Stiefel–Whitney class
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 dimension of the vector space fiber of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist (n−i+1) 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, S^{1}×R is zero.
The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a Z/2Z-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 (Milnor 1970).
Introduction
General presentation
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
here X is the base space of the bundle E, and Z/2Z (often alternatively denoted by Z_{2}) is the commutative ring whose only elements are 0 and 1. The component of w(E) in H^{i}(X; Z/2Z) is denoted by w_{i}(E) and called the i-th Stiefel–Whitney class of E. Thus w(E) = w_{0}(E) + w_{1}(E) + w_{2}(E) + ⋅⋅⋅, where each w_{i}(E) is an element of H^{i}(X; Z/2Z).
The Stiefel–Whitney class w(E) 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 w(E) and w(F) are equal. (Here isomorphic means that there exists a vector bundle isomorphism E → F which covers the identity id_{X} : X → X.) While it is in general difficult to decide whether two real vector bundles E and F are isomorphic, the Stiefel–Whitney classes w(E) and w(F) can often be computed easily. If they are different, one knows that E and F are not isomorphic.
As an example, over the circle S^{1}, there is a line bundle (i.e. a real vector bundle of rank 1) that is not isomorphic to a trivial bundle. This line bundle L is the Möbius strip (which is a fiber bundle whose fibers can be equipped with vector space structures in such a way that it becomes a vector bundle). The cohomology group H^{1}(S^{1}; Z/2Z) has just one element other than 0. This element is the first Stiefel–Whitney class w_{1}(L) of L. Since the trivial line bundle over S^{1} has first Stiefel–Whitney class 0, it is not isomorphic to L.
However, two real vector bundles E and F which have the same Stiefel–Whitney class are not necessarily isomorphic. This happens for instance when E and F are trivial real vector bundles of different ranks over the same base space X. It can also happen when E and F have the same rank: the tangent bundle of the 2-sphere S^{2} and the trivial real vector bundle of rank 2 over S^{2} have the same Stiefel–Whitney class, but they are not isomorphic. However, if two real line bundles over X have the same Stiefel–Whitney class, then they are isomorphic.
Origins
The Stiefel–Whitney classes w_{i}(E) get their name because Eduard Stiefel and Hassler Whitney discovered them as mod-2 reductions of the obstruction classes to constructing n − i + 1 everywhere linearly independent sections of the vector bundle E restricted to the i-skeleton of X. Here n denotes the dimension of the fibre of the vector bundle F → E → X.
To be precise, provided X is a CW-complex, Whitney defined classes W_{i}(E) in the i-th cellular cohomology group of X with twisted coefficients. The coefficient system being the (i−1)-st homotopy group of the Stiefel manifold of (n−i+1) linearly independent vectors in the fibres of E. Whitney proved W_{i}(E) = 0 if and only if E, when restricted to the i-skeleton of X, has (n−i+1) linearly-independent sections.
Since π_{i−1}V_{n−i+1}(F) is either infinite-cyclic or isomorphic to Z/2Z, there is a canonical reduction of the W_{i}(E) classes to classes w_{i}(E) ∈ H^{i}(X; Z/2Z) which are the Stiefel–Whitney classes. Moreover, whenever π_{i−1}V_{n−i+1}(F) = Z/2Z, the two classes are identical. Thus, w_{1}(E) = 0 if and only if the bundle E → X is orientable.
The w_{0}(E) class contains no information, because it is equal to 1 by definition. Its creation by Whitney was an act of creative notation, allowing the Whitney sum Formula w(E_{1} ⊕ E_{2}) = w(E_{1})w(E_{2}) to be true. However, for generalizations of manifolds (namely certain homology manifolds), one can have w_{0}(M) ≠ 1: it only needs to equal 1 mod 8.
Definitions
Throughout, H^{i}(X; G) denotes singular cohomology of a space X with coefficients in the group G. The word map means always a continuous function between topological spaces.
