  We consider the problem of describing the Pontryagin algebra (loop homology)
of moment-angle complexes and manifolds. The moment-angle complex Z_K is a cell
complex built of products of polydiscs and tori parametrised by simplices in a
finite simplicial complex K. It has a natural torus action and plays an
important role in toric topology. In the case when K is a triangulation of a
sphere, Z_K is a topological manifold, which has interesting geometric
structures.
  Generators of the Pontryagin algebra H_*(\Omega Z_K) when K is a flag complex
have been described in the work of Grbic, Panov, Theriault and Wu. Describing
relations is often a difficult problem, even when K has a few vertices. Here we
describe these relations in the case when K is the boundary of a pentagon or a
hexagon. In this case, it is known that Z_K is a connected sum of products of
spheres with two spheres in each product. Therefore H_*(\Omega Z_K) is a
one-relator algebra and we describe this one relation explicitly, therefore
giving a new homotopy-theoretical proof of McGavran's result. An interesting
feature of our relation is that it includes iterated Whitehead products which
vanish under the Hurewicz homomorphism. Therefore, the form of this relation
cannot be deduced solely from the result of McGavran.
