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Thom Spectrum
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p\colon E \to B be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber E_b is an ''n''-dimensional real vector space. We can form an ''n''-sphere bundle \operatorname(E) \to B by taking the one-point compactification of each fiber and gluing them together to get the total space. Finally, from the total space \operatorname(E) we obtain the Thom space T(E) as the quotient of \operatorname(E) by ''B''; that is, by identifying all the new points to a single point \infty, which we take as the basepoint of T(E). If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
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 function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start 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 abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#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 \partial 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 manifold, piecewise linear and topological manifolds. A ''cobordism'' between manifolds M and N is a compact manifold W whose boundary is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Operation
Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled ''Universal homology groups''. Steenrod held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children. Work Thanks to Lefschetz and others, the cup product structure of cohomology was understood by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiefel–Whitney Class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to ''n'', where ''n'' is the rank of the vector bundle. If the Stiefel–Whitney class of index ''i'' is nonzero, then there cannot exist (n-i+1) everywhere linearly independent sections of the vector bundle. A nonzero ''n''th Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, S^1 \times\R, is zero. The Stiefel–Whitney class w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (mathematics)
In mathematics, an identity is an equality (mathematics), equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variable (mathematics), variables) produce the same value for all values of the variables within a certain domain of discourse. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same function (mathematics), functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Formally, an identity is a universally quantified equality. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Isomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity; that is, : \begin f(a+b)&= f(a) + f(b),\\ f(ab) &= f(a)f(b), \\ f(1_R) &= 1_S, \end for all ''a'', ''b'' in ''R''. These conditions imply that additive inverses and the additive identity are also preserved. If, in addition, is a bijection, then its inverse −1 is also a ring homomorphism. In this case, is called a ring isomorphism, and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings have exactly the same properties. If ''R'' and ''S'' are s, then the corresponding notion is that of a homomorphism, defined as above except without the third condition ''f''(1''R'') = 1''S''. A homomorphism between (unital) rings need not be a ring homomorphism. The composition of two rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oriented Bundle
In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: ''E'' →''B'', an orientation of ''E'' means: for each fiber ''E''''x'', there is an orientation of the vector space ''E''''x'' and one demands that each trivialization map (which is a bundle map) :\phi_U : \pi^(U) \to U \times \mathbf^n is fiberwise orientation-preserving, where R''n'' is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of ''E'', which is the real general linear group ''GL''n(R), can be reduced to the subgroup consisting of those with positive determinant. If ''E'' is a real vector bundle of rank ''n'', then a choice of metric on ''E'' amounts to a reduction of the structure group to the orthogonal group ''O''(''n''). In that situation, an orientation of ''E'' amounts to a reduction from ''O''(''n'') to the special orthogonal group ''SO''(''n''). A vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Construction Of The Thom Space
Construction are processes involved in delivering buildings, infrastructure, industrial facilities, and associated activities through to the end of their life. It typically starts with planning, financing, and design that continues until the asset is built and ready for use. Construction also covers repairs and maintenance work, any works to expand, extend and improve the asset, and its eventual demolition, dismantling or decommissioning. The construction industry contributes significantly to many countries' gross domestic products (GDP). Global expenditure on construction activities was about $4 trillion in 2012. In 2022, expenditure on the construction industry exceeded $11 trillion a year, equivalent to about 13 percent of global GDP. This spending was forecasted to rise to around $14.8 trillion in 2030. The construction industry promotes economic development and brings many non-monetary benefits to many countries, but it is one of the most hazardous industries. For exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Cohomology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If ''P'' is a single-point space, then with the usual definitions the integral homology group :''H''0(''P'') is isomorphic to \mathbb (an infinite cyclic group), while for ''i'' ≥ 1 we have :''H''''i''(''P'') = . More generally if ''X'' is a simplicial complex or finite CW complex, then the group ''H''0(''X'') is the free abelian group with the connected components of ''X'' as generators. The reduced homology should replace this group, of rank ''r'' say, by one of rank ''r'' − 1. Otherwise the homology groups should remain unchanged. An ''ad hoc'' way to do this is to think of a 0-th homology class not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left-hand Side And Right-hand Side Of An Equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. definition and example of abbreviation More generally, these terms may apply to an or ; the right-hand side is everything on the right side of a test operator in an [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Of A Vector Bundle
In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: ''E'' →''B'', an orientation of ''E'' means: for each fiber ''E''''x'', there is an orientation of the vector space ''E''''x'' and one demands that each trivialization map (which is a bundle map) :\phi_U : \pi^(U) \to U \times \mathbf^n is fiberwise orientation-preserving, where R''n'' is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of ''E'', which is the real general linear group ''GL''n(R), can be reduced to the subgroup consisting of those with positive determinant. If ''E'' is a real vector bundle of rank ''n'', then a choice of metric on ''E'' amounts to a reduction of the structure group to the orthogonal group ''O''(''n''). In that situation, an orientation of ''E'' amounts to a reduction from ''O''(''n'') to the special orthogonal group ''SO''(''n''). A vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
