Pontryagin-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.
One way to construct this space is as follows. Let
be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber Eb is a n-dimensional real vector space. We can form an n-sphere bundle Sph(E) → B by taking the one-point compactification of each fiber and gluing them together to get the total space. Finally, from the total space Sph(E) we obtain the Thom space T(E) as the quotient of Sph(E) by B; that is, by identifying all the new points to a single point , which we take as the basepoint of T(E). If B is compact, then T(E) is the one-point compactification of E.
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