Computation
Floer homologies are generally difficult to compute explicitly. For instance, the symplectic Floer homology for all surface symplectomorphisms was completed only in 2007. The Heegaard Floer homology has been huge success story in this regard: researchers have exploited its algebraic structure to compute it for various classes of 3-manifolds and indeed found combinatorial algorithms for computation of much of the theory. It is also connected it to existing invariants and structures and many insights into 3-manifold topology have resulted.
Read more about this topic: Floer Homology
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