Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?
Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the infinite-dimensional case the sources I know avoid gluing. For proving that the Morse boundary operator squares to zero Abbondandolo-Majer use either an argument involving cellular filtrations or the graph transform method.
Unfortunately, in the context I want to consider, namely proving that some Morse-like theory is isomorphic to singular homology, both of these approaches do not work. (The latter approach works for showing that my Morse-like theory has a boundary operator that squares to zero, though.)
Reading through the finite-dimensional case ("Morse homology" of Schwarz) I realized that there are a lot of arguments which make it difficult to generalize the gluing procedure to cases where the target is not locally compact. For instance, in a lot of indirect arguments, the Rellich compact embedding theorem is used, which fails if the target is infinite-dimensional.
So, is there any instance where gluing in an infinite-dimensional context is proved? Are there certain obstacles not yet overcome?