# Relating the Chern classes of vector bundles with structure groups $G_1$, $G_2$, and $G_1\times G_2$

Let $$E$$ be a complex vector bundle whose structure group $$G$$ is a product: $$G=G_1\times G_2$$. If the fiber of $$E$$ is $$n$$-dimensional, then are the Chern classes of $$E$$ related to the Chern classes of vectors bundles with structure groups $$G_1$$ and $$G_2$$, also with $$n$$-dimensional fibers and over the same base?

In case it makes a difference, I'm especially interested in the case $$G_1=U(1)$$ and $$G_2=SU(N)$$, but the more general the better.

I'm aware of the Whitney sum formula (which expresses the Chern classes of a Whitney sum $$E_1\oplus E_2$$ of two vector bundles in terms of the Chern classes of the individual vector bundles $$E_1$$ and $$E_2$$), and I'm aware that the Whitney sum of $$G_1$$ and $$G_2$$ principal bundles is a $$G_1\times G_2$$ principal bundle , but I don't know if these observations have anything to say about my question.

Reference:

1. Page 22 in Cohen (1998), The Topology of Fiber Bundles Lecture Notes (http://math.stanford.edu/~ralph/fiber.pdf)

If $$E \to X$$ is a complex vector bundle with structure group $$G=U(1)\times SU(n-1) \subset U(n)$$ there exists complex line bundle $$E_1$$ (with structure group $$U(1)$$) and a complex vector bundle $$E_2$$ with structure group $$SU(n-1)$$ such that $$E$$ is isomorphic to $$E_1 \oplus E_2$$.

This makes the Chern classes of $$E$$ computable in the terms of chern classes of $$E_1$$ and $$E_2$$.

For the statement, consider the $$G$$ principle bundle $$P\to X$$ associated to $$E$$. The proection \begin{align*} pr_1:G \to U(1) \end{align*} induces a $$U(1)$$ principle bundle $$P_1 := (P \times U(1))/G$$. Where the right action of $$G$$ on $$P\times U(1)$$ is given by \begin{align*} (p,z)\cdot g:= (p\cdot g, pr_1(g)^{-1}\cdot z). \end{align*}

Analogously $$pr_2:G \to SU(n-1)$$ defines a $$SU(n-1)$$ principle bundle $$P_2$$. There is an ismorphism of $$G$$ principle bundles $$P \overset{\sim}{\to} P_1 \times P_2$$. Take $$E_1,E_2$$ to be the associated vector bundles respectively.

This corresponds to a map of classifying spaces $$B(G_1\times G_2) \to BG_1 \times BG_2$$ which is homotopy inverse to $$BG_1\times BG_2 \to B(G_1 \times G_2)$$.

• Thank you for the answer! I'll need to spend some time processing it to make sure I understand everything. Oct 13, 2021 at 0:11
• If I understand correctly, you've demonstrated the existence of vector bundles $E_1$ and $E_2$ which satisfy the conditions I described, and whose Chern classes can be used to compute the Chern classes of $E$ though the Whitney sum formula, correct? And pardon my ignorance, but are the associated bundles $E_1$ and $E_2$ uniquely determined by the principal-bundle part of what you described? Its fine if they're not, just want to be sure I understand. Oct 14, 2021 at 1:26
• @ChiralAnomaly Yes that is correct. I am not sure if I understand your second question. Starting with your vector bundle $E$ with structure group $G$ one has explicit constructions for $P$, $P_1$,$P_2$, $E_1$, $E_2$, in that sense they are uniquely determined by these constructions, but I am not sure if that is what you mean. Are you more familiar with transition functions (cocycles) for vector bundles than with principal bundles? Oct 14, 2021 at 7:57
• Yes, that's what I meant by my second question in the comment, and your reply clears it up for me. My background is physics, and my understanding of fiber bundles is still pretty shallow -- enough to appreciate the definitions, but not enough to stand on my own yet, so everything you wrote is helpful. Thanks again! Oct 14, 2021 at 14:18