Relative homology and Poincaré duality for group pairs

DOI10.1016/0022-4049(78)90012-9zbMath0392.20032OpenAlexW2042574265MaRDI QIDQ1251723

Beno Eckmann, Robert Bieri

Publication date: 1978

Published in: Journal of Pure and Applied Algebra (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0022-4049(78)90012-9

zbMATH Keywords

HNN-extensionsamalgamated productsPoincaré duality pairsdualizing modulescap productsgroup pairs(co)homology functorscobordism for Poincaré duality groupshomological dualityMayer-Vietoris sequences for amalgams and HNN-extensions

Mathematics Subject Classification ID

Homological methods in group theory (20J05) Other groups related to topology or analysis (20F38) Extensions, wreath products, and other compositions of groups (20E22) Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations (20E06) Fundamental groups and their automorphisms (group-theoretic aspects) (20F34) Cobordism in PL-topology (57Q20)

Related Items (41)

A relative cohomological invariant for group pairs ⋮ Three-dimensional Poincaré duality groups which are extensions ⋮ Centralizer in PD(3) groups ⋮ Splittings of Poincaré duality groups ⋮ Relative cohomology theory for profinite groups ⋮ Duality systems of groups and PD(n)-systems of groups ⋮ On Poincaré duality for pairs (\(G\),\(W\)) ⋮ Group systems, groupoids, and moduli spaces of parabolic bundles ⋮ Twisted cohomology pairings of knots. I: Diagrammatic computation ⋮ A spectral sequence for Dehn fillings ⋮ Central extensions and bounded cohomology ⋮ Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifolds ⋮ Bounded cohomology of classifying spaces for families of subgroups ⋮ Cohomology of group theoretic Dehn fillings. II ⋮ Decomposition theorems for group pairs ⋮ Hyperbolic groups with boundary an n-dimensional Sierpinski space ⋮ Plane motion groups and virtual Poincaré duality of dimension two ⋮ Pro-\(\mathcal {C}\) completions of orientable \(\mathrm{PD}^3\)-pairs ⋮ Cohomology and the Bowditch boundary ⋮ Poincaré duality pairs of dimension three ⋮ Quasi-Projective and Relative Cohomological Dimension of Groups ⋮ Some properties of E(G,W,F_TG) and an application in the theory of splittings of groups ⋮ Adapted splittings for pairs \((G,W)\) ⋮ ${\mathcal{B}$-BOUNDED COHOMOLOGY AND APPLICATIONS ⋮ A combination theorem for \(PD(n)\)-pairs ⋮ Volumes of \(\operatorname{SL}_n(\mathbb{C})\)-representations of hyperbolic 3-manifolds ⋮ Homomorphisms of nonzero degree betweenPD_n-groups ⋮ Pro-\(p\) completions of Poincaré duality groups. ⋮ Cohomological characterization of relative hyperbolicity and combination theorem. ⋮ Relative property A and relative amenability for countable groups ⋮ Relative ends and duality groups ⋮ On groups with \(S^2\) Bowditch boundary ⋮ Classification of pro-\(p\mathrm{PD}^2\) pairs and the pro-\(p\) curve complex ⋮ On the Tits alternative for \(PD(3)\) groups ⋮ On the ends of pairs of groups ⋮ Schur multipliers and second quandle homology ⋮ Group pairs with periodic cohomology ⋮ On the hierarchy of cohomological dimensions of groups ⋮ Seifert fibre spaces and Poincaré duality groups ⋮ Centralizers and normalizers of subgroups of \(\text{PD}_3\)-groups and open \(\text{PD}_3\)-groups. ⋮ Splitting theorems for certain \(PD^3\)-groups

Cites Work

• Homological criteria for finiteness
• Amalgamated free products of groups and homological duality
• Normal subgroups in duality groups and in groups of cohomological dimension 2
• A homological finiteness criterion
• On groups of finite cohomological dimension and duality groups over a ring
• A remark on subgroups of infinite index in Poincaré duality groups
• Relative version of a theorem of Stallings
• Mayer-Vietoris sequences for HNN-groups and homological duality
• Groups with homological duality generalizing Poincaré duality
• Pairs of relative cohomological dimension one
• Poincaré 2-complexes. I
• Guppen mit Poincaré-Dualität
• Cobordism for Poincaré duality groups
• On a Cohomology Theory for Pairs of Groups
• Groups of cohomological dimension one
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