An analogue of Ganea’s theorem for connected sums

Let M and N be simply-connected n-dimensional Poincaré Duality complexes. A condition is given on M and N which allows for the based loops on the connected sum \(M\#N\) to be expressed as a product of the based loops on M, the based loops on N, and an explicitly identified complementary factor. This is analogous to Ganea’s decomposition of the based loops on a wedge. A generalization is given for a connected sum of k Poincaré Duality complexes for any \(k\ge 2\). The required condition holds, for instance, for products of spheres. Examples are given that are of particular interest in toric topology.