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Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology

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Mallick,  Abhishek       
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Dai, I., Hedden, M., Mallick, A., & Stoffregen, M. (2024). Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology. Journal of Topology, 17(4): e70008. doi:10.1112/topo.70008.


Cite as: https://hdl.handle.net/21.11116/0000-0010-3CC0-2
Abstract
We show that a large class of satellite operators are rank-expanding; that is, they map some rank-one subgroup of the concordance group onto an infinite linearly independent set. Our work constitutes the first systematic study of this property in the literature and partially affirms a conjecture of the second author and Pinzón-Caicedo. More generally, we establish a Floer-theoretic condition for a family of companion knots to have infinite-rank image under satellites from this class. The methods we use are amenable to patterns which act trivially in topological concordance and are capable of handling a surprisingly wide variety of companions. For instance, we give an infinite linearly independent family of Whitehead doubles whose companion knots all have negative τ-invariant. Our also results recover and extend several theorems in this area established using instanton Floer homology.