U torsion structure in Heegaard Floer homology

Abstract

We construct infinite families of Brieskorn spheres with three singular fibers for which the difference between the largest and the second largest length of $U$-torsion in their Heegaard Floer homology $H{F}^{-}$ increases to infinity as the family parameter increases. In particular, these families give a negative answer to the strong geography restriction problem by Alfieri and Binns \cite[Question 1.10]{AB24} which was added to the recent K3 list \cite[Problem 3.55, Remark(5)]{K3}. As a consequence of the results by Alfieri and Binns, these families give the examples of integer homology spheres that are neither surgery on a knot nor large surgery on links.

arXiv comment35 pages, 2 figures