Pontryagin classes of a tensor product of bundles

Question

This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$.

Specifically, given two real vector bundles $E$, $F$ over the same base space $B$ (paracompact, or whatever you need for your answer), is there a known formula for the total Pontryagin class $p(E\otimes F)$ in terms of $p(E)$ and $p(F)$? If not, is there at least a formula for classes with small degree, analogous to the formula for $c_1$ of a tensor product of complex line bundles?
The particular example I have in mind is where $E=F$ is the realification of the tautological complex line bundle $\eta$ over $\mathbb{CP}^n$, where $n\geq 4$ so that $\eta \otimes_\mathbb{R} \eta$ has potentially non-zero $p_1$ and $p_2$.

I've checked the usual references (Milnor-Stasheff, various Hatchers) but they don't seem to tackle this particular problem. Maybe there is an argument using Chern-Weil theory and a formula for the curvature form of a tensor product of bundles with connection.

Answer 1

As Qiaochu Yuan alluded to in the comments, we have $(E\otimes_{\mathbb{R}} F)\otimes_{\mathbb{R}}\mathbb{C} \cong E_{\mathbb{C}}\otimes_{\mathbb{C}}F_{\mathbb{C}}$ where $E_{\mathbb{C}} = E\otimes_{\mathbb{R}}\mathbb{C}$ and $F_{\mathbb{C}} = F\otimes_{\mathbb{R}}\mathbb{C}$, so

$$p_i(E\otimes_{\mathbb{R}}F) = (-1)^ic_{2i}((E\otimes_{\mathbb{R}}F)\otimes_{\mathbb{R}}\mathbb{C}) = (-1)^ic_{2i}(E_{\mathbb{C}}\otimes_{\mathbb{C}}F_{\mathbb{C}}).$$

Therefore, you can obtain a formula for the $i^{\text{th}}$ Pontryagin class of a tensor product (of real vector bundles) if you can find a formula for the $2i^{\text{th}}$ Chern class of a tensor product (of complex vector bundles). This can be achieved using the multiplicativity of the Chern character, as is done in this answer.

Example: For real vector bundles $E$ and $F$ of ranks $r$ and $s$ respectively, we have

\begin{align*} p_1(E\otimes_{\mathbb{R}}F) &= -c_2(E_{\mathbb{C}}\otimes_{\mathbb{C}}F_{\mathbb{C}})\\ &= -rc_2(F_{\mathbb{C}}) - sc_2(E_{\mathbb{C}}) - \binom{r}{2}c_1(F_{\mathbb{C}})^2 - \binom{s}{2}c_1(E)^2 - (rs-1)c_1(E_{\mathbb{C}})c_1(F_{\mathbb{C}})\\ &= rp_1(F) + sp_1(E) - \binom{r}{2}c_1(F_{\mathbb{C}})^2 - \binom{s}{2}c_1(E)^2 - (rs-1)c_1(E_{\mathbb{C}})c_1(F_{\mathbb{C}}). \end{align*}

In particular, modulo $2$-torsion, we have $p_1(E\otimes_{\mathbb{R}}F) = rp_1(F) + sp_1(E)$.