William S. Massey. A Basic Course in Algebraic Topology. GTM No. 127. Springer Verlag, rst edition, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# 1 (E ) of a locally connected topos E bounded over S . # Partially supported by the Natural Sciences and Engineering Research Council of Canada. Supported by the Australian Research Council. Since a well known device for calculating knot groups in topology is the van Kampen theorems [34, 29, 33, 6], we decided to look for suitable analogues of the van Kampen theorems for toposes. An abstract categorical framework for van Kampen theorems was given by Brown and Janelidze [7] using the notion of extensive category [18] and expressed in terms of coverings. This led us to formulate and prove a ....

William S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.

....structures from having sufficiently large posteriors to be classified via the MAP approach. 3. CUBICAL HOMOLOGY Cubical homology is ideally suited for digital images, due to its ability to handle voxels or pixels directly. Whereas homology is by now a standard tool of algebraic topology (cf. [6]) cubical homology is more recent [7, 8] Homology aims at counting holes in a topological space. For threedimensional image data (as investigated in this note) three non trivial homology groups H 0 , H 1 and H 2 exist. The number of connected components, tunnels and voids present in the image ....

W. S. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics. Springer-Verlag, 1991.

....high frequency noise, they preserved the relative size of the triangles, i.e. vertices were not allowed to drift in parameter space. 2. 2 Manifolds and Non Manifolds In order to better understand what follows, let us first consider the definition of an n manifold and a non manifold surface [17]. Definition: an n dimensional surface is an n manifold if all of its points have an open neighborhood homeomorphic to . Definition: a n dimensional surface is an n manifold with bound aries, if every point has an open neighborhood homeomorphic to either or . Note that for (1) corresponds to ....

W. Massey. A Basic Course in Algebraic Topology. Springer Verlag, 1991.

....of so(N) then W so(N) R;m 2 ( Phi ffi oe) 6.4.3. Proof of proposition 6.42. The idea is to show that all relevant information about a normalized marked surface M can be read from the numbers hM; Li for various L s. Keeping exercise 6. 13 and the classification of 2D surfaces (see e.g. [30]) in mind we see that (for a connected normalized surface) it is enough to read its degree, whether it is orientable, and the number of markings on each of its boundary components. Exercise 6.52. Let M be a disk with n aligned tangents marked on its boundary (so that deg M = n Gamma 1) Prove ....

W. S. Massey, A basic course in algebraic topology, Springer-Verlag GTM 127, New-York 1991.

....continuous mappings f : P E and g : P Theta [0; 1] B such that ffi f = g( Delta; 0) Then there should be a lifting b g : P Theta [0; 1] E of g, meaning a continuous mapping with g = ffi b g, such that b g( Delta; 0) f . This is similar to the lifting of paths in covering surfaces [2, 33], but now we are working with continuous families of paths parameterized by the polyhedron P . To understand what this means consider the simple case where E = B Theta F for some topological space F . Here B is the base and F is the fiber . In this case the fibration property is automatic, ....

W. Massey. A basic course in algebraic topology. Springer-Verlag, 1991.

....structures from having sufficiently large posteriors to be classified via the MAP approach. 3. CUBICAL HOMOLOGY Cubical homology is ideally suited for digital images, due to its ability to handle voxels or pixels directly. Whereas homology is by now a standard tool of algebraic topology (cf. [6]) cubical homology is more recent [7, 8] Homology aims at counting holes in a topological space. For threedimensional image data (as investigated in this note) three non trivial homology groups H 0 , H 1 and H 2 exist. The number of connected components, tunnels and voids present in the image ....

W. S. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics. Springer-Verlag, 1991.

....= imff = H 1 (M) which implies fi j 0. Hence ker = imfi = 0, and thus is injective. This implies that H 1 (M ; N) 0. But by Poincar e duality for manifolds with boundary, H 1 (M ; N) H n (M ) H n (M ) where for the second isomorphism we have used the fact that H n (M) is free (cf. [14]) We conclude that H n (M ; Z) 0. If M is nonorientable (which, by part (a) and a covering space argument, can happen if and only if N is nonorientable) essentially the same argument shows H n (M ; Z=2) vanishes. ....

W.S. Massey, A basic course in algebraic topology. Graduate Texts in Mathematics, 127, Springer-Verlag, New York, 1991.

....at most four edges, any edge is common to at most two squares. This approach, generalized to an arbitrary dimension, is the Cubical Complex structure presented in Section 2. We emphasize that our approach is little related to the concept of singular cubical complexes presented in the Massey s book [16] which are not directly suitable for computation. A presentation of cubical homology in a sense closer to the one chosen here was previously given by Ehrenborg and Hetyei [7] Next, we need to extract algebraic information from the geometric objects we deal with. This is computing homology of ....

W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York 1991.

