# About theorem 16.2 in Switzer's Algebraic Topology

I have some difficulties understanding a very precise point of Switzer's proof of the existence and unicity of chern classes, which is Theorem 16.2 in his book. Unfortunatly there are many notations that I have introduce to write 100% of what is needed for my question to be self-contained.

So here we are : the theorem is

16.2. Theorem. Suppose $$h^*$$ is a cohomology theory with products such that for each $$n$$ there are elements $$x_n\in h^2(CP^n)$$ satisfying

1. $$h^*(CP^n)=h^*(pt)[x_n]/x_n^{n+1}$$.
2. The inclusion $$i:CP^n \to CP^{n+1}$$ gives $$i^*x_{n+1} = x_n$$.

Then for each $$U(n)$$-bundle $$\xi$$ over a CW-complex $$X$$ there are uniquely defined elements $$c_0(\xi), c_1(\xi),\cdots, c_n(\xi)$$ with $$c_i(\xi) \in h^{2i}(X)$$ depending only on the isomorphism class of $$\xi$$ and satisfying

1. if $$\xi\to X$$ is a bundle and $$f:Y\to X$$ a map, then $$c_i(f^*\xi) =f^*c_i(\xi)$$
2. $$c_0(\xi) = 1$$
3. If $$\gamma \to CP^n$$ is the Hopf $$U(1)$$-bundle over $$CP^n$$, then $$c_1(\gamma) = x_n$$
4. If $$\xi$$ is a $$U(m)$$-bundle and $$\eta$$ is a $$U(n)$$-bundle, both over X, then... (the convolution formula).

To prove this, the first step is to introduce the projectivisation $$P(\xi)$$ of a $$U(n)$$-bundle $$\xi$$, and to show with Leray-Hirsch that $$h^*(P(\xi))=h^*(X)[1, y, y^2, \cdots, y^{n-1}]$$ as modules over $$h^*(X)$$, where $$y$$ is found by taking the pullback of a generator of $$h^2(CP^\infty)$$ through $$P(\xi)\to CP^{\infty}$$ the classifying map of the tautological line bundle over $$P(\xi)$$.

Then the classes $$c_i(\xi)$$ are defined to be the coefficients for the linear dependence formula $$y^n=(-1)^{n+1}c_n(\xi)\cdot 1+(-1)^n c_{n-1}(\xi)y+\cdots+c_1(\xi)y^{n-1}$$

My issue is the following : to show the third point of the theorem, Switzer says the following

To prove 3. we simply observe that $$P(\gamma) = CP^n$$ and $$\lambda_\gamma = \gamma$$, $$y_\gamma = x_n$$, so $$c_1\gamma = x_n$$.

where

• $$\lambda_\xi$$ stands for the tautological line bundle over $$P(\xi)$$
• $$\gamma$$ is the standard tautological bundle over $$CP^\bullet$$ as stated in the hypothesis of the theorem
• by $$y_\gamma$$ I guess that it is meant "the $$y$$ such that there is this basis obtained by the Leray-Hirsch theorem"

I can't see how this has a sense. Indeed, in the case of $$\gamma$$, the bundle to which we apply Leray-Hirsch theorem is $$CP^n\to CP^n$$, and so the basis is just $$(1)$$ over $$h^*(CP^n)$$... So there is no $$y$$ in this case. My concrete question would be : why does Switzer say that $$y_\gamma=x_n$$ ? Also, with this in mind, the linear dependence formula in the case of $$\gamma$$ gives nothing so I can't see why this could tell how $$c_1(\gamma)$$ can be computed...

In your definitions it says $$y := f^* x_\infty$$ where $$x_\infty$$ is the generator of $$h^2(\mathbb{CP}^\infty)$$ and $$f:P(\xi) \to \mathbb{CP}^\infty$$ is the classifying map of the tautological line bundle over $$P(\xi)$$.
If $$\xi = \gamma$$ is the tautological line bundle over $$X= \mathbb{CP}^n$$ you have that $$f:\mathbb{CP}^n \to \mathbb{CP}^\infty$$ is inclusion into the colimit. Now $$y=f^*x_\infty = x_n$$ by the two assumptions on the cohomology theory. The linear independence formula now says $$y^1 = (-1)^2 c_1(\gamma)$$. So $$x_n$$ is the first chern class of $$\gamma$$.