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Satellites of infinite rank in the smooth concordance group

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Abstract

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group \({\mathcal {C}}\). Our criterion applies widely; notably to many unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.

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Acknowledgements

The authors owe a debt of gratitude to Paul Kirk, with whom the first author learned and explored the ideas and questions central to the paper, and whose mentorship and guidance showed the second author the way forward from her time as a graduate student. The second author is eternally grateful to Tye Lidman for many very fruitful conversations, and to him, Allison Miller, and Chuck Livingston for many helpful comments on an early draft. Matthew Hedden was partially supported by NSF CAREER Grant DMS-1150872, NSF Grant DMS-1709016 and an Alfred P. Sloan Research Fellowship. Juanita Pinzón-Caicedo was partially supported by NSF Grant DMS-1664567, she is also grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support while a portion of this work was prepared for publication.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48823, USA

    Matthew Hedden

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Juanita Pinzón-Caicedo

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  1. Matthew Hedden
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Correspondence to Matthew Hedden.

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Hedden, M., Pinzón-Caicedo, J. Satellites of infinite rank in the smooth concordance group. Invent. math. 225, 131–157 (2021). https://doi.org/10.1007/s00222-020-01026-w

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