hi(X; F).
    [Show full text]
  • Principal Bundles, Vector Bundles and Connections
    Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/.
    [Show full text]
  • Chapter 3 Connections
    Chapter 3 Connections Contents 3.1 Parallel transport . 69 3.2 Fiber bundles . 72 3.3 Vector bundles . 75 3.3.1 Three definitions . 75 3.3.2 Christoffel symbols . 80 3.3.3 Connection 1-forms . 82 3.3.4 Linearization of a section at a zero . 84 3.4 Principal bundles . 88 3.4.1 Definition . 88 3.4.2 Global connection 1-forms . 89 3.4.3 Frame bundles and linear connections . 91 3.1 The idea of parallel transport A connection is essentially a way of identifying the points in nearby fibers of a bundle. One can see the need for such a notion by considering the following question: Given a vector bundle π : E ! M, a section s : M ! E and a vector X 2 TxM, what is meant by the directional derivative ds(x)X? If we regard a section merely as a map between the manifolds M and E, then one answer to the question is provided by the tangent map T s : T M ! T E. But this ignores most of the structure that makes a vector 69 70 CHAPTER 3. CONNECTIONS bundle interesting. We prefer to think of sections as \vector valued" maps on M, which can be added and multiplied by scalars, and we'd like to think of the directional derivative ds(x)X as something which respects this linear structure. From this perspective the answer is ambiguous, unless E happens to be the trivial bundle M × Fm ! M. In that case, it makes sense to think of the section s simply as a map M ! Fm, and the directional derivative is then d s(γ(t)) − s(γ(0)) ds(x)X = s(γ(t)) = lim (3.1) dt t!0 t t=0 for any smooth path with γ_ (0) = X, thus defining a linear map m ds(x) : TxM ! Ex = F : If E ! M is a nontrivial bundle, (3.1) doesn't immediately make sense because s(γ(t)) and s(γ(0)) may be in different fibers and cannot be added.
    [Show full text]
  • On Fibre Bundles and Differential Geometry
    Lectures On Fibre Bundles and Differential Geometry By J.L. Koszul Notes by S. Ramanan No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1 Tata Institute of Fundamental Research, Bombay 1960 Contents 1 Differential Calculus 1 1.1 .............................. 1 1.2 Derivationlaws ...................... 2 1.3 Derivation laws in associated modules . 4 1.4 TheLiederivative... ..... ...... ..... .. 6 1.5 Lie derivation and exterior product . 8 1.6 Exterior differentiation . 8 1.7 Explicit formula for exterior differentiation . 10 1.8 Exterior differentiation and exterior product . 11 1.9 The curvature form . 12 1.10 Relations between different derivation laws . 16 1.11 Derivation law in C .................... 17 1.12 Connections in pseudo-Riemannian manifolds . 18 1.13 Formulae in local coordinates . 19 2 Differentiable Bundles 21 2.1 .............................. 21 2.2 Homomorphisms of bundles . 22 2.3 Trivial bundles . 23 2.4 Induced bundles . 24 2.5 Examples ......................... 25 2.6 Associated bundles . 26 2.7 Vector fields on manifolds . 29 2.8 Vector fields on differentiable principal bundles . 30 2.9 Projections vector fields . 31 iii iv Contents 3 Connections on Principal Bundles 35 3.1 .............................. 35 3.2 Horizontal vector fields . 37 3.3 Connection form . 39 3.4 Connection on Induced bundles . 39 3.5 Maurer-Cartan equations . 40 3.6 Curvatureforms. 43 3.7 Examples ......................... 46 4 Holonomy Groups 49 4.1 Integral paths . 49 4.2 Displacement along paths . 50 4.3 Holonomygroup ....................
