Periodic Projective Resolutions

Wall, C. T. C.

doi:10.1112/plms/s3-39.3.509

Cited by 57 publications

(22 citation statements)

References 18 publications

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“…a free Γ-CW complex) that is homotopy equivalent to a sphere. From this we obtain a characterization of groups which act freely and properly on R n × S m which extends a conjecture formulated by Wall [33] for groups of finite virtual cohomological dimension: Corollary 1.3. A discrete group Γ acts freely and properly on R n × S m for some m, n > 0 if and only if Γ is a countable group with periodic cohomology.…”

Section: Introductionmentioning

confidence: 67%

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Periodic Complexes and Group Actions

Ádem¹,

Smith²

2001

The Annals of Mathematics

137

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Abstract. In this paper we show that the cohomology of a connected CW -complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m ; we construct non-standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m ; and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p) 3 .

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Section: Introductionmentioning

confidence: 67%

“…This result represents a proof of a generalized version of a conjecture due to Wall [33] for groups of finite virtual cohomological dimension (that case was verified in [12]). More recently this has been extended to a larger class of discrete groups (see [22]).…”

Section: Periodic Complexesmentioning

confidence: 97%

Periodic Complexes and Group Actions

Ádem¹,

Smith²

2001

The Annals of Mathematics

137

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“…We refer the reader to [137]. Here Wall shows that given a generator α ∈ H n+1 (G, Z) and an exact sequence…”

Section: Discussion Of Proofmentioning

confidence: 99%

“…Any Swan complex is finitely dominated, and the finiteness obstruction [X] ∈ K 0 (ZG) vanishes if and only if X has the homotopy type of a finite CW -complex. This can be effectively computed, [137], [91], [42], although the algebraic number theory can be quite involved. One qualitative result is that if G is a group of period n + 1, there is always a finite Swan complex of dimension 2n + 1.…”

Section: Existence Of Space Formsmentioning

confidence: 99%

Topological Transformation Groups

James

1984

General Topology and Homotopy Theory

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“…In order to have compatibility with the maps in the arithmetic sequence, we will use the Z/2-module structure on /~o(ZG)induced by [P] ~-[ P * ] for any projective module P. Then our final invariant for detecting the surgery obstruction, defined on the kernel of the primary invariants, is the 6-invariant: We remark that both the finiteness obstruction ( [26,28,46]) and the 6-invariant (see [15,Sections 4,9]) are computable using Reidemeister torsion.…”

Section: Theorem a Let G Be A 2-hyperelementary Group Then The Sum mentioning

confidence: 99%

On the computation of the projective surgery obstruction groups

Hambleton¹,

Madsen²

1993

K-Theory

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Abstract. The computation of the projective surgery obstruction groups LP. (ZG), for G a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. For G a 2-hyperelementary group, the LP.(ZG) are detected by restriction to certain subquotients of G, and a complete set of invariants is given for oriented surgery obstructions.Key words. Surgery on manifolds, Hermitian K-theory. O. IntroductionThe projective surgery obstruction groups were first introduced by S. P. Novikov 1-30] in the context of Hermitian K-theory and the topology of infinite cycle covers of compact manifolds. These groups arise as the codimension 1 summands in a splitting theorem for the Wall surgery obstruction groups of a Laurent polynomial extension 1,39, 12], or more generally in the classification of noncompact manifolds [40,41,27,33]. The algebraic description of projective L-theory and the splitting theorem were given a systematic exposition by A. A. Ranicki 1-34, 36], including a definition of the lower L-groups by analogy with the lower K-groups of Bass.With the extensive development of 'bounded' or 'controlled' topology in the last decade, the role of projective and lower surgery obstruction groups has increased in importance. For example, the concrete structure of these groups is relevant to the classification of linear representations of finite groups up to topological conjugacy ~5, 6, 18] and the recent survey article I13] describes other applications. The purpose of the present paper is to provide a reference for further computations in this area.

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Periodic Projective Resolutions

Cited by 57 publications

References 18 publications

Periodic Complexes and Group Actions

Periodic Complexes and Group Actions

Topological Transformation Groups

On the computation of the projective surgery obstruction groups

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