K-theory

K-theory is a mathematical framework that studies properties of spaces by examining bundles of objects, like vector spaces, attached to each point of the space. It helps classify and understand how these bundles can be combined, compared, or transformed, revealing deep structural information about the space itself. Think of it as a tool to organize and analyze the ways objects can be smoothly assigned across a shape, providing insights into geometry, topology, and even connections to physics. In essence, K-theory captures the intrinsic, global features of spaces through algebraic structures.