This paper introduces topological invariants for closed, oriented three-manifolds equipped with Spin$^{c}$ structures. These invariants are defined using Heegaard splittings and relate to Lagrangian Floer homology for the g-fold symmetric product of a surface Σ. The invariants include Floer homology groups such as $\widehat{HF}(Y,\mathfrak{s})$, $HF^{+}(Y,\mathfrak{s})$, $HF^{-}(Y,\mathfrak{s})$, $HF^{\infty}(Y,\mathfrak{s})$, and $HF_{\mathrm{red}}(Y,\mathfrak{s})$. These groups are defined using the symmetric product of Σ and certain totally real submanifolds associated with the Heegaard splitting. The invariants are shown to be topological invariants of the three-manifold and the Spin$^{c}$ structure, independent of the Heegaard splitting, the choice of attaching circles, the basepoint, and the complex structures used in their definition. The paper also discusses the algebraic structure of these invariants, including actions that decrease the relative degree by two and one, and their relationship with gauge theory and four-manifold invariants. The invariants are constructed using holomorphic disks and their moduli spaces, with careful consideration of smoothness and compactness. The paper establishes the invariance of these homology groups under Heegaard moves, including isotopies, handleslides, and stabilizations, and provides a detailed analysis of the algebraic and topological properties of the invariants.This paper introduces topological invariants for closed, oriented three-manifolds equipped with Spin$^{c}$ structures. These invariants are defined using Heegaard splittings and relate to Lagrangian Floer homology for the g-fold symmetric product of a surface Σ. The invariants include Floer homology groups such as $\widehat{HF}(Y,\mathfrak{s})$, $HF^{+}(Y,\mathfrak{s})$, $HF^{-}(Y,\mathfrak{s})$, $HF^{\infty}(Y,\mathfrak{s})$, and $HF_{\mathrm{red}}(Y,\mathfrak{s})$. These groups are defined using the symmetric product of Σ and certain totally real submanifolds associated with the Heegaard splitting. The invariants are shown to be topological invariants of the three-manifold and the Spin$^{c}$ structure, independent of the Heegaard splitting, the choice of attaching circles, the basepoint, and the complex structures used in their definition. The paper also discusses the algebraic structure of these invariants, including actions that decrease the relative degree by two and one, and their relationship with gauge theory and four-manifold invariants. The invariants are constructed using holomorphic disks and their moduli spaces, with careful consideration of smoothness and compactness. The paper establishes the invariance of these homology groups under Heegaard moves, including isotopies, handleslides, and stabilizations, and provides a detailed analysis of the algebraic and topological properties of the invariants.