Bordered Heegaard Floer homology

2018/06/13 by Robert Lipshitz, Peter Ozsváth, Dylan Thurston +1 · 7 citations
Mathematics · Medicine · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Botulinum Toxin and Related Neurological Disorders

paper · doi:10.1090/memo/1216

Abstract

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) is a module over the algebra and the other of which (type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal A_∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal A_∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tensor product of the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> module of one piece and the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> module from the other piece is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove italic HF With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext class="MJX-tex-mathit" mathvariant="italic">HF</mml:mtext> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat \textit HF</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove italic HF With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext class="MJX-tex-mathit" mathvariant="italic">HF</mml:mtext> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat \textit HF</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

Citations

• Holomorphic disks and knot invariants
• Holomorphic disks and topological invariants for closed three-manifolds

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Related

• Cornered Heegaard Floer homology
• Floer homology beyond borders
• Bordered Floer homology and contact structures
• Heegaard Floer homologies and contact structures
• Bordered Floer homology with integral coefficients for manifolds with torus boundary