On four-dimensional Poincarè duality cobordism groups
| Autor: | Fulvia Spaggiari, Friedrich Hegenbarth, Alberto Cavicchioli |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
General Mathematics
Dimension (graph theory) homology with local coefficients cobordism group 01 natural sciences Omega Surgery obstruction Combinatorics total surgery obstruction symbols.namesake spectral sequence Canonical map 0101 mathematics Poincaré duality Mathematics Exact sequence Image (category theory) 010102 general mathematics Cobordism Wall group four-manifold homotopy type 010101 applied mathematics obstruction theory Poincarè complexes symbols Poincarè complexes four-manifold cobordism group homotopy type Wall group spectral sequence obstruction theory homology with local coefficients total surgery obstruction |
| Popis: | This paper continues the study of four-dimensional Poincare duality cobordism theory from our previous work Cavicchioli et al. (Homol. Homotopy Appl. 18(2):267–281, 2016). Let P be an oriented finite Poincare duality complex of dimension 4. Then, we calculate the Poincare duality cobordism group $$\Omega _{4}^{{\text {PD}}}(P)$$ . The main result states the existence of the exact sequence $$0 \rightarrow L_4 (\pi _1 (P))/A_4 (H_2 (B\pi _1 (P), L_2)) \rightarrow {{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P) \rightarrow \mathbb Z_8 \rightarrow 0$$ , where $${{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P)$$ is the kernel of the canonical map $${\Omega }_{4}^{\mathrm{PD}}(P) \rightarrow H_4 (P, \mathbb Z) \cong \mathbb Z$$ and $$A_4 : H_4 (B\pi _1, \mathbb L) \rightarrow L_4 (\pi _1 (P))$$ is the assembly map. It turns out that $${\Omega }_{4}^{\mathrm{PD}}(P)$$ depends only on $$\pi _1 (P)$$ and the assembly map $$A_4$$ . This does not hold in higher dimensions. Then, we discuss several examples. The cases in which the canonical map $$\Omega _{4}^{{\text {TOP}}}(P) \rightarrow \Omega _{4}^{{\text {PD}}}(P)$$ is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In fact, we establish an exact sequence where s is Ranicki’s total surgery obtruction map. In the above cases, there are $${\text {PD}}_4$$ -complexes X which cannot be homotopy equivalent to manifolds. |
| Databáze: | OpenAIRE |
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