Editing Thom space (section)

==Construction of the Thom space==

One way to construct this space is as follows. Let

:<math>p\colon E \to B</math>

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]] <math>E_b</math> is an ''n''-dimensional real [[vector space]]. We can form an ''n''-[[sphere bundle]] <math>\operatorname{Sph}(E) \to B</math> by taking the [[one-point compactification]] of each fiber and gluing them together to get the total space.{{Elucidate|What are the open sets of the new total space?|date=November 2014}} Finally, from the total space <math>\operatorname{Sph}(E)</math> we obtain the '''Thom space''' <math>T(E)</math> as the quotient of <math>\operatorname{Sph}(E)</math> by ''B''; that is, by identifying all the new points to a single point <math>\infty</math>, which we take as the [[Pointed space|basepoint]] of <math>T(E)</math>. If ''B'' is compact, then <math>T(E)</math> is the one-point compactification of ''E''.

For example, if ''E'' is the trivial bundle <math>B\times \R^n</math>, then  <math>\operatorname{Sph}(E)</math> is <math>B\times S^n</math> and, writing <math>B_+</math> for ''B'' with a disjoint basepoint, <math>T(E)</math> is the [[smash product]] of <math>B_+</math> and <math>S^n</math>; that is, the ''n''-th reduced [[suspension (topology)|suspension]] of <math>B_+</math>.

Alternatively,{{fact|date=July 2024}} since ''B'' is paracompact, ''E'' can be given a Euclidean metric and then <math>T(E)</math> can be defined as the quotient of the unit disk bundle of ''E'' by the unit <math>(n-1)</math>-sphere bundle of ''E''.