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Given an irreducible closed 3--manifold $Y$, we show that its twistedHeegaard Floer homology determines whether $Y$ is a torus bundle over thecircle. Another result we will prove is, if $K$ is a genus 1 null-homologousknot in an $L$--space, and the 0--surgery on $K$ is fibered, then $K$ itself isfibered. These two results are the missing cases of earlier results due to thesecond author.

Two applications of twisted Floer homology