Bundles over Connected Sums

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 as a pullback of a bundle over one of the summands is the same as if it had that bundle as a summand. See Theorem 3.3. An application appears in Ho et al. (Q J Math72:163–197, 2021). Examples are given, including one where the total space of the pullback is not homotopy equivalent to a connected sum 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 by Wall’s theorem. It is a connected sum of an even number of copies of \(S^3 \times S^4\) with a 7-manifold whose homology is \(\mathbb {Z}/k\) in degree 4 (and \(\mathbb {Z}\) in degrees 0 and 7 and zero in all other degrees).