Tangle Floer homology and cobordisms between tangles

摘要

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed Spin(c) structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an isotropic set of Spin(c) structures. Associated with every A-module M we construct a functor