Mod 2 instanton homology and 4-manifolds with boundary

Using instanton homology with coefficients in $Z/2$ we construct a homomorphism $q_2$ from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton $h$--invariant and the Heegaard Floer correction term $d$. If an oriented homology $3$--sphere $Y$ bounds a smooth, compact, negative definite $4$--manifold without $2$--torsion in its homology then $q_2(Y)\ge0$, with strict inequality if the intersection form is non-standard.Comment: 88 pages, 2 figure

Monopoles over 4-manifolds containing long necks, I

We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper1.abs.htm

Compactness and gluing theory for monopoles

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–spheres arXiv: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 The other two parts consist of previously unpublished material.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