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.



A binary operation on vector bundles that adds Chern classes?
Let's work with virtual bundles. Your question is equivalent to the following:


If we fix a $k \geq 1$, does the map $BU \times BU \rightarrow K(\mathbb{Z},2k)$ representing $c\_k \otimes 1 + 1 \otimes c\_k$ factor through some map $BU \times BU \rightarrow BU$ composed with the map $BU \rightarrow K(\mathbb{Z},2k)$ representing $c\_k$. Recall that $BU$ has a CW structure with cells in every even dimension, and that $c\_k$ is represented by a single cell. From this description of $c\_k$, the second map $BU \rightarrow K(\mathbb{Z},2k)$ can be constructed by extending the map $BU\_{2k}/BU\_{2k-1} \rightarrow K(\mathbb{Z},2k)$ ($BU\_{2k}$ meaning the 2k-skeleton) which on the summand corresponding to $c\_k$ represents a generator of $\pi\_{2k}(K(\mathbb{Z},2k))$ and elsewhere is constant, to $BU/BU\_{2k-1}$ and then precomposing with the quotient $BU \rightarrow BU/BU\_{2k-1}$.


Similarly, $c\_k \otimes 1 + 1 \otimes c\_k$ can be represented in the same manner where the nontrivial maps in the bouquet of spheres correspond to the cells $c\_k \otimes 1$ and $1 \otimes c\_k$.


For a CW complex X with cells in only even dimensions and a single 0-cell, there is a one to one correspondence between $[X,BU]\_\*$ and the direct product generated by by the non-basepoint cells of $X$. This is given by letting the coordinate corresponding to the d-cell, which we name e, take $[f]:X \rightarrow BU$ to the class $S^d \xrightarrow{e} X\_d/(X\_{d-1}-e) \xrightarrow{f} BU$ (see Atiyah's K-theory Proposition 2.5.2).


In this case we consider $BU \times BU/ (BU \times BU)\_{2k-1}$, and we choose an element of the product which in the coordinates corresponding to $c\_k \otimes 1$ and $1 \otimes c\_k$ is represented by the generator of $\pi\_{2k}(BU)$. Let's call the corresponding map $\theta$ from $BU \times BU \rightarrow BU \times BU/ (BU \times BU)\_{2k-1} \rightarrow BU$. Up to homotopy we then have a factorization of $c\_k \otimes 1 + 1 \otimes c\_k$ as $c\_k \circ \theta$ because the maps $\pi\_{2k}(BU \times BU/ (BU \times BU)\_{2k-1}) \rightarrow \pi\_{2k}(K(\mathbb{Z},2k))$ are the same. This relies on the (I think true) statement that generator of $\pi\_{2k}(BU)$ under the composite $\pi\_{2k}(BU) \rightarrow\pi\_{2k}(BU /BU\_{2k-1}) \rightarrow H\_{2k}(BU /BU\_{2k-1})$ lands on the cell which represents $c\_k$.


$\bf{Edit:}$ Unfortunately, I've read that this is untrue. Instead it lands on some multiple of the dual of the Chern class. Denote this multiple $a\_k$.


So to apply the operation to the virtual bundles $E,F$ you take the composite $\theta \circ (E,F)$. As well as having the property $c\_k(\theta \circ (E,F))=a\_k (c\_k (E)+c\_k(F))$, we also have that $c\_l=0$ for $0<l<k$.