....same set of vertices. The tiles are in ### correspondence with all the tile sets that can be constructed in the original mesh, and when quadrisected, each one has the same connectivity as the corresponding tile set. Our algorithm, motivated by the concept of covering surface in Algebraic Topology [7], is based on a theorem that states that a triangular mesh is a quadrisected mesh if and only if it is equivalent to the quadrisection of one connected components of its covering mesh. Figure 3 illustrates this construction for a simple quadrisected mesh. The connected components of the covering ....

W.S. Massey. A basic course in algebraic topology. SpringerVerlag, New York-Berlin, 1991. ISBN 0-387-97430-X.

....# 3v # 1v 1v 2v 3 e31 # # # e12 e23 f #v # 1 =v # 2v # 2 =v # 3v # 3 =v # 1 AB Figure 6: Notation used for tile construction. A: A tile set. B: The corresponding tile. Our Loop inverse subdivision algorithm, motivated by the concept of covering surface in Algebraic Topology [12], is based on a theorem that states that a triangular mesh is a quadrisected mesh if and only if it is equivalent to the quadrisection of one connected components of its covering mesh. Figure 5 illustrates this construction for a simple quadrisected mesh. The connected components of the covering ....

W.S. Massey. A basic course in algebraic topology. SpringerVerlag, New York-Berlin, 1991. ISBN 0-387-97430-X.

....In the sequel we will essentially deal with sequential relabelling sequences but the reader should keep in mind that such sequences may be done in a distributed way. I. Litovsky, Y. M etivier and E. Sopena 15 7 Coverings and k Coverings The notion of covering is well known in algebraic topology [10] and has also been studied in Graph Theory [11, 12] where it is in particular related to the notion of uniform emulation [13, 14] Concerning the theory of distributed computations, coverings of graphs have been used in particular for deriving impossibility results [2, 15] In the rst subsection ....

W. S. Massey. A basic course in algebraic topology. Springer-Verlag, 1991. Graduate texts in mathematics.

....1.3. Background material for Gromov Hausdor limits and Ricci curvature can be found in Chapter 1 Sections A C, Chapter 3 Sections A B, Chapter 5 Section A of [Gr] and in Chapters 9 10 of [Pe2] Background material on covering spaces and fundamental groups can be found in Chapters 1 2 of [Sp] and [Ma]. 2 Background and Examples In Sections 2 and 3 we consider compact length spaces. No curvature condition is assumed. See [Gr, Chapter 1] for basic results about length spaces (called path metric spaces) Recall also (c.f. Gr, Chapter 3A] the following de nition of the Gromov Hausdor distance ....

....we are limiting ourselves to compact length spaces, their universal covers may well be non compact. Recall that a space, Y , is semi locally simply connected (or semi locally one connected) if for all y 2 Y there is a neighborhood U of y such that 1 (U; y) 1 (Y; y) is trivial ( Sp, p 78] [Ma, p 142]) That is, any curve in U is contractible in Y . This is weaker than saying that U is simply connected. For a metric space Y , let r(Y ) denote the maximal number r such that every closed curve in a ball of radius r in Y is homotopic to zero in Y , the semi locally simply connectivity radius. ....

W. Massey, A basic course in algebraic topology, GTM 127, Springer-Verlag, 1991.

....FRANCIS EDWARD SU 1. Introduction The Borsuk Ulam theorem and the Brouwer xed point theorem are well known theorems of topology with a very similar avor. Both are non constructive existence results with somewhat surprising conclusions. Most topology textbooks that cover these theorems (e.g. [4], 5] 6] do not mention the two are related although, in fact, the Borsuk Ulam theorem implies the Brouwer Fixed Point Theorem. The theorems themselves are often proved using the machinery of algebraic topology or the concept of degree of a map. That one theorem implies the other can therefore ....

W.S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York, 1991.

....2.2] We then prove that oriented manifolds with this property and only one end have trivial codimension one G homology where G is any abelian group [Prop 2.1] This section of the paper is purely topological and uses Poincare Lefschetz Duality and the Universal Coecient Theorem (c.f. Mun] and [Mas]) In the second section of the paper, we eliminate the topological conditions in Propositions 2.1 using the properties of manifolds with nonnegative Ricci curvature [Props 3.1 3.3] In particular, we use the Splitting Theorem [ChGl] and a theorem in [So2] In addition to proving Theorems 1.1 ....

W. S. Massey, A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991.

....We then prove that orientable manifolds with this property and only one end have trivial 3 codimension one G homology where G is any abelian group [Prop 2.1] This section of the paper is purely topological and uses Poincare Lefschetz Duality and the Universal Coecient Theorem (c.f. Mun] and [Mas]) In the second section of the paper, we eliminate the topological conditions in Propositions 2.1 using the properties of manifolds with nonnegative Ricci curvature [Props 3.1 3.3] In particular, we use the Splitting Theorem [ChGl] and a theorem in [So2] In addition to proving Theorems 1.1 ....

