Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Stiefel–Whitney class
Summary
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to n, where n is the rank of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist everywhere linearly independent sections of the vector bundle. A nonzero nth Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, , is zero.
The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a -characteristic class associated to real vector bundles.
In algebraic geometry one can also define analogous Stiefel–Whitney classes for vector bundles with a non-degenerate quadratic form, taking values in etale cohomology groups or in Milnor K-theory. As a special case one can define Stiefel–Whitney classes for quadratic forms over fields, the first two cases being the discriminant and the Hasse–Witt invariant .
For a real vector bundle E, the Stiefel–Whitney class of E is denoted by w(E). It is an element of the cohomology ring
where X is the base space of the bundle E, and (often alternatively denoted by ) is the commutative ring whose only elements are 0 and 1. The component of in is denoted by and called the i-th Stiefel–Whitney class of E. Thus,
where each is an element of .
The Stiefel–Whitney class is an invariant of the real vector bundle E; i.e., when F is another real vector bundle which has the same base space X as E, and if F is isomorphic to E, then the Stiefel–Whitney classes and are equal.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.