Factorisation and holomorphic blocks in 4d

We study N = 1 theories on Hermitian manifolds of the form M-4 = S-1 x M-3 with M-3 a U(1) fibration over S-2, and their 3d N = 2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D-2 x T-2 and D-2 x S-1. We prove that when the 4d and 3d anomalies are cancelled, the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D-2 x T-2 and D-2 x S-1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.