Stiefel–Whitney class

In mathematics, specifically in algebraic topology and in differential geometry and topology, boots -Whitney classes are a special type of characteristic classes that are mapped to real vector bundles. They are named after Eduard Stiefel and Hassler Whitney.

Basic idea and motivation

Boots -Whitney classes are characteristic classes, they are topological invariants of vector bundles over smooth manifolds, two isomorphic vector bundles have the same Stiefel- Whitney classes. The Stiefel- Whitney classes provide therefore a way to verify that two vector bundles are different on a smooth manifolds, however, can not be decided with their help that two vector bundles are isomorphic ( because non- isomorphic vector bundles have the same Stiefel- Whitney classes can ).

In the topology of the differential geometry and algebraic geometry, it is often important to determine the maximum number of linearly independent sections of a vector burst. The Stiefel- Whitney classes provide obstacles for the existence of such cuts.

In the case of the tangent bundle of a differentiable manifold, the first and second Stiefel -Whitney class is the ( single ) obstructions to orientability and the existence of a spin structure.

Axiomatic access

The Stiefel- Whitney classes are invariants of real vector bundles over a topological space. Each vector bundle over are Kohomologieklassen

For assigned, ie the i-th Stiefel -Whitney class of the vector bundle.

One can describe the Stiefel- Whitney classes by the following axioms, which they clearly define.

Axiom 1: If a differentiable map and a vector bundle is over, then for.

Axiom 2: If and are vector bundles over the same topological space, then.

Axiom 3: For each vector bundle over a path-connected space is the producer of. For every n -dimensional vector bundle is for everyone. For the " Möbius band", that is, the non-trivial 1-dimensional vector bundle over the circle is the producer of.

Boots -Whitney classes as characteristic classes

Be the Grassmann manifold and the tautological bundle. The cohomology ring of the Grassmann manifold with coefficients can be interpreted as polynomial

Pose with producers.

To a vector bundle with fiber can define a unique up to homotopy mapping that is superimposed by a bundle map in the tautological bundle over.

The i-th Stiefel -Whitney class is then given as

Slice

When an n-dimensional vector bundle has k linearly independent sections, then:

The converse is not true. For example, let the closed orientable surface of genus g, and the tangent bundle. Then disappear boots -Whitney classes, but only the torus is parallelizable, has for every vector field on a zero. ( The case is the hairy ball theorem, the general case follows from the theorem of Poincaré - Hopf. )

And orientability

Be a CW complex. One has a canonical isomorphism. Under this isomorphism the 1 -th Stiefel -Whitney class of a vector bundle corresponding to the homomorphism which maps to when the vector bundle along this closed path is the homotopy class of a closed path orientable. ( Otherwise, the homotopy class of the closed path is on mapped. Notice that there is above the circle of only two non- equivalent -dimensional vector bundles. , The homotopy class of the closed path is exactly then mapped if the withdrawn vector bundle is not trivial. )

In particular, a vector bundle is orientable if and only if.

One-dimensional vector bundle

Be a CW complex. The -dimensional vector bundle over form a group with the tensor product as a link. The 1 -th Stiefel -Whitney class is a group isomorphism

Kobordismustheorie

Set ( Pontryagin ): If a compact differentiable n -manifold, the edge is a compact differentiable manifold n 1, then for all.

Set ( Thom ): If the boot -Whitney classes are trivial for a compact differentiable n- manifold, ie for all, then the boundary of a compact differentiable n 1- manifold is.