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DETECTING STEINER AND LINEAR ISOMETRIES OPERADS

Published online by Cambridge University Press:  19 May 2020

JONATHAN RUBIN*
Affiliation:
University of California Los Angeles, Los Angeles, CA90095, USA e-mail: [email protected]
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Abstract

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We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.

MSC classification

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

References

REFERENCES

Balchin, S., Barnes, D. and Roitzheim, C., N -operads and associahedra. Preprint arXiv:1905.03797.Google Scholar
Balchin, S., Bearup, D., Pech, C. and Roitzheim, C., Equivariant homotopy commutativity for G = Cpgr. Preprint arXiv:2001.05815.Google Scholar
Blumberg, A. J. and Hill, M. A., Operadic multiplications in equivariant spectra, norms, and transfers, Adv. Math. 285 (2015), 658708.CrossRefGoogle Scholar
Blumberg, A. J. and Hill, M. A., Incomplete Tambara functors, Algebr. Geom. Topol. 18(2) (2018), 723766.CrossRefGoogle Scholar
Bonventre, P. and Pereira, L. A., Genuine equivariant operads. Preprint arXiv:1707.02226.Google Scholar
Elmendorf, A. D., Kriz, I., Mandell, M. A. and May, J. P., Rings, modules, and algebras in stable homotopy theory. With an appendix by Cole, M.. Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997), xii+249. ISBN: 0-8218-0638-6Google Scholar
Guillou, B. J. and May, J. P., Equivariant iterated loop space theory and permutative G-categories, Algebr. Geom. Topol. 17(6) (2017), 32593339.Google Scholar
Gutiérrez, J. J. and White, D., Encoding equivariant commutativity via operads, Algebr. Geom. Topol. 18(5) (2018), 29192962.CrossRefGoogle Scholar
Hill, M. A. and Hopkins, M. J., Equivariant multiplicative closure, in Algebraic topology: applications and new directions (Tillmann, U., Galatius, S. and Sinha, D., Editors), Contemporary Mathematics, vol. 620 (American Mathematical Society, Providence, RI, 2014), 183–199. ISBN: 978-0-8218-9474-3CrossRefGoogle Scholar
Hill, M. A. and Hopkins, M. J., Equivariant symmetric monoidal structure. Preprint arXiv:1610.03114.Google Scholar
Hill, M. A., Hopkins, M. J. and Ravenel, D. C., On the nonexistence of elements of Kervaire invariant one, Ann. Math. (2) 184(1) (2016), 1262.Google Scholar
May, J. P., The geometry of iterated loop spaces. Lectures Notes in Mathematics, vol. 271 (Springer-Verlag, Berlin-New York, 1972), viii+175.CrossRefGoogle Scholar
May, J. P., Equivariant homotopy and cohomology theory. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou and S. Waner. CBMS Regional Conference Series in Mathematics, vol. 91. Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 1996), xiv+366. ISBN: 0-8218-0319-0CrossRefGoogle Scholar
Rubin, J., Combinatorial N operads. Preprint arXiv:1705.03585.Google Scholar
Rubin, J., Categorifying the algebra of indexing systems. Preprint arXiv:1909.11739.Google Scholar
White, D., Monoidal Bousfield localizations and algebras over operads. arXiv:1404.5197.Google Scholar