Boots Whitney Classes

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

Basic idea and motivation

Stiefel-Whitney classes are distinctive 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 thus provide a way of verifying that two vector bundles are different over a smooth manifold. However, it cannot be used to decide whether two vector bundles are isomorphic, since non-isomorphic vector bundles can have the same Stiefel-Whitney classes.

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

In the case of the tangential bundle of a differentiable manifold , the first and second Stiefel-Whitney classes are the (only) obstructions against orientability and the existence of a spin structure .

Axiomatic approach

The Stiefel-Whitney classes are invariants of real vector bundles over a topological space . Each vector bundle over to Kohomologieklassen${\ displaystyle X}$${\ displaystyle V}$${\ displaystyle X}$

${\ displaystyle w_ {i} (V) \ in H ^ {i} (X, \ mathbb {Z} / 2 \ mathbb {Z})}$

for assigned is called the i-th Stiefel-Whitney class of the vector bundle . ${\ displaystyle i = 0,1,2, \ ldots}$${\ displaystyle w_ {i} (V)}$${\ displaystyle V}$

The Stiefel-Whitney classes can be described by the following axioms, which clearly define them.

Axiom 1 : If a continuous map and a vector bundle is over , then for . * Stands for the return transport . ${\ displaystyle f: Y \ rightarrow X}$${\ displaystyle V}$${\ displaystyle X}$${\ displaystyle w_ {i} (f ^ {*} V) = f ^ {*} w_ {i} (V)}$${\ displaystyle i = 0,1,2, \ ldots}$

Axiom 2 : If and vector bundles are over the same topological space then is . Here is the cup product . ${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle X}$${\ displaystyle w_ {k} (V \ oplus W) = \ sum _ {i = 0} ^ {k} w_ {i} (V) \ cup w_ {ki} (W)}$${\ displaystyle \ cup}$

Axiom 3 : For every vector bundle over a path-connected space is the generator of . For every n-dimensional vector bundle is for all . For the " Möbius strip ", i.e. H. the nontrivial 1-dimensional vector bundle above the circle is the generator of . ${\ displaystyle V}$${\ displaystyle X}$${\ displaystyle w_ {0} (V)}$${\ displaystyle H ^ {0} (X, \ mathbb {Z} / 2 \ mathbb {Z}) \ cong \ mathbb {Z} / 2 \ mathbb {Z}}$${\ displaystyle V}$${\ displaystyle w_ {i} (V) = 0}$${\ displaystyle i> n}$${\ displaystyle V}$${\ displaystyle S ^ {1}}$${\ displaystyle w_ {1} (V)}$${\ displaystyle H ^ {1} (S ^ {1}, \ mathbb {Z} / 2 \ mathbb {Z}) \ cong \ mathbb {Z} / 2 \ mathbb {Z}}$

Stiefel-Whitney Classes as Characteristic Classes

Let be the Graßmann manifold and the tautological bundle. The cohomology ring of the Graßmann manifold with coefficients can be described as a polynomial ring ${\ displaystyle BG}$ ${\ displaystyle G_ {n} (\ mathbb {R} ^ {\ infty})}$${\ displaystyle \ gamma ^ {n} \ rightarrow BG}$${\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}$

${\ displaystyle H (BG; \ mathbb {Z} / 2 \ mathbb {Z}) = \ mathbb {Z} / 2 \ mathbb {Z} \ left [w_ {1}, w_ {2}, w_ {3} , \ ldots, w_ {n} \ right]}$

over represent with producers . ${\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}$${\ displaystyle w_ {i} \ in H ^ {i} (BG; \ mathbb {Z} / 2 \ mathbb {Z})}$

For a vector bundle with fibers , it is possible to define a mapping that is unique except for homotopy , which is superimposed on the tautological bundle by a bundle mapping . ${\ displaystyle \ pi \ colon E \ to X}$${\ displaystyle V \ simeq \ mathbb {R} ^ {n}}$${\ displaystyle f \ colon X \ to BG}$${\ displaystyle F \ colon E \ to \ gamma ^ {n}}$${\ displaystyle BG}$

