Papers › On lens space surgeries from the Poincaré homology sphere
On lens space surgeries from the Poincaré homology sphere
Jacob Caudell
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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 ≥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.
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