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Bordered Heegaard Floer Homology

By: Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston

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Bordered Heegaard Floer Homology

By: Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston

The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ^HF of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ^HF. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

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Page Count

294

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470428881

ISBN 13

9781470428884

Naturality and Mapping Class Groups in Heegard Floer Homology

By: András Juhász,Dylan P. Thurston,Ian Zemke

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Naturality and Mapping Class Groups in Heegard Floer Homology

By: András Juhász,Dylan P. Thurston,Ian Zemke

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Page Count

174

Publisher

American Mathematical Society

Published Date

ISBN 10

1470449722

ISBN 13

9781470449728

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

By: Sergey Fomin,Professor Dylan Thurston

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Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

By: Sergey Fomin,Professor Dylan Thurston

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realization in terms of suitable decorated Teichmüller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. The authors' model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from the authors' previous paper, removing unnecessary assumptions on the surface.

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Page Count

110

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470429675

ISBN 13

9781470429676