Question about Chern Character in Hatcher's book

I have a question about an argument from Allen Hatcher's script Vector Bundles and K-Theory in Cor. 4.4 (see page 110). Here the excerpt:

We consider a vector bundle $$E \to S^{2n}$$, Then for Chern classes we know (by cosidering cohomology groups of $$S^{2n}$$) that $$c_1(E) =... c_{n-1}(E)=0$$.

Futhermore by definitionof Chern character we have $$ch(E)= dim E + s_n(c_1,\ldots, c_n)/n!$$

My question is why holds

$$s_n(c_1,\ldots, c_n)/n!=\pm nc_n(E)/n!$$? (*)

The author refers to a recursion formula from page 63:

$$s_n= \sigma_1 s_{n-1} - \cdots +(-1)^{n-1}n\sigma_n$$.

where $$\sigma_k$$ are the $$k$$-th symmetric polynomials.

What I don't understand is why $$s_n(c_1,\ldots, c_n)/n!=\pm nc_n(E)/n!$$ and not $$s_n(c_1,..., c_n)/n!=c_n^n(E)/n!$$?

Indeed, here the symmetric polynomials are considered in variebles $$t_i:= c_i(E)$$ therefore $$\sigma_1= \sum c_i(E)= c_n(E)$$ and $$\sigma_k=0$$ for $$k >1$$ since all summands of $$\sigma_k$$ the containa factor $$c_j$$ with $$j \neq n$$. But this contracicts (*). Where is the error in my reasonings?

Thank you.

I think you are confused about what $$s_n$$ means here. The notation $$s_n(c_1,\dots,c_n)$$ does not mean we are substituting the $$c_i$$ for the variables $$t_i$$ in the symmetric polynomial $$t_1^n+\dots+t_n^n$$. Rather, $$s_n$$ is defined as the polynomial which, when inputted the elementary symmetric polynomials in $$t_1,\dots,t_n$$, outputs $$t_1^n+\dots+t_n^n$$. That is, $$s_n(\sigma_1(t_1,\dots,t_n),\dots,\sigma_n(t_1,\dots,t_n))=t_1^n+\dots+t_n^n.$$ In particular, this means that when we apply the formula $$s_n= \sigma_1 s_{n-1} - \dots +(-1)^{n-1}n\sigma_n$$ to $$s_n(c_1,\dots,c_n)$$, we are substituting $$c_i$$ for $$\sigma_i$$, not for $$t_i$$. Since $$c_i=0$$ for $$0, we get just $$s_n(c_1,\dots,c_n)=(-1)^{n-1}nc_n.$$

I always find it helps to think about these things in terms of the Splitting Principle:

For every complex vector bundle $$E \to X$$ of rank $$n$$ (where maybe $$X$$ has to be paracompact) there is a "splitting space" $$S(X)$$ and a continuous function $$f\colon S(X) \to X$$ such that $$f^*(E) \cong \oplus_{i=1} ^n L_i$$ where $$L_i \to S(X)$$ is a complex line bundle and $$f^*\colon H^*(X) \to H^*(S(X))$$ is injective.

If we let $$x_i = c_1(L_i)$$ then $$f^*(c_k(E)) = \sigma_k(x_1,\dots,x_n)$$

where $$\sigma_k\in \mathbb{Z}[x_1,\dots,x_n]$$ is the $$k$$-th elementary symmetric polynomial. In particular $$\sigma_1 = x_1 + \dots + x_n$$ doesn't correspond to $$c_n(E)$$, it's $$c_1(E)$$.

The polynomial $$x_1^k + \dots + x_n^k\in \mathbb{Z}[x_1,\dots,x_n]$$ is symmetric so The Theory implies it can be expressed as a polynomial in the elementary symmetric poynomials, i.e. there is an $$s_k \in \mathbb{Z}[y_1,\dots, y_n]$$ (called a "Netwon Polynomial") such that $$s_k(\sigma_1, \dots, \sigma_n) = x_1^k + \dots + x_n^k$$. Then if $$c_1(E),\dots,c_{n-1}(E)$$ vanish so will $$\sigma_1, \dots \sigma_{n-1}\in H^*(S(X))$$, so the recursion formula reduces to

$$s_n = (-1)^n n \sigma_n$$