approach was the basis of Manolescu's 2013 construction of Pin (2)-equivariant Seiberg–Witten Floer homology, with which he disproved the Triangulation Conjecture for manifolds of dimension 5 and higher. Analytic foundations Many of these Floer homologies have not been completely and rigorously constructed, and many conjectural equivalences have not been proved. Technical difficulties come up in the analysis involved, especially in constructing compactified moduli spaces of pseudoholomorphic curves. Hofer, in collaboration with Kris Wysocki and Eduard Zehnder, has developed new analytic foundations via their theory of polyfolds and a "general Fredholm theory". While the polyfold project is not yet fully completed, in some important cases transversality was shown using simpler methods. 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 a success story in this regard: researchers have exploited its algebraic structure to compute it for various classes of 3-manifolds and have found combinatorial algorithms for computation of much of the theory. It is also connected to existing invariants and structures and many insights into 3-manifold topology have resulted. References Footnotes Books and surveys Research articles Project Euclid External links Mathematical physics 3-manifolds Gauge theories Morse theory Homology theory Symplectic topology