Applications of the transfer to stable homotopy theory

Kahn, Daniel S.; Priddy, Stewart

doi:10.1090/s0002-9904-1972-13076-3

Cited by 125 publications

(91 citation statements)

References 13 publications

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“…Also ii) implies i). The proof of ii) uses formal properties of the transfer [1], [13], [7], including the Fröbenius reciprocity and the double coset formula. We follow the proof of wreath product theorem ( [10], Theorem 3.8).…”

Section: In Favourable Cases K(s)mentioning

confidence: 99%

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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Section: In Favourable Cases K(s)mentioning

confidence: 99%

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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“…Let τ π and τ p be the transfer maps of the bundles π and p [2,6,11]. The next lemma follows from [7].…”

Section: Preliminaries and Calculations With Transfermentioning

confidence: 99%

Characteristic classes and transfer relations in cobordism

Bakuradze¹,

Jibladze²,

Vershinin³

2002

Proc. Amer. Math. Soc.

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“…In [10] Snaith showed that this is false if we take G*(X) = H*(X; Z) © G*(X), however, this does not mean that there cannot be a different G*(X) related to G*(X) in some more complex way. Now, it is well known [6] that every generalized cohomology theory admits a transfer homomorphism for finite coverings such that transformations of cohomology theories commute with the tranfer. We can ask: does the transfer in G(X), defined here for 2-coverings, extend to a transfer defined for all finite coverings and possessing all the usual properties of a transfer homomorphism (e.g.…”

Section: G(x) = H°(x;z)®g(x)mentioning

confidence: 99%

“…PROOF. From the definition of the transfer in a generalized cohomology theory in [6] and from the definition of N it follows that we only need to show that the…”

Section: Multiplicativementioning

confidence: 99%

The Evens-Kahn Formula for the Total Stiefel-Whitney Class

Kozłowski¹

1984

Proceedings of the American Mathematical Society

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ABSTRACT. Let G(X) denote the (augmented) multiplicative group of classical cohomology ring of a space X, with coefficients in Z/2. The (augmented) total Stiefel-Whitney class is a natural homomorphism w. KO(X) -► G(X). We show that the functor G( ) possesses a 'transfer homomorphism'for double coverings such that w commutes with the transfer. This is related to a question of G. Segal. As a special case, we obtain a formula for the total Stiefel-Whitney class of a representation of a finite group induced from a (real) representation of a subgroup of index 2, which is analogous to the one obtained by Evens and Kahn for the total Chern class.

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Applications of the transfer to stable homotopy theory

Cited by 125 publications

References 13 publications

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

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

Characteristic classes and transfer relations in cobordism

The Evens-Kahn Formula for the Total Stiefel-Whitney Class

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