Stiefel–Whitney class
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In mathematics, the Stiefel–Whitney class arises as a type of characteristic class associated to real vector bundles . It is denoted by w(E), taking values in , the cohomology groups with mod 2 coefficients. The component of w(E) in is denoted by wi(E) and called the ith Stiefel-Whitney class of E, so that . As an example, over the circle, S1, there is a line bundle that is topologically non-trivial: that is, the line bundle associated to the Möbius band, usually thought of as having fibres [0,1]. The cohomology group
has just one element other than 0, this element being the first Stiefel-Whitney class, w1, of that line bundle.
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[edit] Axioms
Throughout, Hi(X;G) denotes singular cohomology of a space X with coefficients in the group G.
- Naturality: w(f * E) = f * w(E) for any bundle and map f:X' − > X
- w0(E) = 1 in .
- w1(γ1) is the generator of (normalization condition). Here, γn is the canonical line bundle.
- (Whitney product formula).
Some work is required to show that such classes do indeed exist and are unique (at least for paracompact spaces X); see section 3.5 and 3.6 in Husemoller or section 8 in Milnor and Stasheff.
[edit] Line bundles
Let X be a paracompact space, and let Vectn(X) denote the set of real vector bundles over X of dimension n for some fixed positive integer n. For any vector space V, let Grn(V) denote the Grassmannian . Set . Define the tautological bundle by ; this is a real bundle of dimension n, with projection given by . For any map , the induced bundle . Since any two homotopic maps have f * γn and g * γn isomorphic, the map given by is well-defined, where [X;Grn] denotes the set of homotopy equivalence classes of maps . It's not difficult to prove that this map α is actually an isomorphism (see Sections 3.5 and 3.6 in Husemoller, for example). As a result, Grn is called the classifying space of real n-bundles.
Now consider the space Vect1(X) of line bundles over X. For n = 1, the Grassmannian Gr1 is just , where the nonzero element of acts by . The quotient map is therefore a double cover. Since is contractible, we have for i > 1 and ; that is, . Hence is the Eilenberg-Maclane space . Hence for any X, with the isomorphism given by , where η is the generator . Since is also a bijection, we have another bijection . This map w1 is precisely the Stiefel-Whitney class w1 for a line bundle. (Since the corresponding classifying space for complex bundles is a , the same argument shows that the Chern class defines a bijection between complex line bundles over X and .) For example, since , there are only two line bundles over the circle up to bundle isomorphism: the trivial one, and the open Möbius strip (i.e., the Möbius strip with its boundary deleted). If Vect1(X) is considered as a group under the operation of tensor product, then α is an isomorphism: for all line bundles .
[edit] Higher dimensions
The bijection above for line bundles implies that any functor θ satisfying the four axioms above is equal to w. Let be an n-bundle. Then ξ admits a splitting map, a map for some space X' such that is injective and for some line bundles . Any line bundle over X is of the form g * γ1 for some map g, and θ(g * γ1) = g * θ(γ1) = 1 + w1(g * γ1) by naturality. Thus θ = w on Vect1(X). It follows from the fourth axiom above that
Since f * is injective, θ = w Thus the Stiefel-Whitney class is the unique functor satisfying the four axioms above.
Although the map is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle TSn for n even. With the canonical embedding of Sn in , the normal bundle ν to Sn is a line bundle. Since Sn is orientable, ν is trivial. The sum is just the restriction of to Sn, which is trivial since is contractible. Hence . But is not trivial; its Euler class , where [Sn] denotes a fundamental class of Sn and χ the Euler characteristic.
[edit] Stiefel–Whitney numbers
If we work on a manifold of dimension n, then any product of Stiefel-Whitney classes of total degree n can be paired with the -fundamental class of the manifold to give an element of , a Stiefel-Whitney number of the vector bundle. For example, if the manifold has dimension 3, there are three linearly independent Stiefel-Whitney numbers, given by . In general, if the manifold has dimension n, the number of possible independent Stiefel-Whitney numbers is the number of partitions of n.
The Stiefel-Whitney numbers of the tangent bundle of a smooth manifold are called the Stiefel-Whitney numbers of the manifold, and are important invariants.
[edit] Properties
- If Ek has sections which are everywhere linearly independent then .
- wi(E) = 0 whenever i > rank(E).
- The first Stiefel-Whitney class is zero if and only if the bundle is orientable. In particular, a manifold M is orientable if and only if w1(TM) = 0.
- The bundle admits a spin structure if and only if both the first and second Stiefel-Whitney classes are zero.
- For an orientable bundle, the second Stiefel-Whitney class is in the image of the natural map (equivalently, the so-called third integral Stiefel-Whitney class is zero) if and only if the bundle admits a spinc structure.
- All the Stiefel-Whitney numbers of a smooth compact manifold X vanish if and only if the manifold is a boundary (unoriented) of a smooth compact manifold.
[edit] Integral Stiefel-Whitney classes
The element is called the i + 1 integral Stiefel-Whitney class, where β is the Bockstein homomorphism, corresponding to reduction modulo 2, :
For instance, the third integral Stiefel-Whitney class is the obstruction to a Spinc structure.
[edit] References
- D. Husemoller, Fibre Bundles, Springer-Verlag, 1994.
- J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.