Dehn surgeries and rational homology balls

Abstract

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsv\'ath and Szab\'o's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldon's theorem, we classify which integral surgeries along torus knots of the form $T_{kq\pm 1,q}$ bound rational homology balls.Comment: 32 pages, many figures. Several minor improvements and corrections. Comments are welcome

Embedding 3‐manifolds in spin 4‐manifolds

“…In fact, Neumann and Raymond show that, up to reversing the orientation, every lens space bounds a spin linear plumbing of spheres [28, Lemma 6.3]; furthermore, an easy induction shows that the number of spheres in the plumbing is at most p − 1, giving an alternative proof of the proposition above. Now Proposition 2.8 (1) shows that ε(L(p, q)) 1. Moreover, it has been observed by Rasmussen that every lens space in Lisca's list † [24] bounds a rational homology ball that admits a handle decomposition with one handle of each index 0, 1, and 2 (see [3]).…”

“…Combining this directly with Theorem 2.6 and Theorem 2.5 gives the following result. † As observed by several authors, the case gcd(m, k) = 2 should be included in type (1) in the definition of R in that paper. This naturally leads to considering which lens spaces with p even have embedding number 1.…”

“…By the 10/8-Theorem [17], 5 4 |σ(X)| + 2 = 5 4 |1 − n + σ(Z)| + 2 = 5 4 (n − 1 − σ(Z)) + 2, (1) and in particular 7 − 4n − 4b 2 (Z) 5σ(Z) − 5n 5b 2 (Z) − 5n, from which we obtain b 2 (Z) 1 9 n + 7 9 . Let Z = # m S 2 × S 2 \ Z.…”

“…. , H r represent a free basis for π 1 (X (1) ) (where X (1) denotes the union of the 0-handle and the 1-handles of X), and the attaching circles for the other 2-handles are null-homotopic in π 1 (X (1) ).…”

Handle decompositions of rational homology balls and Casson–Gordon invariants

“…. , v, so does Y ([1]; see also [2]). We claim that if each n j is odd, n j and n k are pairwise coprime for each j = k, and a ≡ v (mod 2), then H 1 (Y ) is cyclic.…”