Bordered Manifolds with Torus Boundary and the Link Surgery Formula
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Beschreibung
In this memoir, we develop a theory of bordered \mathrm{HF}^- using the link surgery formula of Manolescu and Osváth. We interpret their link surgery complexes as type-\mathrm{D} modules over an associative algebra \mathcal{K}, which we introduce. We prove a connected sum formula, which we interpret as an A_\infty-tensor product over our algebra \mathcal{K}. Topologically, this connected sum formula may be viewed as a formula for gluing along torus boundary components.
We discuss several important examples. As a basic example, if K_1 and K_2 are knots in S^3, and Y is obtained by gluing the complements of K_1 and K_2 together using an orientation reversing diffeomorphism of their boundaries, then our theory may be used to compute \mathrm{CF}^-(Y) from \mathrm{CFK}^\infty(K_1) and \mathrm{CFK}^\infty(K_2). By computing the type-\mathrm{D} modules for rationally framed solid tori, our theory gives a version of the link surgery formula for rationally framed links. As a final example, we use our theory to derive the Heegaard Floer homology of all 3-manifolds which bound the plumbing of a tree of disk bundles over 2-spheres.
Autor*in
Ian Zemke
University of Oregon, Eugene, USA
Themen in »Bordered Manifolds with Torus Boundary and the Link Surgery Formula«
Heegaard Floer homology Dehn surgery link surgery link Floer homology link surgery complex bordered Heegaard Floer homology