An analogue of Ganea’s theorem for connected sums

Abstract 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 $$M\#N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>#</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> 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 $$k\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The required condition holds, for instance, for products of spheres. Examples are given that are of particular interest in toric topology.