# The Stiefel-Whitney classes of a tangent bundle as a manifold

For any smooth manifold $$M$$, the tangent bundle $$TM$$ as a manifold is always orientable. In other words, the first Stiefel-Whitney class $$w_1$$ of the manifold $$TM$$ always vanishes.

Question: Does the manifold $$TM$$ also have vanishing higher Stiefel-Whitney classes $$w_{i>1}$$? If not, how can we compute them provided we know the Stiefel-Whitney classes of $$M$$?

p.s. I'm concerned about $$w_2$$ in particular.

The characteristic classes of a tangent bundle $$TM$$, treated as a manifold itself, are defined using the double tangent bundle $$TTM$$. Given the projection $$\pi: TM \to M$$, the double tangent bundle splits as $$\pi^* TM \oplus \pi^* TM$$, and we have $$w_1(TTM) = 2 w_1(\pi^* TM) \equiv 0,$$ so $$TM$$ is orientable as you've asserted.
Similarly, $$w_2(TTM) = 2w_2(\pi^* TM) + w_1(\pi^* TM)^2 \equiv \pi^* w_1(TM)^2,$$ so whether it vanishes depends on $$w_1(TM)^2$$ in the cohomology ring of $$M$$.
You can work out what the other Stiefel-Whitney classes $$w_i(TTM)$$ are, using the formula $$w(TTM) = \pi^* w(TM)^2$$, where $$w = 1 + w_1 + w_2 + \cdots$$ is the total Stiefel-Whitney class.