Bordered Floer homology for manifolds with torus boundary via immersed curves

Abstract

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is such a manifold, we show that the type D structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper C upper F upper D With caret left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {CFD}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be viewed as a set of immersed curves decorated with local systems in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper M"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H upper F With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {HF}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H upper F With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {HF}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, it follows that a prime rational homology sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H upper F With caret left-parenthesis upper Y right-parenthesis greater-than 5"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat {HF}(Y)&gt;5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

Topics & Concepts