ABSTRACT

A. Floer introduced Morse theoretic homological invariants that transformed the study of low dimensional topology and symplectic geometry. A dramatic application of the ideas of Floer homotopy theory appeared in the work of Lipshitz and Sarkar on the homotopy theoretic foundations of M. Khovanov’s homological invariants of knots and links. The Floer homotopy theory used was a type of Hamiltonian Floer theory for the cotangent bundle. The Khovanov chain complex is generated by all possible configurations of resolutions of the crossings of a link diagram. Monopole Floer homology is similar in nature to Floer’s Instanton homology theory, but it is based on the Seiberg-Witten equations rather than the Yang-Mills equations. The chapter describes the Floer homotopy theoretic methods of Kragh and of Abouzaid-Kragh in the study of the symplectic topology of the cotangent bundle of a closed manifold, and how they were useful in studying Lagrangian immersions and embeddings inside the cotangent bundle.