  A principal bundle over the connected sum of two manifolds need not be
diffeomorphic or even homotopy equivalent to a non-trivial connected sum of
manifolds. We show however that the homology of the total space of a bundle
formed a pullback of a bundle over one of the summands is the same as if it had
that bundle as a summon. An application appears in a paper by Ho, Jeffrey,
Selick, Xia in the Special Issue of the Quarterly Journal of Mathematics in
honour of Sir Michael Atiyah. Examples are given, including one where the total
space of the pullback is not homotopy equivalent to a connected some with that
as a summand and some in which it is. Finally, we describe the homology of the
total space of a principal $U(1)$ bundle over a $6$-manifold of the type
described in Wall's classification. It is a connected sum of an even number of
copies of $S^3\times S^4$ with a $7$-manifold whose homology is $Z/k$ in degree
$4$ (and $Z$ in degrees $0$ and $7$, and zero in all other degrees.
