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Turkish Journal of Mathematics

WEIPING LI

DOI

-

Abstract

To an integral homology 3-sphere Y, we assign a well-defined {\mathbb Z}-graded (monopole) homology MH_*(Y, I_{\eta}(\Theta; \eta_0)) whose construction in principle follows from the instanton Floer theory with the dependence of the spectral flow I_{\eta}(\Theta; \eta_0), where \Theta is the unique U(1)-reducible monopole of the Seiberg-Witten equation on Y and \eta_0 is a reference perturbation datum. The definition uses the moduli space of monopoles on Y \times {\mathbb R} introduced by Seiberg-Witten in studying smooth 4-manifolds. We show that the monopole homology MH_*(Y, I_{\eta}(\Theta; \eta_0)) is invariant among Riemannian metrics with same I_{\eta}(\Theta; \eta_0). This provides a chamber-like structure for the monopole homology of integral homology 3-spheres. The assigned function MH_{SWF}: \{I_{\eta}(\Theta; \eta_0)\} \to \{MH_*(Y, I_{\eta}(\Theta; \eta_0))\} is a topological invariant (as Seiberg-Witten-Floer Theory).

First Page

125

Last Page

160

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