This requires also the use of some of the results of Hodge theory, about the Hodge Laplacian.
Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.
If the first Betti number of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and symplectic isotopy of symplectomorphisms coincide.
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Henri Poincaré's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds(which he called "varieties"), giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-century knot theory, developed by Maxwell and others), and Betti numbers and raised a question, today known as the Poincaré conjecture, based his new concept of the fundamental group.