In Characteristic Classes p. 157, Milnor and Stasheff construct the Chern classes. Let $\xi:E \to B$ be a complex vector bundle of dimension $n$. We proceed inductively:
- The top Chern class $c_n$ is defined to be the Euler class of the underlying real bundle; i.e., $c_n := e(\xi_{\mathbb{R}})$.
- One forms a new bundle $\xi_0$ over $E_0 := E$ minus zero vectors (of fibers), which we define fiber-wise: let $F$ be the fiber (considered embedded in $E$) of $\xi$ and $v \in F$, then $\xi_0^{-1}(v) := F/\mathbb{C}v$. This is a complex bundle of dimension $n-1$.
- In particular, $\xi_0$ is a real vector $2(n-1)$-bundle, hence we have the exact Gysin sequence $$\dots \rightarrow H^{i-2n+2}(B) \rightarrow H^{i}(B) \rightarrow H^{i}(E_0) \rightarrow H^{i-2n+3}(B) \rightarrow \dots$$ The first map is multiplication by the Euler class, and the second is induced by $\pi_0 := $ the restriction of $\xi$ to $E_0$.
- For $i$ small compared to $n$, $H^{i-2n+2}(B) = H^{i-2n+3}(B) = 0$ so $\pi_0$ induces an isomorphism.
- Now we define $c_{n-1} := \pi_0^{* \,-1}(c_n)$. This is well-defined by (4).
Can I construct Conner-Floyd classes this way? If not, why not?
I am doubtful: an orientation is needed for the Gysin sequence. In the above case, I think $\xi_0$ got its orientation in a "canonical" way from $\xi$. Is this still the case when we consider other cohomology theories? (Maybe the right answer to this question is that I should be less ignorant of orientations.)
I ask this question because the references I find all cite "Grothendieck's method." I like this approach--it emphasizes that one only needs the Euler class, which itself facilitates the Thom isomorphism, thus revealing the 'closeness' of the Thom isomorphism and Chern classes. This construction from Milnor-Stasheff similarly makes it very clear that one 'only needs' the Euler class.
I highlight that last bit to invite corrections to my thinking, extra commentary, ...