The Arf invariant is a mathematical concept used in the field of topology, particularly in the study of homology and cohomology of manifolds. It provides a way to classify certain types of surfaces, specifically those that are non-orientable, by assigning a binary value (0 or 1) based on their properties. This invariant helps in understanding the structure and characteristics of these surfaces. In more technical terms, the Arf invariant is associated with quadratic forms over the field of finite fields. It plays a crucial role in distinguishing between different types of knot theory and linking numbers, offering insights into the relationships between various topological spaces.