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Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...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 e ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extraordinary Cohomology Theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Quantum Field Theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. Overview In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Manifold
In mathematics, a piecewise linear (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 theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). Life and career René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares''), was w ... [...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 (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics ( critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface M (more generally, a manifold). If f is the function M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quintuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defined inductively using the construction of an ordered pair. Mathematicians usually write tuples by listing the elements within parentheses "" and separated by a comma and a space; for example, denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "nbsp; or angle brackets "⟨ ⟩". Braces "" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term ''tuple'' can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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