Geometry & Topology, Vol. 7 (2003) Paper no. 26, pages 889--932.

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

Ciprian Manolescu

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.

Keywords. 3--manifolds, Floer homology, Seiberg--Witten equations, Bauer--Furuta invariant, Conley index

AMS subject classification. Primary: 57R58. Secondary: 57R57.

DOI: 10.2140/gt.2003.7.889

E-print: arXiv:math.DG/0104024

Submitted to GT on 2 May 2002. Paper accepted 5 December 2003. Paper published 10 December 2003.

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Ciprian Manolescu
Department of Mathematics, Harvard University
1 Oxford Street, Cambridge, MA 02138, USA
Email: manolesc@fas.harvard.edu

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