Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on K^*_{orb}(\XX) \otimes \C , the orbifold K-theory of any almost complex presentable orbifold \XX. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here H^*_{CR}(\XX) is the Chen-Ruan cohomology of \XX. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryajin product (the latter is also called the fusion product in string theory).Comment: 34 pages. Final version to appear in Trans. of AMS. Improve the expositions and Change of the title thanks the referee

Exact triangles in monopole homology and the Casson-Walker invariant

We establish the exact triangle in Seiberg-Witten-Floer theory relating the monopoloe homologies of any two closed 3-manifolds which are obtained from each other by $\pm 1$-surgery. We also show that the sum of the modified version of the Seiberg-Witten invariants for any closed rational homology 3-sphere $Y$ over all $Spin^c$ structures equals to $\frac 12 |H_1(Y, \Z)| \lambda (Y)$ where $\lambda (Y)$ is the Casson-Walker invariant.Comment: 35 pages, 3 figure

Equivariant Seiberg-Witten Floer Homology

This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with related gluing theorems, needed in the proof of the topological invariance of the equivariant version of the Floer homology.Comment: 162 pages, LaTex, 1 figure, diagrams (xypic

Exact triangles in Seiberg-Witten Floer theory. Part IV: Z-graded monopole homology

Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.Comment: 34 page

Exact triangles in Seiberg-Witten Floer theory. Part III: proof of exactness

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.Comment: 69 pages, 4 figure

Exact triangles in Seiberg-Witten Floer theory. Part II: geometric limits of flow lines

This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)Comment: 71 pages, 3 figure