Abstract
We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez–Dolan homotopy cardinality, by constructing a unique additive common generalization after restriction to an odd prime p. This is achieved by \(\ell \)-adically extrapolating to height \(n=-1\) the sequence of Euler characteristics associated with the Morava K(n) cohomology theories for (any) \(\ell \mid p-1\). We compute this sequence explicitly in several cases and incorporate in the theory some folklore heuristic comparisons between the Euler characteristic and the homotopy cardinality involving summation of divergent series.
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Notes
A solution using a different technique was claimed in the preprint [4], but apparently it has a mistake. The proposed approach is however still interesting and merits further investigation.
By [15], for \(\pi \)-finite double loop-spaces, the K(n)-homology itself can not see the Postnikov invariants.
While \(\mathbb {Z}_p^n\) is not quite finitely generated, replacing it with \((\mathbb {Z}/p^N)^n\) for \(N\gg 0\) does not change the first (arbitrarily many) first terms of both sides of the equation, so we can still apply the formula to it.
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Acknowledgements
Most of the results presented here were obtained during my first Ph.D. project back in 2017 under the supervision of Tomer Schlank (who was the one to suggest to “somehow use Morava K-theories”). they were subsequently announced in the traschromatic homotopy theory conference held in Regensburg that year and presented in several talks since then, but various distractions delayed the writing process. It should be noted that at the time, Lurie’s [21] was not published yet and much of the general higher semiadditive context was still missing as well. Thus, in a sense, the present time is more ripe for communicating these ideas, which may partially excuse the long delay. I am deeply grateful to Tomer Schlank for his help with this project, as well as his mentoring over the passing years. I also thank him for useful comments on an earlier draft of this paper. I wish to extend my gratitude also to Maxime Ramzi for reviving my interest in this project through many stimulating conversations motivating me to finally write this paper. I would also like to thank John Baez for his numerous excellent expositions of these and other higher categorical ideas, that got me into thinking about this project in the first place. I would like to thank all the present and former members of the seminarak group for stimulating conversations and encouragement, with special thanks to Shai Keidar and Shaul Ragimov for providing the elegant argument in the proof of the second part of Proposition 3.5. Finally, I would like to that the anonymous referee for several useful comments.
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Yanovski, L. Homotopy cardinality via extrapolation of Morava–Euler characteristics. Sel. Math. New Ser. 29, 81 (2023). https://doi.org/10.1007/s00029-023-00886-3
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DOI: https://doi.org/10.1007/s00029-023-00886-3