Heegaard Floer theory for 3-manifolds, tangles and cobordisms between them

Part 1: Heegaard Floer homology is an invariant of 3-manifolds defined by Ozsvath and Szabo (2004) by counting holomorphic disks in a configuration space of points moving and colliding in them. We outline their construction and give a few examples.

Part 2: Juhasz introduced an extention of the hat Heegaard Floer homology for 3-manifolds with boundary (together with an extra structure over the boundary) called sutured manifolds. We gave a framework that generalizes Juhasz's construction and brings different flavors of Heegaard Floer homology for closed 3-manifolds, knots and links under the same roof. We will provide a new description of sutured manifolds as "Tangles" and define a notion of cobordism between them. Associated with cobordisms between tangles, we outline the definition of maps between Heegaard Floer homologies of the corresponding tangles. This construction generalizes Ozsvath and Szabo's cobordism maps for 4-dimensional cobordisms between closed 3-manifolds.