Volume 15, issue 5 (2015)

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The Morava $K$–theory of $BO(q)$ and $MO(q)$

Nitu Kitchloo and W Stephen Wilson

Algebraic & Geometric Topology 15 (2015) 3047–3056
Abstract

We give an easy proof that the Morava $K\phantom{\rule{0.3em}{0ex}}$–theories for $BO\left(q\right)$ and $MO\left(q\right)$ are in even degrees. Although this is a known result, it had followed from a difficult proof that ${BP}^{\ast }\left(BO\left(q\right)\right)$ was Landweber flat. Landweber flatness follows from the even Morava $K\phantom{\rule{0.3em}{0ex}}$–theory. We go further and compute an explicit description of $K{\left(n\right)}_{\ast }\left(BO\left(q\right)\right)$ and $K{\left(n\right)}_{\ast }\left(MO\left(q\right)\right)$ and reconcile it with the purely algebraic construct from Landweber flatness.

Keywords
Morava $K$—theory, BO, characteristic classes
Mathematical Subject Classification 2010
Primary: 55R45, 55N15
Secondary: 55N20, 55N22