# Congruence relations with Pontryagin classes

I'm reading the paper Rational Analogs in projective planes by Zhixu Su. I am trying to work out the example to calculate the form $$e_2$$ for dimension 16 in the paper. I am however not sure how to proceed from the step before the last one to the last one. I tried grouping things with $$p_j=\sum_{i_1<..., the jth Pontryagin class, without success. In the expression, $$k$$ represents the dimension of the manifold. Here are the steps that are featured in the paper:

$$e_2 = \sigma_2 (e^{x_1} + e^{-x_1}-2,x^{x_2} + e^{-x_2}-2,...) \\ =\sum_{j,k}(e^{x_j}+e^{-x_j}-2)(e^{x_k}+e^{-x_k}-2) \\ =\sum_{j,k}(x_j^2+\frac{x_j^4}{12}+\frac{x_j^6}{360}+\frac{x_j^8}{20160} + O(x_j^9))(x_k^2+\frac{x_k^4}{12}+\frac{x_k^6}{360}+\frac{x_k^8}{20160} + O(x_k^9)) \\ =\sum_{j,k}(x_j^2x_k^2 + \frac{x_j^4x_k^4}{144}+\frac{x_j^2x_k^6}{360}+\frac{x_j^6x_k^2}{360})+\text{terms of degree other than 8 and 16} \\ =p_2+\frac{p_2^2}{720}+\frac{p_4}{360}$$

Where we set beforehand $$p_1=p_3=0$$.

• Do you mean $j$s instead of $k$s in the RHS of your formula for $p_j$? – user10354138 Jun 12 at 14:37
• @user10354138, sorry, what $ks$? – Alonso Perez Lona Jun 12 at 16:25
• All occurrence of $k$ in $p_j=\sum_{i_1<\dots<i_k}^{k/2}x_{i_1}^2\dots x_{i_k}^2$, RHS doesn't depend on $j$ but on this mysterious $k$? – user10354138 Jun 12 at 16:27
• Oh, true, thanks for pointing out. – Alonso Perez Lona Jun 12 at 16:30

Hmm.. you are using $$k$$ for two different things, the dummy in $$\sum_{j,k}$$, and the dimension of manifold? Anyway, let's forget about the dimension of manifold and let every letter appearing in $$\sum_{\dots}$$ be dummy.
We have, by Newton-Girard formulae (and imposing $$p_1=p_3=0$$), \begin{align*} \sum_j x_j^4 &= p_1^2 - 2p_2=-2p_2\\ \sum_j x_j^6 &= p_1(p_1^2-2p_2)-p_2p_1+3p_3\\ &=p_1^3-3p_1p_2+3p_3=0\\ \sum_j x_j^8 &= p_1(p_1^3-3p_1p_2+3p_3)-(p_1^2 - 2p_2)p_2+p_3p_1-4p_4\\ &=p_1^4-4p_1^2p_2+4p_1p_3 + 2p_2^2-4p_4\\ &=2p_2^2-4p_4 \end{align*} So $$\sum_{j and $$\sum_{j Hence \begin{align*} &\sum_{j