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Perspectives in Analysis, Geometry, and Topology pp 283–329Cite as

Birkhäuser

Floer Homology on the Extended Moduli Space

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Part of the book series: Progress in Mathematics ((PM,volume 296))

Abstract

Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant in the form of a relatively \(\mathbb{Z}/8\mathbb{Z}\)-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah–Floer conjecture for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU (2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting.

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Notes

  1. 1.

    A subset of a topological space is comeager if it is the intersection of countably many open dense subsets. Many authors use the term “Baire second category,” which, however, denotes more generally subsets that are not meager, i.e., not the complement of a comeager subset. See, for example, [48, Chap. 7.8].

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Acknowledgments

We would like to thank Yasha Eliashberg, Peter Kronheimer, Peter Ozsváth, Tim Perutz, and Michael Thaddeus for some very helpful discussions during the preparation of this paper. Especially, we would like to thank Ryszard Rubinsztein for pointing out an important mistake in an earlier version of this paper (in which topological invariance was stated as a conjecture).

The first author was partially supported by NSF grant DMS-0852439 and a Clay Research Fellowship. The second author was partially supported by the NSF grants DMS-060509 and DMS-0904358.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA, 90095, USA

    Ciprian Manolescu

  2. Mathematics-Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Christopher Woodward

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  1. Ciprian Manolescu
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Correspondence to Ciprian Manolescu .

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Editors and Affiliations

  1. Université Pierre et Marie Curie and Institut Universitaire de France Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005, Paris, France

    Ilia Itenberg

  2. Institut des Hautes Études Scientifiques Le Bois-Marie, 35, route de Chartres, Bures-sur-Yvette, 91440, France

    Burglind Jöricke

  3. Department of Mathematics, Stockholm University, Stockholm, SE-10691, Sweden

    Mikael Passare

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Manolescu, C., Woodward, C. (2012). Floer Homology on the Extended Moduli Space. In: Itenberg, I., Jöricke, B., Passare, M. (eds) Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol 296. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8277-4_13

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