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“School of Mathematics”

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Paper   IPM / M / 16150
School of Mathematics
  Title:   A refinement of sutured Floer homology
  Author(s):  Eaman Eftekhary (Joint with A. Alishahi)
  Status:   Published
  Journal: J. Symplectic Geom.
  Vol.:  13
  Year:  2015
  Pages:   609-743
  Supported by:  IPM
  Abstract:
We introduce a refinement of the Ozsváth-Szabó complex associated to a balanced sutured manifold (X,τ) by Juhász []. An algebra \Ringτ is associated to the boundary of a sutured manifold and a filtration of its generators by \Ht2(X,∂X;\Z) is defined. For a fixed class \spinc of a \SpinC structure over the manifold \ovl X, which is obtained from X by filling out the sutures, the Ozsváth-Szabó chain complex \CFT(X,τ,\spinc) is then defined as a chain complex with coefficients in \Ringτ and filtered by \SpinC(X,τ). The filtered chain homotopy type of this chain complex is an invariant of (X,τ) and the \SpinC class \spinc ∈ \SpinC(\ovl X). The construction generalizes the construction of Juhász. It plays the role of \CFT(X,\spinc) when X is a closed three-manifold, and the role of
\CFKT(Y,K;\spinc)=

\relspinc ∈ \spinc 
\CFKT(Y,K,\relspinc),
when the sutured manifold is obtained from a knot K inside a three-manifold Y. Our invariants generalize both the knot invariants of Ozsváth-Szabó and Rasmussen and the link invariants of Ozsváth and Szabó. We study some of the basic properties of the corresponding Ozsváth-Szabó complex, including the exact triangles, and some form of stabilization.

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