Heegaard Floer Homology

Heegaard Floer homology has grown to be a vast subject in low-dimensional topology since its inception in the early 2000s. It is thus impossible to cover in four lectures its numerous features, variants, applications, and connections to other theories. These lecture notes are aimed at students and researchers who are not yet acquainted with the theory, and represent my attempt at providing the context of how various ideas fit in it, in a short amount of time. As a result, we will inevitably be vague at times and omit details that the expert will find important. We will also have to omit or glide over many significant contributions by members of the Floer theory community, and the reader is encouraged to fill in these gaps by consulting other references.