Four-Dimensional Complexes with Fundamental Class

This paper continues the study of 4-dimensional complexes from our previous work Cavicchioli et al. (Homol Homotopy Appl 18(2):267–281, 2016; Mediterr J Math 15(2):61, 2018. https://doi.org/10.1007/s00009-018-1102-3) on the computation of Poincaré duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal PD 4-complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincaré duality in all dimensions. Such complexes with partial Poincaré duality properties, which we call SFC 4-complexes, are very interesting to study and can be classified, up to homotopy type. For this, we introduce the concept of resolution, which allows us to state a condition for a SFC 4-complex to be a PD 4-complex. Finally, we obtain a partial classification of SFC 4-complexes. A future goal will be a classification in terms of algebraic SFC 4-complexes similar to the very satisfactory classification result of PD 4-complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).