# Pontryagin classes of a tensor product of bundles

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.

• Complexification is a monoidal functor: it sends tensor products of real vector bundles to tensor products of complex vector bundles. So if you know how to solve this problem for Chern classes then you know how to solve it for Pontryagin classes. May 17, 2015 at 0:45
• Good point, I neglected to notice that. With that in mind it's elementary to compute $p_1(\eta\otimes_\mathbb{R}\eta)=-c_2((\eta\otimes_\mathbb{C}\eta) \oplus (\bar{\eta}\otimes_\mathbb{C}\bar{\eta}))=4c_1(\eta)^2$, where $\bar\eta$ is the conjugate bundle, and $p_2=0$. It seems that in general there are formulas for the total chern class of a tensor product of bundles, such as in arxiv.org/pdf/1012.0014.pdf but they don't seem very easy to work with, even in small degrees. Maybe it's more practical to work with the chern character instead. I guess my question is answered. May 18, 2015 at 10:12
• It is indeed more practical to work with the Chern character. May 18, 2015 at 18:57

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 $$p_1(E\otimes_{\mathbb{R}}F) = rp_1(F) + sp_1(E)$$.