Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0

Abstract

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of four-manifolds with boundary which generalizes the Bauer-Furuta stable homotopy invariant of closed four-manifolds.

v4, added errata: In Section 9, the Coulomb-Neumann condition should be replaced by a double Coulomb condition, as in Khandhawit's paper (arxiv:1401.7590). Other minor errors are fixed. The main results are unchanged. Version 3 was published in Geom. Topol. 7(2003) 889-932; current version contains appended errata. v5: added item (4) to the errata

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