    [Show full text]
  • Sheaf Cohomology
    Sheaf Cohomology Gabriel Chˆenevert Payman Kassaei September 24, 2003 In this lecture, we define the cohomology groups of a topological space X with coefficients in a sheaf of abelian groups F on X in terms of the derived functors of the global section functor Γ(X; ¢ ). Then we introduce Cechˇ coho- mology with respect to an open covering of X, which permits to make explicit calculations, and discuss under which conditions it can be used to compute sheaf cohomology. 1 Derived functors We first need to review some homological algebra in order to be able to define sheaf cohomology using the derived functors of the global sections functor. Let A be an abelian category, that is, roughly, an additive category in which there exist well-behaved kernels and cokernels for each morphism, so that, for example, the notion of an exact sequence in A makes sense. If X is a fixed object in A and Ab denotes the category of abelian groups, then we have a contravariant functor Hom( ¢ ;X): A ¡! Ab: It is readily seen to be left exact, that is, for any short exact sequence f g 0 ¡! A ¡! B ¡! C ¡! 0 in A , the sequence g¤ f ¤ 0 ¡! Hom(C; X) ¡! Hom(B; X) ¡! Hom(A; X) is exact in Ab. Definition 1.1. An object I of A is said to be injective if the functor Hom( ¢ ;I) is exact. 1 Since Hom( ¢ ;I) is always left exact, we see that an object I of A is injective if and only if for each exact sequence 0 ! A ! B and morphism A ! I, there exists a morphism B ! I making the following diagram commute.
    [Show full text]
  • RESEARCH STATEMENT My Research Areas Are
    RESEARCH STATEMENT SANJEEVI KRISHNAN My research areas are directed algebraic topology (Section x1) and applied topology (Section x2). The appli- cations span network optimization, image analysis, distributed computing, and computability theory. Current nascent work also spans data analysis and Lorentzian geometry (Section x3). Future plans (Section x4), ex- tending past and existing work, involve unifying local-to-global methods in algebraic topology and statistics in a principled and tractable manner. 1. Directed Algebraic Topology Directed algebraic topology is an algebraic topology for spaces with extra order-theoretic structure, like the state spaces of dynamical systems. My relevant work in the field includes generalizations of topological lattices with convenient categorical properties, an identification of directed spaces admitting directed universal covers with order-theoretic properties convenient for state space analysis, general simplicial and cubical approximation theorems, the construction of (co)homology theories, and the discovery of a relevant Poincar´eDuality. The relevant work is listed in reverse chronological order. 1.1. Directed Poincar´eDuality. Flow-cut dualities in network optimization resemble Poincar´eDualities. A complete reformulation of the former in terms of the latter allows for general duality theorems of aesthetic and practical interest in optimization theory. 1.1.1. Background. Poincar´eDuality in the classical setting takes the form of isomorphisms p ∼ H (X; O ⊗R F) = Hn−p(X; F) for module-valued sheaves F on weak homology n-manifolds X [3] and even generalizes to a weak equivalence between sheaves of spectra [2] representing generalized local cycles and cocycles. A proof of the former relies on basic homological algebra while a proof of the latter relies on a local-to-global Whitehead Theorem for simplicial presheaves.
    [Show full text]
  • INTERACTION BETWEEN CECH COHOMOLOGY and GROUP COHOMOLOGY 1. Cech Cohomology Let I Be Some Index Set and Let U = {U I : I ∈ I}
    INTERACTION BETWEEN CECH COHOMOLOGY AND GROUP COHOMOLOGY TAYLOR DUPUY Abstract. [Summer 2012] These are old notes and they are just being put online now. They are incomplete and most likely riddled with errors that I may never fix. There are important technical hypotheses missing that make which make certain proofs invalid as stateded below. For example we sometimes need the sheaf of abelian group to be quasi-coherent modules to make things work. If someone reads this and wants to be awesome and email me I would appreciate it. {Taylor Abstract. [Original] Given a sheaf of groups acting on an abelian sheaf of groups we define a group cohomology theory. A special case of a theorem proves that twisted homomorphism in certain group cohomologies when paired with cocycles in cech cohomology give rise to 1-cocycles of vector bundles. 1. Cech Cohomology Let I be some index set and let U = fUi : i 2 Ig be a collection of open sets S with X = i2I Ui. Let G be a sheaf of groups on X. Q −1 A collection (gij) 2 i;j2I Γ(Ui \ Uj; G) with gij = gji is said to be a cocycle provided (1.1) gijgjkgki = 1 1 on Ui \ Uj \ Uk. The collection of cocycles will be denoted by Z (U; X; G). A Q collection gij 2 i;j2I Γ(Ui \ Uj; G) is called a coboundary if there exists some collection "i 2 G(Ui) for each i 2 I such that −1 gij = "i"j : 1 We will denote the collection of coboundaries by B (U; X; G).
    [Show full text]