The complex oriented cohomology of extended powers

Hunton, John

doi:10.5802/aif.1627

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…Then h * (X Ω × S E) is isomorphic to a direct sum of h * (E/S ′ )'s, shifted in degree, where S ′ runs over the stabilizers of a set of orbit representatives for the action of S on B Ω .' For specific choices of S and E stronger results are known-see [19] for some recent results in the case when S = C p and E is contractible.…”

Section: Remarksmentioning

confidence: 98%

On the Integral Cohomology of Wreath Products

Leary

1997

Journal of Algebra

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Under mild conditions on the space X, we describe the additive structure of the integral cohomology of the space X p = EC in terms of the cohomology of X. C p pWe give weaker results for other similar spaces, and deduce various corollaries concerning the cohomology of finite groups. ᮊ 1997 Academic Press INTRODUCTIONLet S be a group with a fixed action on a finite set ⍀. By the wreath product G X S of a group G with S we mean a split extension with kernel G ⍀ , quotient S, and with the S-action on G ⍀ given by permuting the copies of G. Our main interest is the integral cohomology of finite groups of the form G X S. We work in greater generality, however, because it is no more difficult to study the cohomology of spaces of the formwhere E is an S-free S-CW-complex, and X is a CW-complex of finite type. The mod-p cohomology of certain such spaces plays a crucial role in w x Steenrod's definition of the reduced power operations 29 . Building on work of Steenrod, Nakaoka described the cohomology of such spaces with w x coefficients in any field 23 . The point about working over a field is that then the cellular cochain complex for X is homotopy equivalent to the cohomology of X, viewed as a complex with trivial differential. If the integral cohomology of X is free, then a similar result holds in this case.Ž . Evens used this to study the cohomology of the classifying space of the Ž . Lie group U m X ⌺ in the course of his work on Chern classes of n w x induced representations 13 . The study of the integral cohomology in the U Ž . case when H X is not free is much harder. The pioneers in this case were Evens and Kahn, who made a partial study of the important special 184 0021-8693r97 $25.00Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. COHOMOLOGY OF WREATH PRODUCTS 185case of X p = EC . We complete the study of this case in Section 4 C p p below, which could be viewed as both an extension of and a simplification w x of 15, Sect. 4 . Many, but not all, of our results are corollaries of this work. Our paper has the following structure.In Section 1 we give some algebraic background. Most of this material is well-known, although we have not seen Lemma 1.4 stated explicitly before, and we believe that Lemma 1.1 is original. This lemma, which compares spectral sequences coming from double complexes consisting of ''the same groups,'' but with ''different differentials,'' is the key to our extension of w x the Evens᎐Kahn results 15 . Theorem 2.1 is a statement of the result of Nakaoka mentioned above, which for interest's sake we have made more general than the original. A weak version of this theorem could be stated U Ž . as ''if H X is free, then the Cartan᎐Leray spectral sequence for U Ž ⍀ . H X = E collapses at the E -page.'' A similarly weakened version of S 2 Ž . Theorem 2.2 would say that over the integers or any PID , this spectral < < sequence collapses at the E -page, r s 2 q ⍀ , without any condition onThe Cartan᎐Leray spectral sequence may be obtained from a double complex. In...

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Section: Remarksmentioning

confidence: 98%

On the Integral Cohomology of Wreath Products

Leary

1997

Journal of Algebra

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“…In this paper we consider the complex oriented cohomology of homotopy orbit spaces X p hG = EG× G X p . Several authors have computed these cohomology groups, [15], [11], [12], [10], however we are particularly interested in the ring structure and thereby explicit formulas for the transfer. Thus we are led to consider Fibrins reciprocity, the relation between cup products and transfer: T r * (x)y = T r * (xρ * (y)) c Geometry & Topology Publications (formula (i) of Section 2) where ρ : EG×X p → X p hG is the covering projection and T r * : E * (X p ) → E * (X p hG ) is the associated transfer homomorphism.…”

Section: Introductionmentioning

confidence: 99%

“…Hunton [12] has shown that if K(s) * (X) is concentrated in even dimensions then so is K(s) * (X p hπ ). We adopt the stronger hypothesis that X is good and derive a stronger result, following the argument of [10] Theorem 7.3 for classifying spaces.…”

Section: Transfer and K(s)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transfer and complex oriented cohomology rings

Bakuradze¹,

Priddy

2003

Algebr. Geom. Topol.

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For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (i.e. T r * (1)) and transferred Chern classes for p-fold covers. Applications to the classifying spaces of p-groups are given.

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Transferred characteristic classes and generalized cohomology rings

Bakuradze

2013

J Math Sci

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The complex oriented cohomology of extended powers

Cited by 4 publications

References 24 publications

On the Integral Cohomology of Wreath Products

On the Integral Cohomology of Wreath Products

Transfer and complex oriented cohomology rings

Transferred characteristic classes and generalized cohomology rings

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