The exact sequence of a localization for Witt groups. II. Numerical invariants of odd-dimensional surgery obstructions

Pardon, William

doi:10.2140/pjm.1982.102.123

Cited by 19 publications

(2 citation statements)

References 28 publications

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“…or a maximal order in the quaternion algebra F k for some k, in the four cases of (1.2). The/Y groups above were obtained for G abelian in [2],/~ in [4], [15] and /_( modulo an extension problem in [-15]. The remaining ones are now in [11].…”

Section: (A) Q(~k)mentioning

confidence: 99%

The surgery obstruction groups for finite 2-groups

Hambleton

Milgram

1980

Invent Math

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In this paper we establish an effective method for calculating the oriented surgery obstruction groups Lh,(ZG) for G a finite group of 2-primary order. We show that these groups depend explicitly on the rational representations of G and certain facts about the reduced projective class group Ko(ZG), and prove that most of the relevant structure of Ko (ZG) Our main concern, however, is with studying the map dk+l in (*). The involution on/s is given by [P] ~ -[P*] where P* is the dual module to P, and only the 2-torsion part of glo(ZG ) matters in (,). We define a finite abelian 2-group with involution We(G ) which depends only on the rational representations of G and a involution preserving map tp: We(G) ---,,gio(ZG ) which is onto the 2-primary part of K0(ZG ). Then we prove that dk+ ~ factors, as a composite L~+ I(ZG) a~+l , Hk(Z/2 ' We(G)) ~, , Hk(Z/2 '/~o(ZG))*

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Section: (A) Q(~k)mentioning

confidence: 99%

The surgery obstruction groups for finite 2-groups

Hambleton

Milgram

1980

Invent Math

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“…The problem of computing the projective surgery obstruction groups L~(ZG) has already been extensively studied for the case where G is a finite hyperelementary group (see, for example, [2,3,7,20,23,24,32,44]). These computations may be considered as a further development and application of the classical theory of quadratic forms over fields.…”

Section: O Introductionmentioning

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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Semi-invariants in surgery

Davis¹,

Ranicki²

1987

K-Theory

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The exact sequence of a localization for Witt groups. II. Numerical invariants of odd-dimensional surgery obstructions

Cited by 19 publications

References 28 publications

The surgery obstruction groups for finite 2-groups

The surgery obstruction groups for finite 2-groups

On the computation of the projective surgery obstruction groups

Semi-invariants in surgery

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