Whitney polygons, symbol homology and cobordism maps

Details Whitney polygons, symbol homology and cobordism maps Whitney polygons, symbol homology and cobordism maps

2011-04-26

arxiv.org/abs/1104.4884v2

Mathematics

Geometric Topology

42 pages, 2 figures; tightened and improved introduction, improved exposition, notational inaccuracies and notational errors c

Scientific paper

We define a new homology theory we call symbol homology by using decorated moduli spaces of Whitney polygons. By decorating different types of moduli spaces we obtain different flavors of this homology theory together with morphisms between them. Each of these flavors encodes the properties of a different type of Heegaard Floer homology. The morphisms between the symbol homologies enable us to push properties from one Floer theory to a different one. Furthermore, we obtain a new presentation of Heegaard Floer theory in which maps correspond to multiplication from the right with suitable elements of our symbol homology. Finally, we present the construction of cobordism maps in knot Floer theories and apply the tools from symbol homology to give an invariance proof.

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