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4-manifolds
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic R4
In mathematics, an exotic \R^4 is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space \R^4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures \R^4, as was shown first by Clifford Taubes. Prior to this construction, non-diffeomorphic smooth structures on spheres exotic sphereswere already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and remains open as of 2024). For any positive integer ''n'' other than 4, there are no exotic smooth structures \R^n; in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to \R^n is diffeomorphic to \R^n. Small exotic R4s An exotic \R^4 is cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Form (4-manifold)
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure. Definition using intersection Let M be a closed manifold, closed 4-manifold (PL manifold, PL or smooth manifold, smooth). Take a Triangulation (topology), triangulation T of M. Denote by T^* the Poincare_duality#Dual_cell_structures, dual cell subdivision. Represent classes a,b\in H_2(M;\Z/2\Z) by 2-cycles A and B modulo 2 viewed as unions of 2-simplices of ''T'' and of T^*, respectively. Define the intersection form modulo 2 :\cap_: H_2(M;\Z/2\Z) \times H_2(M;\Z/2\Z) \to \Z/2\Z by the formula :a\cap_ b = , A\cap B, \bmod 2. This is well-defined because the intersection of a cycle and a boundary consists of an Parity (mathematics), even number of points (by definition of a cycle and a boundary). If M i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism Theorem
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] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field means a smooth scheme, smooth proper morphism, proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface x^4+y^4+z^4+w^4=0 in complex projective space, complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donaldson's Theorem
In mathematics, and especially differential topology and gauge theory (mathematics), gauge theory, Donaldson's theorem states that a definite quadratic form, definite intersection form (4-manifold), intersection form of a Compact space, compact, orientability, oriented, smooth manifold of dimension 4 is diagonalizable matrix, diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group. History The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986. Idea of proof Donaldson's proof utilizes the moduli space \mathcal_P of solutions to the Yang–Mills_equations#Anti-self-duality_equations, anti-self-duality equations on a principal bundle, principal \operatorname(2)-bundle P over the fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 Manifold
In low-dimensional topology, a branch of mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice. History The E_8 manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the E_8 manifold is not even triangulable as a simplicial complex. Construction The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for E_8. This results in P_, a 4-manifold whose boundary is homeomorphic to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the E_8 manifold. See also * * * References * * {{DEFAULTSORT:E8 Manifold 4-manifolds Geometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dolgachev Surface
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic. Properties The blowup X_0 of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface X_q is given by applying logarithmic transformations of orders 2 and ''q'' to two smooth fibers for some q\ge 3. The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature (1,9) (so it is the unimodular lattice I_). The geometric genus p_g is 0 and the Kodaira dimension is 1. found the first examples of simply-connected homeomorphic but not diffeomorphic 4-manifolds X_0 and X_3. More generally the surfaces X_q and X_r are always homeomorphic, but are not diffeomorphic unless q=r. showed that the Dolgachev surfac ... [...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. These groups are known as Kervaire–Milnor groups. Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive ''where'' and ''when'' events occur. Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity. In 1908, Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions into a single four-dimensional continuum now known as Minkowski space. This interpretation proved vital t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Structure
In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
