In Milnor's "Characteristics classes" there are two notation for the tangent bundle of a smooth manifold $M$. i.e. $\tau_M$ and $DM$. and $DM_x$ for tangent space. Why he uses of two different notation for one notion? I think both of them are $TM$ is new books notation.

Characteristics classes, page 14, Example 2. The tangent bundle $\tau_M$ of a smooth manifold $M$. The total space of $\tau_M$ is the manifold $DM$ consisting of all pairs $(x,v)$ with $x\in M$ and $v$ tangent to $M$ at $x$.


To specify a bundle $\tau_M$, you would have to specify two topological spaces $E$ and $B$, and a projection map $\pi$ between them, verifying some conditions along the way.

So Milnor does exactly this, he specifies $DM$ as the total space, he specifies $M$ as the base space, then $\pi$, and finally goes on to show that the conditions on $\pi$ hold.

$DM$ is the total space of the bundle $\tau_M$. They are not the same thing.


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