The article introduces topological invariants for closed, oriented three-manifolds equipped with a Spin$^c$ structure. These invariants are defined using Heegaard splittings of the three-manifold and are related to Lagrangian Floer homology. The authors define several Floer homology groups, including $\widehat{HF}(Y, \mathfrak{s})$, $HF^+(Y, \mathfrak{s})$, $HF^-(Y, \mathfrak{s})$, $HF^{\infty}(Y, \mathfrak{s})$, and $HF_{\text{red}}(Y, \mathfrak{s})$. These groups are constructed by studying the $g$-fold symmetric product of the genus $g$ Riemann surface $\Sigma$ and using certain totally real submanifolds associated to the Heegaard splitting. The invariants are shown to be topological invariants of the three-manifold and the Spin$^c$ structure, independent of the choice of Heegaard splitting, attaching circles, basepoint, and complex structures. The proof involves detailed constructions and arguments, including the study of holomorphic disks and their properties in the symmetric product.The article introduces topological invariants for closed, oriented three-manifolds equipped with a Spin$^c$ structure. These invariants are defined using Heegaard splittings of the three-manifold and are related to Lagrangian Floer homology. The authors define several Floer homology groups, including $\widehat{HF}(Y, \mathfrak{s})$, $HF^+(Y, \mathfrak{s})$, $HF^-(Y, \mathfrak{s})$, $HF^{\infty}(Y, \mathfrak{s})$, and $HF_{\text{red}}(Y, \mathfrak{s})$. These groups are constructed by studying the $g$-fold symmetric product of the genus $g$ Riemann surface $\Sigma$ and using certain totally real submanifolds associated to the Heegaard splitting. The invariants are shown to be topological invariants of the three-manifold and the Spin$^c$ structure, independent of the choice of Heegaard splitting, attaching circles, basepoint, and complex structures. The proof involves detailed constructions and arguments, including the study of holomorphic disks and their properties in the symmetric product.