Heegaard Floer homology is combinatorially computable, but a convenient computational scheme in general is still missing, especially for $HF^-$ of hyperbolic manifolds. We aim to use the Manolescu-Ozsv {a}th the link surgery formula for computing Heegaard Floer homology of surgeries on links and finding applications on $L$-space surgeries on links. The main difficulty is to reduce the complexity of the algorithms.

We give a polynomial time algorithm to compute the whole package of the completed

Heegaard Floer homology ${\textbf{HF}}^{-}$ of all surgeries on a two-bridge link of slope $q/p$, $L=b(p,q)$, by using nice diagrams and some algebraic rigidity results to simplify the link surgery formula.

We also initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space links, including some chain links and two-bridge links; from them, we obtain many hyperbolic $L$-spaces by integral surgeries, including the Weeks manifold. We give bounds on the ranks of the link Floer homology of $L$-space links and on the coefficients in the multi-variable Alexander polynomials. We also describe the Floer homology of surgeries on any $L$-space link using the link surgery formula of Manolescu and Ozsv {a}th.

As applications, we compute the graded Heegaard Floer homology of surgeries on 2-component $L$-space links in terms of only the Alexander polynomial and the surgery framing. We also give a fast algorithm to classify $L$-space surgeries among them.

Manolescu and Ozsváth have recently developed a formula for calculating the Heegaard Floer homologies of integral surgery on a link. We use their link surgery formula to give a complete calculation (mod 2) of the infinity flavor of Heegaard Floer homology for a torsion Spin^c structure on any closed, orientable three-manifold in terms of the cup product structure on its integral cohomology ring.

In this thesis, we define different versions of unfolded Seiberg-Witten Floer spectra for general 3-manifolds. They generalize Manolescu's and Kronheimer-Manolescu's construction of Floer stable homotopy type. We prove some properties of these new invariants and give some topological applications (Joint works with collaborators.) Along the way, as an application of the Seiberg-Witten Floer spectrum, we study the Pin(2)-equivariant Seiberg-Witten Floer KO-theory and prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres.

In this thesis, we study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a disoriented link cobordism. Furthermore, we give a comparison with Oszvath-Stipsicz-Szabo’s and Manolescu’s constructions of link cobordism maps for an unoriented band move.

We investigate the sheaf-theoretic SL(2,C) Floer cohomology for knots and 3-manifolds presented as surgeries on knots in S^3. We establish a relationship between the 3-manifold invariants, HP(Y) and HP#(Y) for Y a surgery on a small knot in S^3, and the SL(2,C) Casson invariant defined by Curtis. We use this to compute HP for surgeries on the trefoil and the figure-eight knots. We also compute HP for surgeries on two non-small knots, the granny and square knots. For the knot invariant, we prove that the (τ-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant is generically independent of the parameter τ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions posed by Cote-Manolescu.

We study Manolescu's $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of rational homology three-spheres, with applications to the homology cobordism group $\theta^H_3$ in mind. We compute this homology theory for Seifert rational homology three-spheres in terms of their Heegaard Floer homology. We prove Manolescu's conjecture that $\beta=-\bar{\mu}$, the Neumann-Siebenmann invariant, for Seifert integral homology three-spheres. We establish the existence of integral homology spheres not homology cobordant to any Seifert space. We show that there is a naturally defined subgroup of the homology cobordism group, generated by certain Seifert spaces, which admits a $\mathbb{Z}^\infty$ summand, generalizing the theorem of Fintushel-Stern and Furuta on the infinite-generation of the homology cobordism group. In addition to the application of the $\mathrm{Pin}(2)$-theory to Seifert spaces, we apply it to the full homology cobordism group. In this direction, we identify a $\mathbb{F}[U]$-submodule of Heegaard Floer homology, called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles.

