On higher real $K$-theories and finite spectra

Christian Carrick; Michael A. Hill

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arXiv · 2507.07051

On higher real $K$-theories and finite spectra

Christian Carrick·Michael A. Hill

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Abstract

We study higher chromatic height analogues $eo_h$ of the connective real $K$-theory spectrum $ko$. We show that $eo_h$ is an fp spectrum of type $h$ in the sense of Mahowald--Rezk. We use these to study an Euler characteristic for fp spectra introduced by Ishan Levy, and give a partial answer to a question of Levy regarding the algebraic $K$-theory of the category of finite type $h$ spectra. As a corollary, we prove that if the generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ exists, then the $2$-adic valuation of $\prod i_j$ must exceed that of the height $h$.

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Christian Carrick, Michael A. Hill. 2025-07-09. On higher real $K$-theories and finite spectra. https://arxiv.org/abs/2507.07051

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