<i>K</i>(<i>n</i>) Chern approximations of some finite groups

Schuster, Björn

doi:10.2140/agt.2012.12.1695

Cited by 7 publications

(7 citation statements)

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“…The good groups in the sense, that K(s) odd = 0 (this fact is in principle weaker than being good in the sense of Hopkins-Kuhn-Ravenel) also play a role in the literature [19,20,21,26].…”

Section: Preliminariesmentioning

confidence: 99%

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Morava K-theory rings of the extensions of C2 by the products of cyclic 2-groups

Bakuradze¹,

Gachechiladze²

2016

MMJ

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In [19] Schuster proved that mod 2 Morava K-theory K(s) * (BG) is evenly generated for all groups G of order 32. There exist 51 non-isomorphic groups of order 32. In [12], these groups are numbered by 1, · · · , 51. For the groups G 38 , · · · , G 41 , that fit in the title, the explicit ring structure is determined in [5]. In particular, K(s) * (BG) is the quotient of a polynomial ring in 6 variables over K(s) * (pt) by an ideal generated by explicit polynomials. In this article we present some calculations using the same arguments in combination with a theorem of [2] on good groups in the sense of Hopkins-Kuhn-Ravenel. In particular, we consider the groups G 36 , G 37 , each isomorphic to a semidirect product (C 4 × C 2 × C 2 ) ⋊ C 2 , the group G 34 ∼ = (C 4 × C 4 ) ⋊ C 2 and its non-split version G 35 . For these groups the action of C 2 is diagonal, i.e., simpler than for the groups G 38 , · · · , G 41 , however the rings K(s) * (BG) have the same complexity.2010 Mathematics Subject Classification. 55N20; 55R12; 55R40.

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“…The good groups in the sense, that K(s) odd = 0 (this fact is in principle weaker than being good in the sense of Hopkins-Kuhn-Ravenel) also play a role in the literature [19,20,21,26].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [24], [25], [26], [27] the multiplicative structure has only been determined modulo certain indeterminacy. In [19], [20], [21], [22], [23] some artificial generators are suggested in order to obtain explicit relation.…”

Section: Introductionmentioning

confidence: 99%

Morava K-theory rings of the extensions of C2 by the products of cyclic 2-groups

Bakuradze¹,

Gachechiladze²

2016

MMJ

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“…Also the presentation of K(s) * (BG) in terms of a formal group law and a splitting principle is not always convenient. It is proved in [9,10] that the groups G 38 , . .…”

Section: Preliminariesmentioning

confidence: 99%

“…and y 2 s 1 y 2 s 2 = 0 by (19). It is proved in [10] that the group K(s) * (BG) is evenly generated and the Euler characteristic χ s,2 is 16 s /2 + 8 s − 4 s /2.…”

Section: By Frobenius Reciprocity Of the Transfer)mentioning

confidence: 99%

Induced representations, transferred Chern classes and Morava rings K(s)*(BG): Some calculations

Bakuradze

2011

Proc. Steklov Inst. Math.

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The ring structure of Morava K-theory K(s) * (BG) for the 2-group no. 38 of order 32 from the Hall-Senior list is calculated. Previously it was known that K(s) * (BG) is evenly generated and for s = 2 is generated by Chern characteristic classes.Let a finite group G be good in the sense of Hopkins-Kuhn-Ravenel, i.e., K(s) * (BG) is evenly generated by transferred Euler classes [7]. Then, even if the additive structure is calculated, the multiplicative structure is still a delicate task. It is not always determined by representation theory. Also the presentation of K(s) * (BG) in terms of a formal group law and a splitting principle is not always convenient. It is proved in [9, 10] that the groups G 38 , . . . , G 41 of order 32 from the Hall-Senior list [6] are good. For all other groups the corresponding Morava rings were already covered in the literature. Here we will compute K(s) * (BG 38 ), s > 1. For the generating relations we will follow a certain plan proved to be sufficient for the modular p-groups [2] and 2-groups D, SD, QD and Q [4]. Let G = G 38 be the group a, b, c a 4 = b 2 = c 4 = [a, b] = 1, cac −1 = ac 2 , cbc −1 = a 2 b .Let C = a, b, c 2 ∼ = C 4 × C 2 × C 2 be the maximal abelian subgroup of index 2. Consider complex line bundles λ, μ and ν over BC, the pullbacks of the canonical complex line bundles by the projections onto the first, second and third factor, respectively. Define complex plane bundles over BG by the following representations of G:that is, the transferred λ, ν and λν, respectively. The quotient by the center is G/ a 2 , c 2 ∼ = C 2 × C 2 × C 2 . The projections onto factors induce three line bundles α, β and γ, respectively, and det(λ ! ) = αβγ, det(ν ! ) = α, det((λν) ! ) = β.

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“…This note provides a theorem on good groups in the sense of Hopkins-Kuhn-Ravenel [10] and some relevant examples. [15,16,17,18,19]. …”

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Morava K(s)*-rings of the extensions of Cp by the products of good groups under diagonal action

Bakuradze

2015

Georgian Mathematical Journal

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Abstract. This note provides a theorem on good groups in the sense of Hopkins-Kuhn-Ravenel [10] and some relevant examples. [15,16,17,18,19]. Preliminaries and main resultNot so surprising, that the examples of calculations with groups D, SD, QD, Q, M [6], [2,3] show, that even if the additive structure of K(s) * (BG) is calculated, the multiplicative structure is still a delicate task. Just outside of the class of p-groups with maximal cyclic subgroup-already for 2-groups of order 32-one is lead to a complicated ring structures [4].In a previous article [5] the author and S. Priddy obtained formulas relating Chern classes of transfer bundles to transfers of Chern classes. As formal consequences of such formulas one obtains new relations (as well as often much simpler derivations of old ones), and the hope generally is that these combined methods prove sufficient.The main result of this note is Theorem 1.1 on good groups in the sense of Hopkins-Kuhn-Ravenel as follows.Recall from [10] the following definition. a) For a finite group G, an element x ∈ K(s) * (BG) is good if it is a transferred Euler class of a complex subrepresentation of G, i.e., a class of the form T r * (e(ρ)), where ρ is a complex representation of a subgroup H < G, e(ρ) ∈ K(s) * (BH) is its Euler class (i.e., its top Chern class, this being defined since K(s) * is a complex oriented theory), and T r : BG → BH is the transfer map.(b) G is called to be good if K(s) * (BG) is spanned by good elements as a K(s) * module. The good groups in the weaker sense, i.e., K(s) odd = 0, also play a role in the literature [14], [15], [22].2010 Mathematics Subject Classification. 55N20; 55R12; 55R40.

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K(n) Chern approximations of some finite groups

Cited by 7 publications

References 11 publications

Morava K-theory rings of the extensions of C2 by the products of cyclic 2-groups

Morava K-theory rings of the extensions of C2 by the products of cyclic 2-groups

Induced representations, transferred Chern classes and Morava rings K(s)*(BG): Some calculations

Morava K(s)*-rings of the extensions of Cp by the products of good groups under diagonal action

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