Complex sentence: Many of these Floer homologies have not been completely rigorously constructed, and many conjectural equivalences have not been proved, due to the difficulty of the analysis involved, especially in constructing compactified moduli spaces of pseudoholomorphic curves.
Split into simpler sentences: 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.