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For a vector bundle thom space $T$ is defined as $T=E/A$, where $E$ is the total space and $A$ is the set of vectors in $E$ of length $\geq 1$. Alternatively, $T$ is the mapping cone of the associated sphere bundle, i.e., $(S \times I \cup B)/ S \times {1}$. The base is glued to the total space $S$ via the projection map. However, I cannot see that these are equivalent. Any help..

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    We know the cell decomposition of T comes from the base. But how do we know that the attaching map between cells is the projection map of the sphere bundle?2012-07-04

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