Axiomatic definition
The Stiefel-Whitney characteristic class of a finite rank real vector bundle E on a paracompact base space X is defined as the unique class such that the following axioms are fulfilled:
- Normalization: The Whitney class of the tautological line bundle over the real projective space P^{1}(R) is nontrivial, i.e. .
- Rank: w_{0}(E) = 1 ∈ H^{0}(X), and for i above the rank of E, , that is, .
- Whitney product formula: , that is, the Whitney class of a direct sum is the cup product of the summands' classes.
- Naturality: w(f*E) = f*w(E) for any real vector bundle E → X and map , where f*E denotes the pullback vector bundle.
The uniqueness of these classes is proved for example, in section 17.2 – 17.6 in Husemoller or section 8 in Milnor and Stasheff. There are several proofs of the existence, coming from various constructions, with several different flavours, their coherence is ensured by the unicity statement.
Definition via infinite Grassmannians
The infinite Grassmannians and vector bundles
This section describes a construction using the notion of classifying space.
For any vector space V, let Gr_{n}(V) denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian
- .
Recall that it is equipped with the tautological bundle , a rank n vector bundle that can be defined as the subbundle of the trivial bundle of fiber V whose fiber at a point is the subspace represented by Ẃ.
Let f : X → Gr_{n}, be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map f on X
depends only on the homotopy class of the map [f]. The pullback operation thus gives a morphism from the set
of maps X → Gr_{n} modulo homotopy equivalence, to the set
of isomorphism classes of vector bundles of rank n over X.
The important fact in this construction is that if X is a paracompact space, this map is a bijection. This is the reason why we call infinite Grassmannians the classifying spaces of vector bundles.
The case of line bundles
We now restrict the above construction to line bundles, ie we consider the space, Vect_{1}(X) of line bundles over X. The Grassmannian of lines Gr_{1} is just the infinite projective space
- ,
which is doubly covered by the infinite sphere S^{∞} by antipody. This sphere S^{∞} is contractible, so we have
Hence P^{∞}(R) is the Eilenberg-Maclane space K(Z/2Z, 1).
It is a property of Eilenberg-Maclane spaces, that
for any X, with the isomorphism given by f → f*η, where η is the generator
- .
Applying the former remark that α : [X, Gr_{1}] → Vect_{1}(X) is also a bijection, we obtain a bijection
- w_{1} : Vect_{1}(X) → H^{1}(X; Z/2Z);
this defines the Stiefel–Whitney class w_{1} for line bundles.
The group of line bundles
If Vect_{1}(X) is considered as a group under the operation of tensor product, then the Stiefel–Whitney class is an isomorphism: w_{1} : Vect_{1}(X) → H^{1}(X; Z/2Z). That is, w_{1}(λ ⊗ μ) = w_{1}(λ) + w_{1}(μ) for all line bundles λ, μ → X.
For example, since H^{1}(S^{1}; Z/2Z) = Z/2Z, 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).
The same construction for complex vector bundles shows that the Chern class defines a bijection between complex line bundles over X and H^{2}(X; Z), because the corresponding classifying space is P^{∞}(C), a K(Z, 2). This isomorphism is true for topological line bundles, the obstruction to injectivity of the Chern class for algebraic vector bundles is the Jacobian variety.
Properties
Topological interpretation of vanishing
- w_{i}(E) = 0 whenever i > rank(E).
- If E^{k} has sections which are everywhere linearly independent then the top degree Whitney classes vanish: .
- 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 w_{1}(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 H^{2}(M, Z) → H^{2}(M, Z/2Z) (equivalently, the so-called third integral Stiefel–Whitney class is zero) if and only if the bundle admits a spin^{c} structure.
- All the Stiefel–Whitney numbers (see below) of a smooth compact manifold X vanish if and only if the manifold is the boundary of some smooth compact (unoriented) manifold (Warning: Some Stiefel-Whitney class could still be non-zero, even if all the Stiefel Whitney numbers vanish!)
