Studying framed Instanton Floer homology via sutures

Framed instanton Floer homology was constructed by Kronheimer and Mrowka, and has become a useful invariants for 3-manifolds. It has a closed relation with the representation variety of the fundamental group of the 3-manifolds. However, many basic structures and tools are missing for the study of framed instanton Floer homology. In this talk, we will discuss how to use sutured manifolds and sutured Instanton Floer homology to study framed instanton Floer homology. In particular, we will introduce a large surgery formula for instanton theory and present applications to computing the framed instanton Floer homology of some new families of 3-manifolds and applications to the studies of the representation of fundamental groups. This is a join work with Fan Ye.