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The question in the title, to which I add some clarification. Can every class in $H^4(B\mathbb{T})$ be realized as the second Chern class of a principal $SU(2)$ bundle? $B\mathbb{T}$ is the classifying space of a torus which is a product of copies of $U(1)$. Rationally, the two data are equivalent, but I was wondering what we can say about geometric principal bundles.

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    $\begingroup$ I think not. If $\mathbb{T}$ equals $U(1)^r$, then $H^*(B\mathbb{T})$ is isomorphic to $\mathbb{Z}[t_1,\dots,t_r]$. Since every (finite-dimensional, complex) linear representation of $\mathbb{T}$ is a sum of characters, then every $U(n)$-bundle on $B\mathbb{T}$ should be a direct sum of $U(1)$-bundles. In particular, every $SU(2)$-bundle should be of the form $L\oplus L^*$, where $L$ is a $U(1)$-bundle and $L^*$ is the dual. Thus $c_2$ in $\mathbb{Z}[t_1,\dots,t_r]_2$ equals $-c_1(L)^2$ for $c_1(L)$ in $\mathbb{Z}[t_1,\dots,t_r]_1$. So most elements, e.g., $t_1^2$, will not be $c_2$. $\endgroup$
    – Jason Starr
    Commented Oct 4, 2013 at 12:15

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