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 Poincaré 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 Poincaré duality in all dimensions. Such complexes with partial Poincaré 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).
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Acknowledgements
Work performed under the auspices of the scientific group G.N.S.A.G.A. of the C.N.R (National Research Council) of Italy and partially supported by the MIUR (Ministero per la Ricerca Scientifica e Tecnologica) of Italy within the project Strutture Geometriche, Combinatoria e loro Applicazioni. The authors would like to thank the anonymous referee for his/her useful suggestions and remarks, which improved the final version of the paper.
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Cavicchioli, A., Hegenbarth, F. & Spaggiari, F. Four-Dimensional Complexes with Fundamental Class. Mediterr. J. Math. 17, 175 (2020). https://doi.org/10.1007/s00009-020-01618-z
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DOI: https://doi.org/10.1007/s00009-020-01618-z
Keywords
- Poincaré complexes
- four-dimensional complexes
- homotopy type
- Wall group
- spectral sequence
- obstruction theory
- Homology with local coefficients
- Poincaré duality