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Plumbing (mathematics)
In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of disk bundles. It was first described by John Milnor and subsequently used extensively in surgery theory to produce manifolds and normal maps with given surgery obstructions. Definition Let \xi_i=(E_i,M_i,p_i) be a rank ''n'' vector bundle over an ''n''-dimensional smooth manifold M_i for ''i'' = 1,2. Denote by D(E_i) the total space of the associated (closed) disk bundle D(\xi_i)and suppose that \xi_i, M_i and D(E_i)are oriented in a compatible way. If we pick two points x_i\in M_i, ''i'' = 1,2, and consider a ball neighbourhood of x_i in M_i, then we get neighbourhoods D^n_i\times D^n_i of the fibre over x_i in D(E_i). Let h:D^n_1\rightarrow D^n_2 and k:D^n_1\rightarrow D^n_2 be two diffeomorphisms (either both orientation preserving or reversing). The plumbing of D(E_1) and D(E_2) at x_1 and x_2 is d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelizable Manifold
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M. Examples *An example with n = 1 is the circle: we can take ''V''1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every Lie group ''G'' is paralleliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundles
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected) manifolds in each dimension separately: * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Vector Bundle
In mathematics, a stable vector bundle is a ( holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles \mathbfGL_n is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of \mathbb^1 by \mathcal(1) there is an exact sequence0 \to \mathcal(-1) \to \mathcal\oplus \mathcal \to \mathcal(1) \to 0which represents a non-zero element in v \in \text^1(\mathcal(1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Normal Bundle
In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after Michael Spivak. Construction via embeddings Given an embedding of a manifold in Euclidean space (provided by the theorem of Hassler Whitney), it has a normal bundle. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to isotopy, thus the (class of the) bundle is unique, and called the ''stable normal bundle''. This construction works for any Poincaré space ''X'': a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a spherical fibration over ''X''. For more restricted spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Obstruction
In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal (X) \to L_n (\pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n \geq 5: A degree-one normal map (f,b) \colon M \to X is normally cobordant to a homotopy equivalence if and only if the image \theta (f,b)=0 in L_n (\mathbb pi_1 (X). Sketch of the definition The surgery obstruction of a degree-one normal map has a relatively complicated definition. Consider a degree-one normal map (f,b) \colon M \to X. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve (f,b) so that the map f becomes m-connected (that means the homotopy groups \pi_* (f)=0 for * \leq m) for high m. It is a consequence of Poincaré duality that if we can achieve this for m > \lfloor n/2 \rfloor then th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Invariant
In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex ''X'' (more geometrically a Poincaré space), a normal map on ''X'' endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, ''X'' has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold ''M'' to ''X'' matching the fundamental classes and preserving normal bundle information. If the dimension of ''X'' is \ge 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to ''X'' actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov. The cobordism classes of normal maps on ''X'' are called normal invariants. Depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic Sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). The first exotic spheres were constructed by in dimension n = 7 as S^3- bundles over S^4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. Introduction The unit ''n''-sphere, S^n, is the set of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M \hookrightarrow W \quad\mbox\quad N \hookrightarrow W are homotopy equivalences. The ''h''-cobordism theorem gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder ''M'' × , 1 Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
