Four-Dimensional Complexes with Fundamental Class
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Four-Dimensional Complexes with Fundamental Class Alberto Cavicchioli , Friedrich Hegenbarth and Fulvia Spaggiari 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 Poincar´e duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal PD4 -complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincar´e duality in all dimensions. Such complexes with partial Poincar´e duality properties, which we call SFC4 -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 SFC4 -complex to be a PD4 -complex. Finally, we obtain a partial classification of SFC4 complexes. A future goal will be a classification in terms of algebraic SFC4 -complexes similar to the very satisfactory classification result of PD4 -complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008). Mathematics Subject Classification. 57 N 65, 57 R 67, 57 Q 10. Keywords. Poincar´e complexes, four-dimensional complexes, homotopy type, Wall group, spectral sequence, obstruction theory, Homology with local coefficients, Poincar´e duality.
1. Introduction A fundamental result on closed Poincar´e duality complexes of dimension n (shortly, PDn -complexes) is the so-called Disc Theorem due to Wall ([21], §2.4] or [22], §2.9]): If Xis a PDn -complex, n ≥ 4,then Xis homotopy equivalent to K ∪ϕ Dn ,where K is a complex of dimension ≤ n − 1,and ϕ : Sn−1 → Kis the attaching map of an n-cell. Moreover, the presentation is essentially unique, that is, given two presentations K ∪ϕ Dn and K ∪ϕ Dn of X, there exists a homotopy equivalence pairs (M (ϕ ), Sn−1 ) → (M (ϕ), Sn−1 ). Here, M (ϕ ) and M (ϕ) are the mapping cylinders of ϕ and ϕ, respectively. 0123456789().: V,-vol
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Conversely, one may ask the following questions: (1) Let us be given a complex K of dimension ≤ n − 1. Does there exist a map ϕ : Sn−1 → K, such that P = K ∪ϕ Dn is a closed PDn -complex? (2) If there is a map ϕ : Sn−1 → K with P = K ∪ϕ Dn a PDn -complex, can one construct other maps ψ : Sn−1 → K with Q = K ∪ψ Dn a PDn -complex? (3) Fixing K, can one classify all closed PDn -complexes of type K ∪ϕ Dn up to homotopy equivalence? We consider only connected complexes K. Of course, K ∪ϕ Dn depends, up to homotopy equivalence, only on [ϕ] ∈ πn−1 (K). As in our previous papers [6] and [7], we are interested in constructing and classifying PD4 -complexes. The present paper gives some contributions by considering problems (2) and (3) in case n = 4. Furthermore, we treat only the oriented version, i.e., the (co)homology modules with Z-coefficients are untwisted. Here are
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