Hilbert manifolds and perturbations
P. B. Kronheimer, Tomasz Mrowka · Cambridge University Press eBooks · 2007
In finite-dimensional Morse theory, the reconstruction of the ordinary homology of a manifold from a suitable Morse function relies on the familiar notions of transversality in differential topology. Finite-dimensional results such as Sard's theorem have their analogs for infinite-dimensional manifolds modelled on Banach or Hilbert spaces. This chapter introduces suitable Hilbert manifolds, to replace our configuration spaces of smooth pairs ( A , Ф). We then examine with care how to introduce perturbations of the Seiberg–Witten equations. In the next chapter, this framework will allow us to carry over the transversality results, from finite-dimensional Morse theory, to the Morse theory of the Chern–Simons–Dirac functional and its perturbations. Completions and Hilbert manifolds Completions of the configuration spaces Although Sobolev norms were introduced in the proof of the compactness theorems, we have so far considered only smooth configurations. We now introduce the Sobolev completions of C ( X , s X ) and C ( Y , s), and of the corresponding gauge groups. In order to be able to deal with X and Y side by side, we will temporarily introduce M to stand for either one. Thus M will be a compact Riemannian manifold (whose dimension will be 3 or 4) with a spin c structure. The boundary ∂ M may be non-empty.