Stiefel-Whitney classes

Stiefel-Whitney classes are mathematical tools used to study how geometric shapes called vector bundles behave over a space. They capture information about the shape's twisting or non-orientability, like how a Möbius strip has a twist. These classes assign algebraic invariants—called cohomology classes—that help distinguish different types of bundles and identify properties like the presence of twists or holes in the structure. In essence, Stiefel-Whitney classes provide a way to quantify and classify the subtle geometric features of vector bundles in topology.