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Seiberg-Witten Floer Homology and Heegaard Splittings

Marcolli, Matilde (1996) Seiberg-Witten Floer Homology and Heegaard Splittings. International Journal of Mathematics, 7 (5). pp. 671-696. ISSN 0129-167X. doi:10.1142/S0129167X96000359.

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The dimensional reduction of Seiberg-Witten theory defines a gauge theory of compact connected three-manifolds. Solutions of the equations modulo gauge symmetries on a three-manifold Y can be interpreted as the critical points of a functional defined on an infinite dimensional configuration space of U(1)-connections and spinors. The original Seiberg-Witten equations on the infinite cylinder Y × IR are the downward gradient flow of the functional. Thus, it is possible to construct an infinite dimensional Morse theory. The associated Morse homology is the analogue in the context of Seiberg-Witten theory of Floer’s instanton homology constructed using Yang-Mills gauge theory. The construction and the properties of this Seiberg-Witten Floer homology are essentially different according to whether the three-manifold Y is a homology sphere or has non-trivial rational homology. In this work we construct the Seiberg-Witten Floer homology for three-manifolds with b^1(Y ) > 0. We define an associated Casson-like invariant and we prove that it satisfies the expected intersection formula under a Heegaard splitting of the three-manifold.

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Additional Information:© 1996 World Scientific Publishing. Received 9 February 1996; Revised 30 May 1996. I am grateful to Tom Mrowka for the many useful conversations, for his very helpful comments and criticism, and for his patience. I am grateful to Bai-Ling Wang for the invaluable and always stimulating collaboration. I thank Rong Wang for having informed me of his interesting work on the subject, for having pointed out a mistake in a previous version of this work, and for some very helpful suggestions. I am grateful to my advisor Mel Rothenberg for having supervised this work. I thank Stefan Bauer, Rogier Brussee, Ralph Cohen, Rafe Mazzeo, and Dan Pollack for useful and stimulating conversations.
Issue or Number:5
Classification Code:1991 Mathematics Subject Classification: 57R57; 55N35; 58E05
Record Number:CaltechAUTHORS:20170713-092558409
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Official Citation:SEIBERG-WITTEN-FLOER HOMOLOGY AND HEEGAARD SPLITTINGS. M. MARCOLLI. International Journal of Mathematics 1996 07:05, 671-696
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79071
Deposited On:13 Jul 2017 16:39
Last Modified:15 Nov 2021 17:45

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