TL;DR
This paper develops a graph TQFT framework for hat Heegaard Floer homology, enabling the study of cobordisms with graphs and disconnected ends, and applies it to compute fundamental group actions.
Contribution
It introduces a new graph TQFT construction for hat Heegaard Floer homology using sutured Floer TQFT, extending previous theories to disconnected cobordisms.
Findings
Constructed maps for elementary graph cobordisms
Enabled computation of fundamental group actions
Extended TQFT to disconnected cobordisms
Abstract
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juh\'{a}sz's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental group on hat Heegaard Floer homology.
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