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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?

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  • $\begingroup$ The tags don't seem to work properly! Any ideas? $\endgroup$
    – Orbicular
    Commented Jun 29, 2011 at 19:13
  • $\begingroup$ Are you thinking of Floer-type theories, with semi-infinite critical points? In those theories, there are general results about linearized gluing and a standard approach to non-linear gluing (but not a theorem). Or of finite index critical points, as in the Riemannian energy functional? $\endgroup$
    – Tim Perutz
    Commented Jun 30, 2011 at 11:19
  • $\begingroup$ I am thinking of Morse homology on Hilbert manifolds with critical points of finite Morse index. In (symplectic) Floer homology one usually considers maps with locally compact domain which is not the case in the stuff I consider. $\endgroup$
    – Orbicular
    Commented Jun 30, 2011 at 20:20

2 Answers 2

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There's quite a bit of literature on gluing theory for instantons and monopoles, which take the place of flow lines of Morse functions in instanton Floer homology and in Seiberg-Witten Floer homology correspondingly. It is indeed quite a bit harder than the finite dimensional case, and yes, it's an active research topic. I'm no expert and I can't explain the key ideas in a nutshell, but here are some references to get you started.

I think the originator of the theory was Cliff Taubes:

  • Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom. 17, 139-170 (1982; Zbl 0484.53026).
  • Self-dual connections on 4-manifolds with indefinite intersection matrix. J. Differ. Geom. 19, 517-560 (1984; Zbl 0552.53011).

There's a nice exposition of gluing theory for instantons in:

  • S.K. Donaldson, Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics. 147. Cambridge: Cambridge University Press. (2002; Zbl 0998.53057).

Nice expositions for gluing theory for monopoles are to be found in:

  • K.A. Frøyshov Compactness and gluing theory for monopoles. Geometry and Topology Monographs 15. Coventry: Geometry & Topology Publications. (2008; Zbl 1207.57044).
  • P. Kronheimer, Peter and T. Mrowka Monopoles and three-manifolds. New Mathematical Monographs 10. Cambridge: Cambridge University Press. (2007; Zbl 1158.57002).
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I would say, take a look at the paper of Abbondandolo and Majer :http://arxiv.org/abs/math/0403552 or http://www.dm.unipi.it/~abbondandolo/preprints/montreal.pdf The construction is neat and clear.

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