|
Pontrjagin-Thom Construction
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: E \to B be a rank ''n'' real number, real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the Fiber (mathematics)#Fiber in naive set theory, fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
