Construction of The Thom Space
One way to construct this space is as follows. Let
- p : E → B
be a rank k real vector bundle over the paracompact space B. Then for each point b in B, the fiber Fb is a k-dimensional real vector space. We can form an associated sphere bundle Sph(E) → B by taking the one-point compactification of each fiber separately. Finally, from the total space Sph(E) we obtain the Thom complex T(E) by identifying all the new points to a single point, which we take as the basepoint of T(E).
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