We describe two projects involving the construction of Floer homology

theories for knots in lens spaces. In the first project, we propose

definitions of complex manifolds P_M(X,m,n) that could potentially be

used to construct the symplectic Khovanov homology of n-stranded knots

in lens spaces. The manifolds P_M(X,m,n) are defined as moduli spaces

of Hecke modifications of rank 2 parabolic bundles over an elliptic

curve X. To characterize these spaces, we describe all possible Hecke

modifications of all possible rank 2 vector bundles over X, and we

use these results to define a canonical open embedding of P_M(X,m,n)

into M^s(X,m+n), the moduli space of stable rank 2 parabolic bundles

over X with trivial determinant bundle and m+n marked points. We

explicitly compute P_M(X,1,n) for n=0,1,2. For comparison, we present

analogous results for the case of rational curves, for which a

corresponding complex manifold P_M(CP^1,3,n) is isomorphic for n even

to a space Y(S^2,n) defined by Seidel and Smith that can be used to

compute the symplectic Khovanov homology of n-stranded knots in S^3.

In the second project, we describe a scheme for constructing

generating sets for Kronheimer and Mrowka's singular instanton knot

homology for the case of knots in lens spaces. The scheme involves

Heegaard-splitting a lens space containing a knot into two solid tori.

One solid torus contains a portion of the knot consisting of an

unknotted arc, as well as holonomy perturbations of the Chern-Simons

functional used to define the homology theory. The other solid torus

contains the remainder of the knot. The Heegaard splitting yields a

pair of Lagrangians in the traceless SU(2)-character variety of the

twice-punctured torus, and the intersection points of these

Lagrangians comprise the generating set that we seek. We illustrate

the scheme by constructing generating sets for several example knots.

Our scheme is a direct generalization of a scheme introduced by

Hedden, Herald, and Kirk for describing generating sets for knots in

S^3 in terms of Lagrangian intersections in the traceless

SU(2)-character variety for the 2-sphere with four punctures.

In the early 2000s, Ozsv {a}th and Szab {o} introduced a collection of invariants for 3--manifolds and 4--manifolds called Heegaard Floer homology. To a 3--manifold they constructed a group, and to a 4--manifold which cobounds two 3--manifolds, they constructed a homomorphism between the manifolds appearing on the ends. Their invariants satisfy many of the axioms of a TQFT as described by Atiyah, however their construction has some additional restrictions which prevent it from fitting into Atiyah's framework. There is a refinement of Heegaard Floer homology for 3--manifolds containing a knot, due to Ozsv {a}th and Szab {o}, and independently Rasmussen, and a further refinement for 3--manifolds containing links, due to Ozsv {a}th and Szab {o}. It's a natural question as to whether one can define functorial maps associated to link cobordisms.

In this thesis, we describe a package of cobordism maps for Heegaard Floer homology and link Floer homology. The cobordism maps satisfy an appropriate analogy of the axiomatic description of a TQFT formulated by Atiyah. To a ribbon graph cobordism between two based 3--manifolds, we associate a map between the Heegaard Floer homologies of the ends. To a decorated link cobordism, we obtain maps on the link Floer homologies of the ends. The maps associated to decorated link cobordisms reduce to the maps for ribbon graphs, in a natural way.

As applications, we describe several formulas for mapping class group actions on the Heegaard Floer and knot Floer groups. We prove a new bound on a concordance invariant $\Upsilon_K(t)$ from knot Floer homology, and also see how the link cobordism maps give straightforward proofs of other bounds on concordance invariants from knot Floer homology. We also explore the interaction of the maps with conjugation actions on Heegaard Floer homology and link Floer homology, giving connected sum formulas for involutive Heegaard Floer homology and involutive knot Floer homology.

We define four versions of equivariant instanton Floer homology (I+, I-, I∞ and I~) for a class of 3-manifolds and SO(3)-bundles over them including all rational homology spheres.

These versions are analogous to the four flavors of monopole and Heegaard Floer homology theories. This construction is functorial for a large class of 4-manifold cobordisms, and agrees with Donaldson’s definition of equivariant instanton homology for integer homology spheres. Furthermore, one of our invariants is isomorphic to Floer’s instanton homology for admissible bundles, and we calculate I∞(Y,E) in all cases it is defined, away from characteristic 2.

The appendix, possibly of independent interest, defines an algebraic construction of three equivariant homology theories for dg-modules over a dg-algebra, the equivariant homology H+(A,M), the coBorel homology H-(A,M), and the Tate homology H∞(A,M).

The constructions of the appendix are used to define our invariants.