complex numbers structure of Chern classes I have started to read on Chern classes. I cannot quite yet see where the complex numbers and their properties come in as we define Chern classes and other properties. I do see it in later theorems and computation. It seems we could have defined projectivization for any vector bundle for example or even Chern classes. Am I missing something? Where and when do complex field properties really come in?
I am reading Bott and Tu book differential forms in Algebraic Topology. There the construction is based on line bundles and then projectivization and at least in initial steps I do not see any complex field requirement in definitions or proofs. See pp 267 and 270. https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf
 A: For complex line bundles $L$ the first Chern class $c_1(L)$ takes values in $H^2(-, \mathbb{Z})$ and this is quite specific to working over $\mathbb{C}$. One way of thinking about this class is as coming from the exponential sequence
$$1 \to \mathbb{Z} \xrightarrow{2\pi i} \mathbb{C} \xrightarrow{\exp} \mathbb{C}^{\times} \to 1$$
which induces a long exact sequence with connecting homomorphism $H^1(-, \mathbb{C}^{\times}) \to H^2(-, \mathbb{Z})$; this is exactly the first Chern class. Another way of thinking about this class is as a generator of the cohomology of $\mathbb{CP}^{\infty}$, the classifying space of complex line bundles.
Over $\mathbb{R}$ we instead have the first Stiefel-Whitney class $w_1(L)$ of a real line bundle which takes values in $H^1(-, \mathbb{Z}_2)$ and which generates the mod-$2$ cohomology of $\mathbb{RP}^{\infty}$, the classifying space of real line bundles. This comes directly from $\mathbb{R}^{\times}$ being homotopy equivalent to $\mathbb{Z}_2$, with no need for a long exact sequence.
Above, $\mathbb{C}^{\times}$, which is homotopy equivalent to $S^1$, satisfies $\pi_0 \cong 1, \pi_1 \cong \mathbb{Z}$, whereas $\mathbb{R}^{\times}$, which is homotopy equivalent to $S^0$, instead satisfies $\pi_0 \cong \mathbb{Z}_2, \pi_1 \cong 1$. So their homotopy is quite different and this leads to the difference in their characteristic classes.