Uniqueness of the Stiefel–Whitney classes
The bijection above for line bundles implies that any functor θ satisfying the four axioms above is equal to w, by the following argument. The second axiom yields θ(γ^{1}) = 1 + θ_{1}(γ^{1}). For the inclusion map i : P^{1}(R) → P^{∞}(R), the pullback bundle i*γ^{1} is equal to . Thus the first and third axiom imply . Since the map i*: H^{1}(P^{∞}(R); Z/2Z) → H^{1}(P^{1}(R); Z/2Z) is an isomorphism, and θ(γ^{1}) = w(γ^{1}) follow. Let E be a real vector bundle of rank n over a space X. Then E admits a splitting map, i.e. a map f : X′ → X for some space X′ such that is injective and for some line bundles . Any line bundle over X is of the form g*γ^{1} for some map g, and θ(g*γ^{1}) = g*θ(γ^{1}) = g*w(γ^{1}) = w(g*γ_{1}) by naturality. Thus θ = w on . 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.
Non-isomorphic bundles with the same Stiefel–Whitney classes
Although the map w_{1} : Vect_{1}(X) → H^{1}(X; Z/2Z) is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle TS^{n} for n even. With the canonical embedding of S^{n} in R^{n+1}, the normal bundle ν to S^{n} is a line bundle. Since S^{n} is orientable, ν is trivial. The sum TS^{n} ⊕ ν is just the restriction of TR^{n+1} to S^{n}, which is trivial since R^{n+1} is contractible. Hence w(TS^{n}) = w(TS^{n})w(ν) = w(TS^{n} ⊕ ν) = 1. But TS^{n} → S^{n} is not trivial; its Euler class , where [S^{n}] denotes a fundamental class of S^{n} and χ the Euler characteristic.
Related invariants
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 Z/2Z-fundamental class of the manifold to give an element of Z/2Z, 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. They are known to be cobordism invariants. It was proven by Lev Pontryagin that if B is a smooth compact (n+1)–dimensional manifold with boundary equal to M, then the Stiefel-Whitney numbers of M are all zero.^{[1]} Moreover, it was proved by René Thom that if all the Stiefel-Whitney numbers of M are zero then M can be realised as the boundary of some smooth compact manifold.^{[2]}
One Stiefel–Whitney number of importance in surgery theory is the de Rham invariant of a (4k+1)-dimensional manifold,
Wu classes
The Stiefel–Whitney classes w_{k} are the Steenrod squares of the Wu classes v_{k}, defined by Wu Wenjun in (Wu 1955). Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: Sq(v) = w. Wu classes are most often defined implicitly in terms of Steenrod squares, as the cohomology class representing the Steenrod squares. Let the manifold X be n dimensional. Then, for any cohomology class x of degree n-k, . Or more narrowly, we can demand , again for cohomology classes x of degree n-k.^{[3]}
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, Z → Z/2Z:
For instance, the third integral Stiefel–Whitney class is the obstruction to a Spin^{c} structure.
Relations over the Steenrod algebra
Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes of its tangent bundle) are generated by those of the form . In particular, the Stiefel–Whitney classes satisfy the Wu formula, named for Wu Wenjun:^{[4]}
See also
- Characteristic class for a general survey, in particular Chern class, the direct analogue for complex vector bundles
- Real projective space
References
- ↑ Pontrjagin, L. S. (1947). "Characteristic cycles on differentiable manifolds". Math. Sbornik N. S. (in Russian). 21 (63): 233–284.
- ↑ Milnor, J. W.; Stasheff, J. D. (1974). Characteristic Classes. Princeton University Press. pp. 50–53. ISBN 0-691-08122-0.
- ↑ Milnor, J. W.; Stasheff, J. D. (1974). Characteristic Classes. Princeton University Press. pp. 131–133. ISBN 0-691-08122-0.
- ↑ (May 1999, p. 197)
- D. Husemoller, Fibre Bundles, Springer-Verlag, 1994.
- May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2009-08-07
- Milnor, John Willard (1970), With an appendix by J. Tate, "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501
External links
- Wu class at the Manifold Atlas