On lens space surgeries from the Poincar\'e homology sphere

Abstract

Building on Greene's changemaker lattices, we develop a lattice embedding obstruction to realizing an L-space bounding a definite 4-manifold as integer surgery on a knot in the Poincar\'e homology sphere. As the motivating application, we determine which lens spaces are realized by $p/q$-surgery on a knot $K$ when $p/q > 2g(K) -1$. Specifically, we use the lattice embedding obstruction to show that if $K(p)$ is a lens space and $p \geq 2g(K)$, then there exists an equivalent surgery on a Tange knot with the same knot Floer homology groups; additionally, using input from Baker, Hedden, and Ni, we identify the only two knots in the Poincar\'e homology sphere that admit half-integer lens space surgeries. Thus, together with the Finite/Cyclic Surgery Theorem of Boyer and Zhang, we obtain the corollary that lens space surgeries on hyperbolic knots in the Poincar\'e homology sphere are integral.