Nonseparating spheres and twisted Heegaard Floer homology

Abstract

If a 3-manifold Y contains a non-separating sphere, then some twisted Heegaard Floer homology of Y is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in L-spaces.

Thurston norm and cosmetic surgeries

“…The main theorem of this paper is an analogue of [13,Theorem 9.7] and [8,Theorem 1.5]. See also [16].…”

“…As in the proof of [2, Theorem 2.2], using the known non-triviality results for twisted coefficients stated in [8] and the Universal Coefficients Theorem, we can prove the following theorems. (The same results can also be proved via the approach taken in [4,7].…”

Cosmetic surgery in L–space homology spheres

“…Combining number theoretical methods, Wang was able to prove the conjecture for genus one knots in S 3 [20]. Ni, in another direction, used a twisted version of the Heegaard Floer homology to prove the case when K is a nullhomologous knot in a closed three-manifold Y that contains a non-separating sphere [7].…”

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

“…If X is as in Conjecture 5.2, then by Lemmas 2.1 and 5.4 we may assume Y has an S 1 × S 2 summand, so the Monopole or Heegaard Floer homology of Y with certain twisted coefficients is zero [31]. The monodromy induces a map on the Floer homology of Y [20].…”