Bundles over Connected Sums

TL;DR

This paper investigates the homology of total spaces of principal bundles over connected sums, revealing conditions under which these spaces resemble connected sums and providing explicit homology calculations for specific cases.

Contribution

It demonstrates that the homology of certain bundle total spaces over connected sums can be computed as pullbacks, and provides explicit examples and classifications, including for $U(1)$ bundles over 6-manifolds.

Findings

Homology of bundle total spaces can match that of a summand under certain conditions.

Examples show total spaces may or may not be homotopy equivalent to connected sums.

Explicit homology description for $U(1)$ bundles over specific 6-manifolds.

Abstract

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.