Four-Dimensional Complexes with Fundamental Class

Abstract

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 Poincare duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal $${\text {PD}}_4$$ -complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincare duality in all dimensions. Such complexes with partial Poincare duality properties, which we call $${\text {SFC}}_4$$ -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 $${\text {SFC}}_4$$ -complex to be a $${\text {PD}}_4$$ -complex. Finally, we obtain a partial classification of $${\text {SFC}}_4$$ -complexes. A future goal will be a classification in terms of algebraic $${\text {SFC}}_4$$ -complexes similar to the very satisfactory classification result of $${\text {PD}}_4$$ -complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).