Under what type of transformations are characteristic classes and characteristic numbers of a manifold invariant?

Question

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 :)

Answer 1

I assume what you're asking about is whether the Stiefel-Whitney class/numbers of a smooth manifold only depend on the homotopy type; similarly with the Pontryagin numbers of an oriented such manifold; similarly for the Chern classes of a complex manifold.

SW: Yes. These can be defined entirely in terms of Steenrod squares, and indeed they only depend on the cohomology ring of the manifold. They're therefore actually the same for any pair of homotopy equivalent manifolds. (As a curious corollary, homotopy equivalent manifolds are cobordant!)

Pontryagin classes: As you say Milnor has found examples where the Pontryagin classes do not agree, but Novikov proved that the rational Pontryagin classes are topological invariants. The Pontryagin numbers are well-defined up to homeomorphism as a corollary of the above, since torsion can't be taken to something nonzero in top homology via cup product. So Pontryagin numbers are well-defined; again a cute corollary is that homeomorphic smooth manifolds are smoothly oriented bordant. I don't know an elementary proof of this, sadly.

Chern classes aren't even well-defined up to diffeomorphism; a complex structure is a very serious addition to a manifold. (This is why you don't see them discussed in most other sources - smooth manifolds don't come with complex structures on their tangent bundle!) $c_n(M) = \chi(M)$, so that's invariant, but usually you should be able to find diffeomorphic manifolds with different say $c_1(M)$. (I don't have a good example off the top of my head. The best I can do is point out that it's a hard theorem that K3 surfaces, which are defined by the property that $c_1(M) = 0$, are all diffeomorphic.)

Answer 2

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.

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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 $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$.