$U$-action on perturbed Heegaard–Floer homology

This paper has two purposes. First, as a continuation of [27], we apply a similar method to compute the perturbed $HF^+$ for some special classes of fibered three-manifolds in the second highest spin$^c$ structures, including the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves. Second, we establish an adjunction inequality for the perturbed Heegaard–Floer homology, which indicates a potential connection between the $U$-action on the homology group and the Thurston norm of a three-manifold. As an application, we find the $U$-action on the perturbed $HF^+$ of the above classes of fibered three-manifolds is trivial.

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Published 18 September 2012