Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016 pp. 603–619.

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

Authors: Malkhaz Bakuradze (1) and Natia Gachechiladze (1)
Author institution:(1) Iv. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences

Summary:

In 2011, 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 a monograph by Hall and Senior, these groups are numbered by 1, ..., 51. For the groups G₃₈, ..., G₄₁, 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)^*(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 G₃₆, G₃₇, each isomorphic to a semidirect product (C₄×C₂×C₂) ⋊ C₂ , the group G₃₄ ≅ (C₄×C₄) ⋊ C₂ and its non-split version G₃₅. For these groups the action of C₂ is diagonal, i.e., simpler than for the groups G₃₈, ..., G₄₁, however the rings K(s)^*(BG) have the same complexity.

2010 Math. Subj. Class. 55N20; 55R12; 55R40.

Keywords: Transfer, Morava K-theory.

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