Fundamental class, Poincaré duality and finite oriented FC4-complexes

Abstract

Abstract 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 call FC 4 {\mathrm{FC}_{4}} -complexes, defined as follows. An FC 4 {\mathrm{FC}_{4}} -complex is a 4-dimensional finite oriented CW-complex X with a single 4-cell such that H 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 that K ∪ 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 with ℤ {\mathbb{Z}} - and Λ-coefficients. Here Λ denotes the integral group ring of π 1 ⁢ ( K ) {\pi_{1}(K)} . Before, we give a classification of all FC 4 {\mathrm{FC}_{4}} -complexes based on the finite 3-complex K, and make some remarks concerning ℤ {\mathbb{Z}} - and Λ-Poincaré duality.