TITLE: A binary operation on vector bundles that adds Chern classes?
QUESTION [11 upvotes]: Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern class is the sum of the Chern classes of the constituents: $c_{1}(E\oplus F)=c_{1}E+c_{1}F$.
My question is: For $k\geq 1$, are there binary operations $B^{k}(E,F)$ with the property that $c_{k}(B^{k}(E,F))=c_{k}E+c_{k}F$?
I am happy to allow the binary operation to take values in virtual bundles if need be.
Motivation: These operations would of course be pretty useful for contstructing (virtual) vector bundles with desired Chern classes.

REPLY [5 votes]: Such an operation with values in bundles does not exist for $k = 4$ and the base space $\mathbb{HP}^2$. For virtual vector bundles, it depends how you extend the definition of the Chern classes; for the natural extension such that the total Chern class is multiplicative under sums, it also doesn't exist (in fact, this extension is used in the proof).
Pullback along the evident inclusion $\mathbb{HP}^n\hookrightarrow \mathbb{HP}^\infty\cong BSU(2)$ gives a ring homomorphism $R(SU(2))\to K^0(\mathbb{HP}^n)$ preserving the augmentation ideals $I$. Since $\mathbb{HP}^n$ is covered by $n+1$ contractible sets (the standard coordinate charts), we have $I^3_{K^0(\mathbb{HP}^n)} = 0$, and therefore an induced morphism $\phi_n:R(SU(2))/I_{R(SU(2))}^{n+1}\to K^0(\mathbb{HP}^n)$. Now the representation ring $R(SU(2))$ is additively generated by the representations $[V_n]$, of dimension $n+1$, subject to the relations $[V_m]\otimes [V_n] = [V_{|m-n|}] + [V_{|m-n|+2}] + \dots + [V_{m+n}]$. Setting $v = [V_1]-2$, it follows easily that $I_{R(SU(2))}^n/I_{R(SU(2))}^{n+1}$ is generated by $v^n$; multiplicativity of the Atiyah-Hirzebruch spectral sequence then shows that $\phi_n$ is an isomorphism. (This calculation is a special case of the Atiyah-Segal completion theorem.)
Now $I_{R(SU(2))}/I_{R(SU(2))}^3$ is generated by the classes of $V_1-2$ and $V_2-3$, and their images $E_1-2$ and $E_2-3$ define virtual vector bundles over $\mathbb{HP}^2$ whose classes generate $\widetilde K^0(\mathbb{HP}^2)$. Let's calculate their total Chern classes: $E_1$ is just the canonical $2$-dimensional bundle over $\mathbb{HP}^2$, so its total Chern class is $c(E_1) = 1 + x$ where $x\in H^4(\mathbb{HP}^2;\mathbb Z)$ is a generator. Since $E_2$ is $3$-dimensional, only its second Chern class can be nonzero, and one easily calculates $c(E_2) = 1 + c_2(E_2) = 1 + 4x$: we have $E_2\oplus \mathbb C = E_1^{\otimes 2}$, so pulling back along the map $\iota:\mathbb{CP}^2\to \mathbb{HP}^2$, noting that $\iota^*E_{1} = L\oplus L^{-1}$ splits as a sum of the canonical line bundle and its dual and setting $u = c_1(L)$ we have
\begin{align*}
\iota^*c(E) &= (1+u)(1-u) = 1-u^2\\
\implies \iota^*x &= -u^2\implies \iota^*|_{H^4}\text{ is injective}\\[12pt]
\iota^*E_2 &= (\iota^*E_1)^{\otimes 2} - 1 \\
&= (L\oplus L^{-1})^{\otimes 2} - 1 \\
&= L^{\otimes 2} + L^{\otimes -2} + 1\\
\iota^*c(E_2) &= (1 + 2u)(1 - 2u)\\
&= 1 - 4u^2
\end{align*}
The total Chern class can be extended to a natural transformation $K^0(X) \to \bigoplus H^{2*}(X;\mathbb Z)$ by $[E] - [F]\mapsto c(E)c(F)^{-1}$. We immediately obtain
\begin{align*}
c(k\cdot E_1 + l\cdot E_2) &= (1+x)^k(1+4x)^l \\
&= 1 + (k + 4l)x + \left(\frac{k(k-1)}{2} + 4kl + 8l(l-1)\right)x^2\\
&= 1 + (k+4l)x + \left(\frac{(k+4l)(k+4l-1)}{2} - 6l\right)x^2
\end{align*}
In particular, no cohomology class of the form $(3n+2)x^2$ with $n\in\mathbb Z$ can be the fourth Chern class of a (virtual) bundle over $X$, so there is no (virtual) bundle $F$ with $c_4(F) = c_4(E_1^{\oplus 2}) + c_4(E_1^{\oplus 2})$.