Abstract
An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call “Floer simple,” we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope from the interval's interior. As applications, we give a new proof of the classification of Seifert fibered L-spaces over S2, and prove a special case of a conjecture of Boyer and Clay [6] about L-spaces formed by gluing three-manifolds along a torus.
Highlights
An oriented rational homology 3-sphere Y is called an L-space if the Heegaard Floer homology HF (Y ) satisfies HF (Y, s) Z for each Spinc structure s on Y
We find that m + nl ∈ L(Y ) for all n > 0, and that l is a limit point of L(Y ). It follows that Y is Floer simple and D>τ0(Y ) = ∅
If Y1 is Floer simple, it follows from Theorem 1.8 that α is NLS detected by Y2 if and only if α is not in the interior of L(Y )
Summary
An oriented rational homology 3-sphere Y is called an L-space if the Heegaard Floer homology HF (Y ) satisfies HF (Y, s) Z for each Spinc structure s on Y. A compact oriented three-manifold Y with torus boundary is Floer simple if it has some Dehn filling Y (α) whose core Kα is a Floer simple knot in Y (α). It follows that if Y is Floer simple, the Floer homology of any Dehn filling of Y can be determined from the Turaev torsion together with a single α ∈ L(Y ). Given τ (Y ) and a Floer simple filling slope α for Y , it is straightforward to determine L(Y ): the torsion determines the set Dτ(Y ), and L(Y ) is the smallest interval with endpoints in ι−1(D>τ0(Y )) which contains α in its interior
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