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 Mosc. Math. J., 2016, Volume 16, Number 4, Pages 603–619 (Mi mmj611)

Morava $K$-theory rings of the extensions of $C_2$ by the products of cyclic $2$-groups

Iv. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences

Abstract: In 2011, Schuster proved that $\mod2$ 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 a monograph by Hall and Senior, these groups are numbered by $1,…,51$. For the groups $G_{38},…,G_{41}$, which fit in the title, the explicit ring structure is determined in a joint work of M. Jibladze and the author. In particular, $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(\mathrm{pt})$ by an ideal generated by explicit polynomials. In this article we present some calculations using the same arguments in combination with a theorem by the author 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 $(C_4\times C_2\times C_2)\rtimes C_2$, the group $G_{34}\cong(C_4\times C_4)\rtimes C_2$ 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.

Key words and phrases: transfer, Morava $K$-theory.

 Funding Agency Grant Number Volkswagen Foundation 1/84 328 Shota Rustaveli National Science Foundation DI/16/5-103/12 The first named author was supported by Volkswagen Foundation, Ref. 1/84 328 and Rustaveli Foundation grant DI/16/5-103/12.

DOI: https://doi.org/10.17323/1609-4514-2016-16-4-603-619

Full text: http://www.mathjournals.org/.../2016-016-004-001.html
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MSC: 55N20, 55R12, 55R40
Received: December 22, 2014; in revised form February 8, 2016
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Citation: Malkhaz Bakuradze, Natia Gachechiladze, “Morava $K$-theory rings of the extensions of $C_2$ by the products of cyclic $2$-groups”, Mosc. Math. J., 16:4 (2016), 603–619

Citation in format AMSBIB
\Bibitem{BakGac16} \by Malkhaz~Bakuradze, Natia~Gachechiladze \paper Morava $K$-theory rings of the extensions of $C_2$ by the products of cyclic $2$-groups \jour Mosc. Math.~J. \yr 2016 \vol 16 \issue 4 \pages 603--619 \mathnet{http://mi.mathnet.ru/mmj611} \crossref{https://doi.org/10.17323/1609-4514-2016-16-4-603-619} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3598497} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000391211000001} 

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• http://mi.mathnet.ru/eng/mmj/v16/i4/p603

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. M. Bakuradze, N. Gachechiladze, “Some 2-groups from the view of Hilbert–Poincaré polynomials of $K(2)^*(BG)$”, Tbil. Math. J., 10:2 (2017), 103–110