Beno Eckmann

Relative homology and poincaré duality for group pairs

0.1. A PoincarC duality group is a group whose (co)homology fulfills duaiity relations analogous to those of the (co)homology of closed manifolds. More general duality groups have been introduced [‘i] whereby the orientation module 2 (with trivial or nontrivial group action) is replaced by a dualizing module C whose underlying abelian group need not be Z. In the present paper, we investigate pairs of’ groups (G, S) consisting of a group G and a subgroup S whose relative (co)homology fulfills duality relations similar to those well-known for compact manifolds-with-boundary. Such pairs are called Poincurk’ duality paits. Here again, the orientation module 2 can be replaced by a more general dualizing module C, but apart from general statements we will restrict ourselves essentially to the Poincare case.

Der Cohomologie-Ring einer beliebigen Gruppe

Einleitung 1. Es wird in dieser Arbeit einer beliebigen Gruppe (fi und einem Ring J mit Einselement durch ein einfaches algebraisches Verfahren ein Ring P (ffi, J ) zugeordnet ; seine Elemente sind Klassen yon Funkt ionen mehrerer Variabeln aus (fi mit Wer ten in J . Die algebraische Bedeutung dieses Ringes wird nur gelegentlich berfihrt; sie scheint selbst/~ndiges Interesse zu verdienen und auf naheliegende Verallgemeinerungen hinzuweisen. Im Mit te lpunkt unserer Untersuchung aber stehen die Beziehungen yon P ((fi, J ) zur algebraischen Topologie. Wir werden zeigen: P ((fi, J ) h/~ngt zusammen mi t dem Cohomologiering 1) (bezfiglich des Koeffizientenbereiches J ) derjenigen Polyeder / / , deren Fundamenta lgruppe zu ffi isomorph ist oder ffi als homomorphes Bild besitzt ; m. a. W. derjenigen Polyeder, welche eine regul/~re l~berlagerung ~) mit zu (fi isomorpher Gruppe yon Deckt ransformat ionen besitzen.

Projective and Hilbert modules over group algebras, and finitely dominated spaces

The following two remarks came to the attention of the author after the paper had appeared. They do not affect the validity of the results but they simplify some of the statements. Terminology, Sections, notations etc. refer to the above paper. The remarks concern the Bass conjecture for a groupG, as described in Section 2. We here just recall that the Hattori-Stallings rank rP of a finitely generated projective ZG−module P is a Z−valued function of the conjugacy classes in G. The (strong) Bass conjecture (SB) claims that rP (x) = 0 for x 6= 1 ∈ G and thus rP (1) = rk P = dimRR⊗G P .

Social choice and topology a case of pure and applied mathematics

Abstract The existence of a social choice model on a preference space P is a topological, even homotopical problem. It has been solved 50 years ago, under different terminology, by the author and, a little later, jointly with T. Ganea and P.J. Hilton. P must be an H-space and either contractible or homotopy equivalent to a product of Eilenberg-MacLane spaces over the rationals.

Cobordism for Poincaré duality groups

1. Relative homology for pairs. Homology and cohomology for a pair of groups G D S (cf. [6] ) can be extended to pairs (G, S) consisting of a group G and a family of subgroups S = {St}, as follows: If S = 0 one takes the usual (absolute) groups of G. If S =£ 0, let A be the kernel of the G-homomorphism (&.Z(GlS?) ->• Z given by augmentations;^! being a G-module, we put Z/*(G, S; A) = fl*-^; Hom(A, ,4)) and Hk(G, S; A) = Hk_t(G; A® A) where G acts diagonally in Hom(A, A) and A ® A. One has exact sequences

MATHEMATICS : QUESTIONS AND ANSWERS

and which is not primarily erected for this purpose. The applications bear witness to the power of mathematics, but are not its real motivation. The springs of 686 [October MATHEMATICS: QUESTIONS AND ANSWERS This content downloaded from 207.46.13.120 on Wed, 14 Sep 2016 04:15:20 UTC All use subject to http://about.jstor.org/terms