W. S. Massey, A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991.

.... Field Extensions Branched Coverings; Galois extensions Regular Coverings; Galois Groups Groups of Deck transformations; For more on Coverings Spaces in particular, and Topology, in general, we recommend the classic text of Seifert and Threlfall [14] or the more recent book of Massey [12]. The rst appendix in [13] also gives a nice and quick summary. 4. It is a nontrivial result of Minkowski that for any number eld K other than Q , we have jd K j 1. This means that there exists at least one prime number p which is rami ed in K. Thus, we might say that Q is simply connected ....

W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.

....definition: a surface is a two manifold if all of its points have an open neighborhood homeomorphic to . This definition can be extended to two manifold surfaces with boundaries, where every point has an open neighborhood homeomorphic to either or . More information on n manifolds can be found in [11]. If a surface does not satisfy this criteria, it is called a non manifold. Examples of non manifold surfaces include for instance selfintersecting surfaces or T junctions 2 . There are various data structures that describe non manifolds. In the subsequent framework we use a boundary ....

W. Massey. A Basic Course in Algebraic Topology. Springer Verlag, 1991.

....An isotopy is a homotopy h t for which every h t (x) is a homeomorphism. In particular, during an isotopy of a simple curve the image remains simple at every stage [34] Homotopy is an essential tool for classifying manifolds of low dimensions, while isotopy is instrumental in knot theory [25]. Fig. 4 illustrates the difference between homotopy and isotopy. Here, X is interval [0, 1] on the real line and Y is IR 2 . The figure shows a curve deforms into another curve (or a point) For homotopy, it is not necessary that the images of h t (x) remain homeomorphic to each other. The ....

W. Massey. A Basic Course in Algebraic Topology. Springer-Verlag, New York, N.Y., 1991.

....to be able to build geologic models we have to be able to model at least two manifold surfaces with boundaries. A two manifold can be interpreted intuitively as a surface that does not intersect itself. A more rigorous definition is given in Definition 2; additional information can be found in [63]: Definition 1: A homeomorphism is defined as a continuous invertible map whose inverse is also continuous. Definition 2: An n dimensional manifold with boundary is a Hausdorff space such that each point has an open neighborhood homeomorphic to or to . A two manifold with boundary is an ....

....we have seen in Section 2.1 a representation that can model two manifold surfaces is powerful enough to model most geologic models. There is one particular instance of non manifold surfaces that we would like to model: pseudo manifold singularities, which are defined in Definition 3 as well as in [63]. Definition 3: An n dimensional pseudo manifold is an n dimensional finite, regular CWcomplex which satisfies the following three conditions: 1. Every face is a face of some n cell 2. Every dimensional cell is a face of exactly two n cells 3. Given any two n cells and there exist a sequence of ....

W. S. Massey. A Basic Course in Algebraic Topology. Springer Verlag, 1997.

....definition: a surface is a two manifold if all of its points have an open neighborhood homeomorphic to . This definition can be extended to two manifold surfaces with boundaries, where every point has an open neighborhood homeomorphic to either or . More information on n manifolds can be found in [11]. If a surface does not satisfy this criteria, then it is called a nonmanifold. Examples of non manifold surfaces include for instance self intersecting surfaces or T junctions 2 . There are various data structures that describe non manifolds. In the subsequent framework we use a boundary ....

W. Massey. A Basic Course in Algebraic Topology. Springer Verlag, 1991.

....1.3. Background material for Gromov Hausdorff limits and Ricci curvature can be found in Chapter 1 Sections A C, Chapter 3 Sections A B, Chapter 5 Section A of [Gr] and in Chapters 9 10 of [Pe2] Background material on covering spaces and fundamental groups can be found in Chapters 1 2 of [Sp] and [Ma]. 2 Background and Examples In Sections 2 and 3 we consider compact length spaces. No curvature condition is assumed. See [Gr, Chapter 1] for basic results about length spaces (called path metric spaces) Recall also (c.f. Gr, Chapter 3A] the following definition of the Gromov Hausdorff ....

....we are limiting ourselves to compact length spaces, their universal covers may well be non compact. Recall that a space, Y , is semi locally simply connected (or semi locally one connected) if for all y 2 Y there is a neighborhood U of y such that 1 (U; y) 1 (Y; y) is trivial ( Sp, p 78] [Ma, p 142]) That is, any curve in U is contractible in Y . This is weaker than saying that U is simply connected. For a metric space Y , let r(Y ) denote the maximal number r such that every closed curve in a ball of radius r in Y is homotopic to zero in Y , the semi locally simply connectivity radius. ....

W. Massey, A basic course in algebraic topology, GTM 127, Springer-Verlag, 1991.

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William S. Massey. A Basic Course in Algebraic Topology. GTM No. 127. Springer Verlag, rst edition, 1991.

No context found.

W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York, 1991.

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W. Massey, A basic course in algebraic topology, GTM 127, Springer-Verlag, 1991.

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W. S. Massey. A basic course in algebraic topology. Springer-Verlag, 1991. Graduate texts in mathematics.

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