Lagrangian Floer Homology and the Arnold Conjecture on Cotangent Bundles

This monograph explores the rigorous construction of Lagrangian Floer homology within the symplectic framework of cotangent bundles T*M, specifically addressing the Arnold Conjecture regarding the lower bound of intersection points of Lagrangian submanifolds under Hamiltonian isotopies. We systematically develop the analytic foundations of the Floer chain complex CF*(L, L'; H), generated by the critical points of the symplectic action functional, and defined over the Novikov ring to accommodate non-exact symplectic manifolds, though specializing here to the exact Liouville case. Central to this exposition is the analysis of the moduli spaces of pseudo-holomorphic strips M(x, y) satisfying the perturbed Cauchy-Riemann equations. We present a detailed proof of the isomorphism between the Lagrangian Floer homology of the zero section and the singular homology of the free loop space LM, establishing a bridge between symplectic topology and classical algebraic topology. The transfinite nature of the homological algebra involved is discussed in the context of A-structures and the Fukaya category.