This book is devoted to the study
of moduli spaces of Seiberg-Witten monopoles over
spinc Riemannian 4–manifolds with
long necks and/or tubular ends. The original purpose
of this work was to provide analytical foundations
for a certain construction of Floer homology of
rational homology 3–spheres; this is carried
out in [Monopole Floer homology for rational
homology 3–spheresarXiv:08094842].
However, along the way the project grew, and, except
for some of the transversality results, most of the
theory is developed more generally than is needed for
that construction. Floer homology itself is hardly
touched upon in this book, and, to compensate for that,
I have included another application of the analytical
machinery, namely a proof of a "generalized blow-up
formula" which is an important tool for computing
Seiberg–Witten invariants.
The book is divided into three
parts. Part 1 is almost identical to my paper
[Monopoles
over 4–manifolds containing long necks
I, Geom. Topol. 9 (2005) 1–93]. The
other two parts consist of previously unpublished
material. Part 2 is an expository account of gluing
theory including orientations. The main novelties
here may be the formulation of the gluing theorem,
and the approach to orientations. In Part 3 the
analytical results are brought together to prove the
generalized blow-up formula.
