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In this paper we continue our investigations of 4-dimensional complexes in [A. Cavicchioli, F. Hegenbarth, F. Spaggiari,  Four-dimensional complexes with fundamental class , Mediterr. J. Math.  17  (2020), 175]. We study a class of finite oriented 4-complexes which we callFC  4{\mathrm{FC}_{4}}   -complexes, defined as follows. AnFC  4{\mathrm{FC}_{4}}   - complex  is a 4-dimensional finite oriented  CW -complex  X  with a single 4-cell such thatH  4   ⁢   (  X  ,  ℤ  )    ≅  ℤ{H_{4}(X,\mathbb{Z})\cong\mathbb{Z}}    with a fundamental class[  X  ]   ∈    H  4   ⁢   (  X  ,  ℤ  ){[X]\in H_{4}(X,\mathbb{Z})}   . By well-known results of Wall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes  K  does an element[  φ  ]   ∈    π  3   ⁢   (  K  ){[\varphi]\in\pi_{3}(K)}    exist such thatK   ∪  φ    D  4{K\cup_{\varphi}D^{4}}    is a Poincaré complex? Second, if there exists one, how many others can be constructed from  K ? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on  K  and[  φ  ]   ∈    π  3   ⁢   (  K  ){[\varphi]\in\pi_{3}(K)}    to satisfy Poincaré duality withℤ{\mathbb{Z}}   - and Λ-coefficients. Here Λ denotes the integral group ring ofπ  1   ⁢   (  K  ){\pi_{1}(K)}   . Before, we give a classification of allFC  4{\mathrm{FC}_{4}}   -complexes based on the finite 3-complex  K , and make some remarks concerningℤ{\mathbb{Z}}   - and Λ-Poincaré duality.

Fundamental class, Poincaré duality and finite oriented FC 4 -complexes