Symmetric products and a Cartan-type formula for polyhedral products (Q6613455)

In this paper the authors give an analogue of the homological Cartan formula for the additive structrue of the homology of the polyhedral product \(Z(K; (X,A))\) of any family of finite, pointed, path-connected \(CW\)-complexes \((X,A)\) over a simplicial complex \(K\). To obtain this the authors extend the ``geometric'' Cartan type formula of the polyhedral smash product \(\widehat{Z}(K; (U,V))\) for wedge decomposable pairs using a novel interplay between polyhedral products and the infinite symmetric products.\N\NAs an application, the Hilber-Poincaré series for the polyhedral product \(Z(K;(X,A)\) is established in terms of the Hilbert-Poincaré series of the links of simplices in full subcomplexes \(K_J\) of \(K\) and associated with spheres and mod-\(p^r\) Moore spaces.\N\NFor the entire collection see [Zbl 1540.57001].