Suppose M is a $2n$-complex dimensional complex manifold.

a) **Why is Pontryagin class independent of orientation of the bundle?**

```
E.g. $p_i(\tau M) = p_i(\bar{\tau M})$
```

I know that for the top dimension, $p_{2n}(\tau M)=e(\tau M) \cup e(\tau M)$, it can be realized as a square and hence is independent of the actual orientation of $\tau M$, but why is it true for i < 2n?

b) A related question: is $c_{2i}(E) = c_{2i}(\bar{E})$ true in general? Furthermore, is there any general relation between $c(E)$ and $c(\bar{E})$, or $c_i(E)$ and $c_i(\bar{E})$? I know that for line bundle V, V \otimes \bar{V} is trivial, by taking conjugate of the transition function. But is it true for general complex bundle E that E \otimes \bar{E} is trivial? I believe not, I think I read somewhere that it is isomorphic to the bundle Hom(E, E)? What is the implication on characteristic class?

c) **T** or **F**: diffeomorphic vector bundles over the same manifold have the same characteristic classes.
I think the answer in general is no. For example, the Chern class of the canonical line bundle and the conjugate canonical line bundle over $\mathbb{C}P^1$ are different.
However, I remember learning a fact that

```
The classifying maps for two isomorphic bundles are homotopic to each other.
```

And by the functoriality of characteristic class, the characteristic class for the two bundles is just the pullback of the char class of the universal bundle, hence they would be the same...

I am confused. Or maybe the canonical line bundle and its conjugate bundle are not isomorphic after all? I know the scalar multiplication structures on the two bundles are different, but somehow they seem diffeomorphic to me. It would be great if you can help me pick out True and False statements from above. Appreciated!

Characteristic Classes, Lemma 14.9. – Adam Saltz Mar 28 '13 at 5:02