Abstract. Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.

Abstract. We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass [2] for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest [4] may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.

Let C Z denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group C to C Z is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order.

Abstract. A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction termsd and d for certain families of three-manifolds.

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.homology sphere with rank y HF pY q " |H 1 pY ; Zq|, where y HF denotes the hat flavor of Heegaard Floer homology.Theorem 2. Let K be a knot in S 3 whose branched double cover ΣpKq is an L-space. If each summand of the first homology of ΣpKq has square-free order, then K satisfies the cosmetic crossing conjecture.Theorem 2 will be deduced from the Dehn surgery characterization of the unknot in L-spaces [Gai15, KMOS07] (see Theorem 11). It is interesting to juxtapose Theorem 2 with [BFKP12, Theorem 1.1], which implies that if a genus one knot K admits a cosmetic crossing change, then H 1 pΣpKqq is cyclic of order d 2 , for some d P Z.An abundant source of knots that meet the conditions of Theorem 2 are the Khovanov thin knots. These knots derive their definition from reduced Khovanov homology, which associates to an oriented link L in S 3 a bigraded vector spaceThe Ě Kh-thin links are those with their homology supported in a single diagonal δ " j´i of the bigradings. In this case, the dimension of Ě Kh is given by the determinant of the link. By work of Manolescu and Ozsváth [MO08], all quasi-alternating links are Ě Kh-thin, and this class includes all non-split alternating links [OS05]. Of relevance here is the fact that the branched double cover of a Ě Kh-thin link is an L-space, which follows from the spectral sequence from Ě KhpLq to y HF p´ΣpLqq [OS05] and the symmetry of Heegaard Floer homology under orientation reversal [OS04].Because the determinant of a knot is equal to the order of the first homology of its branched double cover we immediately obtain the following corollary. Corollary 3. A ĚKh-thin knot with square-free determinant satisfies the cosmetic crossing conjecture.

We study 4-dimensional homology cobordisms without 3-handles via Heegaard and instanton Floer homologies, character varieties, and Thurston geometries. We provide obstructions to such cobordisms arising from each of these theories, and illustrate some topological applications.

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.