TITLE: Under what type of transformations are characteristic classes and characteristic numbers of a manifold invariant?
QUESTION [6 upvotes]: When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they are not I will be very confused.
Then I hear talk about "topological invariants of smooth manifolds". I think this is one of my difficulties. Does this means that if you have two smooth manifolds and a homeomorphism between these then the invariant is preserved and has nothing to do with the chosen smooth structure? It seems like a strange thing to say.
I believe that Milnor has shown that the integer Pontrjagin classes are NOT topological invariants, and Novikov has proved it is true that the rational pontrjagin classes are. I haven't seen anything indicating either way for Chern classes.
What about characteristic numbers then? The signature of a manifold is a topological invariant so certainly certain combinations of characteristic numbers can be, but is it true in general? The literature is usually a bit advanced so I don't think I could go through it easily but of course these are important basic questions, if anyone could clarify that would be great. 
I would like to know most about Chern/Pontrjagin classes/numbers but if anyone has something else to throw in I'll gladly take it :)

REPLY [3 votes]: The oriented four-dimensional manifold $ \mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ admits a complex structure consistent with it's orientation, namely the one which comes from the blow-up of $\mathbb{CP}^2$ at a point. Note that $$H^2(\mathbb{CP}^2\#\mathbb{CP}^2; \mathbb{Z}) \cong H^2(\mathbb{CP}^2; \mathbb{Z})\oplus H^2(\overline{\mathbb{CP}^2}; \mathbb{Z}) \cong \mathbb{Z}a \oplus \mathbb{Z}b.$$
With it's standard complex structure, $c_1(\mathbb{CP}^2\#\overline{\mathbb{CP}^2}) = 3a - b$.  However, $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ also admits complex structures with first Chern class $3a + b$, $-3a + b$ and $-3a-b$. In particular, the first Chern class is not invariant under diffeomorphism.

The above statement can be deduced from the following general facts.

Suppose $E \to X$, $F \to X$ are real vector bundles and $\phi : E \to F$ is a vector bundle isomorphism. If $J$ is an almost complex structure on $F$, then $\phi^{-1}\circ J\circ\phi$ is an almost complex structure on $E$. Equipping $E$ and $F$ with these almost complex structures, they can be viewed as complex vector bundles and $\phi$ becomes a complex vector bundle isomorphism. In particular, $c_i(E) = c_i(\phi^*F) = \phi^*c_i(F)$.
In the special case where $X$ is a smooth manifold, $E = F = TX$, and $\phi = f_*$ where $f : X \to X$ is a diffeomorphism, we also have $N_{f_*^{-1}\circ J\circ f_*}(V, W) = f_*^{-1}N_J(f_*V, f_*W)$. Therefore $f_*^{-1}\circ J\circ f_*$ is integrable if and only if $J$ is integrable. If they are integrable, then $f : X \to X$ is a biholomorphism.

There is a diffeomorphism $f : \mathbb{CP}^2 \to \mathbb{CP}^2$ which acts by $-1$ on $H^2(\mathbb{CP}^2; \mathbb{Z})$; see the beginning of this answer. As in that answer, one can extend this to a self-diffeomorphism of $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ which can act by $1$ on $a$ and $-1$ on $b$, or $-1$ on $a$ and $1$ on $b$, or $-1$ on $a$ and $-1$ on $b$. Combining with the statements above, we obtain complex structures on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ with first Chern class $3a + b$, $-3a - b$, and $-3a + b$ respectively.
By construction, $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ equipped with any of these complex structures is biholomorphic to the standard $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$.