The class groups of quaternion and dihedral 2‐groups

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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“…Since D(G)=0 for the dihedral groups [7], their /_P-groups are covered by Theorem A. Since G = Z/2 n is abelian its ~-groups appear in [2].…”

Section: W 2 Applicationsmentioning

confidence: 97%

The surgery obstruction groups for finite 2-groups

Hambleton

Milgram

1980

Invent Math

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“…For instance D(2g r~) =0 whenever ~ is a dihedral 2-group, and D(2g ~) is of order 2 whenever rc is a (generalized) quaternion 2-group [3].…”

Section: Ii) If ~ Is a P-group Of Order >=_p2 Then P-ll~[w(x)=o If Pmentioning

confidence: 99%

The finiteness obstructions for nilpotent spaces lie inD(? ?)

Mislin

Varadarajan

1979

Invent Math

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“…It is easy to deduce from the discussion of Cartesian squares in [14] that, under the isomorphisms of (4.6), έ? (Δ n Proof of (4.14).…”

Section: U Amentioning

confidence: 99%

“…If Q n is generalized quaternion ((7.4) (c)), K 0 (ZQ n ) = ZJ2 by [14]. In general, it is necessary to understand the maps in the Rothenberg sequence to know how strong (3.16) is in any given case.…”

Section: ) (A) and [14]) L%(zπ) = Ll(zπ)mentioning

confidence: 99%