Building homology theories (ala Floer)

TL;DR

This paper discusses a unified framework for constructing homology theories in low-dimensional topology, symplectic, and contact geometry, highlighting how the specific context influences algebraic structures.

Contribution

It provides an overview of the common framework used since the 1980s for building homology theories across related mathematical fields.

Findings

Identifies a shared approach to homology construction since the 1980s.

Explains how the nature of the studied situation affects algebraic chain groups.

Connects homology theories in topology, symplectic, and contact geometry.

Abstract

These notes are an expanded version of evening talks at the 2025 Georgia International Topology Conference, and an abbreviated version of talks at Georgia Tech, which were aimed at graduate students. The hope was to indicate a common framework that has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry. In addition to this, we also try to indicate how the specific nature of the situation being studied dictates the algebraic nature of the chain groups used to define the homology.