The i- th Stiefel-Whitney class then results as

${\ displaystyle w_ {i} (E): = f ^ {*} (w_ {i}) \ in H ^ {*} (X; \ mathbb {Z} / 2 \ mathbb {Z}) \ ,.}$

Cuts

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

${\ displaystyle w_ {n-k + 1} (V) = w_ {n-k + 2} (V) = \ ldots = w_ {n} (V) = 0}$.

The reverse is not true. Let, for example, be the closed, orientable surface of genus g and its tangential bundle. Then the Stiefel-Whitney classes disappear , but only the torus can be parallelized, for each vector field has a zero. (The case is the Igel theorem , the general case follows from the Poincaré-Hopf theorem .) ${\ displaystyle S_ {g}}$${\ displaystyle TS_ {g}}$${\ displaystyle w_ {1} (TS_ {g}) = w_ {2} (TS_ {g}) = 0}$${\ displaystyle g \ not = 1}$${\ displaystyle S_ {g}}$${\ displaystyle g = 0}$

w₁ and orientability

Be a CW complex . You have a canonical isomorphism . Under this isomorphism, the 1st Stiefel-Whitney class of a vector bundle corresponds to the homomorphism which maps the homotopy class of a closed path if and only if the vector bundle can be oriented along this closed path. (Otherwise the homotopy class of the closed path is mapped to. Note that there are only two non-equivalent -dimensional vector bundles over the circle . The homotopy class of the closed path is mapped to if and only if the withdrawn vector bundle is over nontrivial.) ${\ displaystyle X}$${\ displaystyle H ^ {1} (X; \ mathbb {Z} / 2 \ mathbb {Z}) = Hom (\ pi _ {1} X, \ mathbb {Z} / 2 \ mathbb {Z})}$${\ displaystyle w_ {1} (E) \ in H ^ {1} (X; \ mathbb {Z} / 2 \ mathbb {Z})}$${\ displaystyle \ pi: E \ rightarrow X}$${\ displaystyle \ pi _ {1} X \ rightarrow \ mathbb {Z} / 2 \ mathbb {Z}}$${\ displaystyle 0}$${\ displaystyle 1}$${\ displaystyle S ^ {1}}$${\ displaystyle n}$${\ displaystyle 1}$${\ displaystyle S ^ {1}}$

In particular, a vector bundle is orientable if and only if . ${\ displaystyle \ pi: E \ rightarrow X}$${\ displaystyle w_ {1} (E) = 0 \ in H ^ {1} (X; \ mathbb {Z} / 2 \ mathbb {Z})}$

One-dimensional vector bundles

Be a CW complex . The -dimensional vector bundles form a group with the tensor product as a link. The 1st Stiefel-Whitney class gives a group isomorphism ${\ displaystyle X}$${\ displaystyle 1}$${\ displaystyle X}$${\ displaystyle Vect ^ {1} (X)}$

${\ displaystyle w_ {1} \ colon Vect ^ {1} (X) \ rightarrow H ^ {1} (X; \ mathbb {Z} / 2 \ mathbb {Z})}$.

Cobordism theory

Theorem ( Pontryagin ): If a compact differentiable n-manifold is the boundary of a compact differentiable n + 1-manifold, then is for all . ${\ displaystyle M}$${\ displaystyle w_ {i} (TM) = 0}$${\ displaystyle i \ geq 1}$

Theorem ( Thom ): If for a compact differentiable n-manifold the Stiefel-Whitney classes are trivial, i. H. for all , then is the boundary of a compact differentiable n + 1-manifold. ${\ displaystyle M}$${\ displaystyle w_ {i} (TM) = 0}$${\ displaystyle i \ geq 1}$${\ displaystyle M}$