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Morava $K$-theory rings of the extensions of $C_2$ by the products of cyclic $2$-groups
Malkhaz Bakuradze,
Natia Gachechiladze 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
$1,\dots,51$. For the groups
$G_{38},\dots,G_{41}$, 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
$G_{38},\dots,G_{41}$, however the rings
$K(s)^*(BG)$ have the same complexity.
Key words and phrases:
transfer, Morava $K$-theory.
MSC: 55N20,
55R12,
55R40 Received: December 22, 2014; in revised form
February 8, 2016
Language: English
DOI:
10.17323/1609-4514-2016-16-4-603-619