Cohomology theories

Cohomology theories are mathematical tools used in topology to study the properties of shapes and spaces. They help classify and understand the structures of these spaces by assigning them algebraic objects, like groups or rings. Essentially, cohomology captures information about a space's features, such as holes or twists, and their relationships. This allows mathematicians to determine if two shapes are similar or to understand more complex spaces in a systematic way. It has applications in various fields, including geometry, physics, and data analysis.