9.3 Stiefel-Whitney classes

Stiefel-Whitney classes are key tools in algebraic topology, providing insights into real vector bundles over topological spaces. These cohomology classes measure bundle twisting and non-orientability, serving as characteristic classes analogous to Chern classes for complex vector bundles.

Defined by four axioms, Stiefel-Whitney classes capture essential topological properties of real vector bundles. They're used to study manifold orientability, spin structures, and cobordism theory, offering valuable information about vector bundles and their base spaces.

Definition of Stiefel-Whitney classes

• Stiefel-Whitney classes are cohomology classes associated to real vector bundles over a topological space, providing a way to measure the twisting and non-orientability of the bundle
• Serve as characteristic classes for real vector bundles, analogous to Chern classes for complex vector bundles, capturing important topological information about the bundle and the base space
• Denoted as $w_i(E) \in H^i(B; \mathbb{Z}/2\mathbb{Z})$, where $E$ is a real vector bundle over a base space $B$ and $i$ is the degree of the cohomology class

Vector bundles over manifolds

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• Real vector bundles are a family of real vector spaces parametrized by points in a topological space (base space), with a continuous structure map
• For a smooth manifold $M$, the tangent bundle $TM$ is a canonical example of a real vector bundle, consisting of tangent spaces at each point of the manifold
• Other examples include normal bundles, universal bundles over Grassmann manifolds, and the Möbius strip (a non-trivial line bundle over the circle)

Axioms for Stiefel-Whitney classes

• Stiefel-Whitney classes satisfy four axioms that uniquely characterize them:
  1. $w_0(E) = 1 \in H^0(B; \mathbb{Z}/2\mathbb{Z})$ for any real vector bundle $E$
  2. $w_i(E) = 0$ for $i > \text{rank}(E)$
  3. Naturality: for any continuous map $f: B' \to B$, $f^*(w_i(E)) = w_i(f^*E)$
  4. Whitney product formula: for vector bundles $E$ and $F$ over $B$, $w(E \oplus F) = w(E) \smile w(F)$, where $\smile$ denotes the cup product
• These axioms ensure that Stiefel-Whitney classes capture essential topological properties of real vector bundles and are well-behaved under natural operations

Universal Stiefel-Whitney classes

• Universal Stiefel-Whitney classes are elements in the cohomology of the classifying space $BO(n)$ for real vector bundles of rank $n$
• For any real vector bundle $E$ of rank $n$ over a space $B$, there exists a classifying map $f: B \to BO(n)$ such that $E \cong f^*\gamma^n$, where $\gamma^n$ is the universal bundle over $BO(n)$
• The Stiefel-Whitney classes of $E$ are then given by the pullback of the universal Stiefel-Whitney classes via the classifying map: $w_i(E) = f^*(w_i(\gamma^n))$

Properties of Stiefel-Whitney classes

• Stiefel-Whitney classes possess several important properties that make them useful tools in studying the topology of real vector bundles and their base spaces
• These properties allow for the computation and comparison of Stiefel-Whitney classes in various settings and provide insight into the geometric and topological features of the bundles

Naturality under pullbacks

• Stiefel-Whitney classes are natural under pullbacks: for a continuous map $f: B' \to B$ and a real vector bundle $E$ over $B$, the Stiefel-Whitney classes of the pullback bundle $f^*E$ over $B'$ are given by $w_i(f^*E) = f^*(w_i(E))$
• This property allows for the comparison of Stiefel-Whitney classes of vector bundles over different base spaces related by a continuous map

Whitney product formula

• For two real vector bundles $E$ and $F$ over the same base space $B$, the Stiefel-Whitney classes of the Whitney sum $E \oplus F$ are given by the cup product of the Stiefel-Whitney classes of $E$ and $F$: $w(E \oplus F) = w(E) \smile w(F)$
• This formula allows for the computation of Stiefel-Whitney classes of direct sums of vector bundles in terms of the Stiefel-Whitney classes of the individual bundles

Stiefel-Whitney numbers

• Stiefel-Whitney numbers are characteristic numbers obtained by evaluating cup products of Stiefel-Whitney classes on the fundamental class of a closed manifold
• For a closed $n$-dimensional manifold $M$ and a partition $I = (i_1, \ldots, i_k)$ of $n$, the Stiefel-Whitney number $\langle w_{i_1}(TM) \smile \cdots \smile w_{i_k}(TM), [M] \rangle \in \mathbb{Z}/2\mathbb{Z}$ is an invariant of the manifold
• Stiefel-Whitney numbers can be used to distinguish between non-homeomorphic manifolds and provide information about the cobordism class of the manifold

Non-vanishing and non-triviality

• The non-vanishing of certain Stiefel-Whitney classes can provide information about the non-triviality of a vector bundle and the topology of the base space
• For example, if $w_1(E) \neq 0$, then the vector bundle $E$ is not orientable, and if $w_2(TM) \neq 0$ for a manifold $M$, then $M$ does not admit a spin structure
• The top Stiefel-Whitney class $w_n(E)$ of a rank $n$ vector bundle $E$ over a connected space $B$ is zero if and only if $E$ is trivial

Computation of Stiefel-Whitney classes

• Computing Stiefel-Whitney classes directly from the definition can be challenging, but several methods and tools are available to simplify the process
• These methods often rely on the properties of Stiefel-Whitney classes, such as naturality and the Whitney product formula, and the relationship between Stiefel-Whitney classes and other cohomology operations

Classifying spaces and maps

• Stiefel-Whitney classes can be computed using the classifying space $BO(n)$ and the universal Stiefel-Whitney classes $w_i(\gamma^n)$
• For a real vector bundle $E$ of rank $n$ over a space $B$, the classifying map $f: B \to BO(n)$ induces a pullback of the universal Stiefel-Whitney classes, giving $w_i(E) = f^*(w_i(\gamma^n))$
• Computing the classifying map and the cohomology of $BO(n)$ can simplify the computation of Stiefel-Whitney classes

Characteristic classes of projective spaces

• Real projective spaces $\mathbb{RP}^n$ and their associated tautological line bundles $\gamma^1_n$ provide a rich source of examples for computing Stiefel-Whitney classes
• The total Stiefel-Whitney class of the tautological line bundle over $\mathbb{RP}^n$ is given by $w(\gamma^1_n) = 1 + a$, where $a \in H^1(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z})$ is the generator of the cohomology ring
• Using the Whitney product formula and naturality, one can compute Stiefel-Whitney classes of vector bundles over projective spaces and their subspaces

Wu's theorem and Steenrod squares

• Wu's theorem relates Stiefel-Whitney classes of the tangent bundle of a smooth manifold to the action of Steenrod squares on the cohomology of the manifold
• For a smooth compact $n$-dimensional manifold $M$, there exist unique cohomology classes $v_i \in H^i(M; \mathbb{Z}/2\mathbb{Z})$, called Wu classes, such that for any $x \in H^{n-i}(M; \mathbb{Z}/2\mathbb{Z})$, $\langle v_i \smile x, [M] \rangle = \langle Sq^i(x), [M] \rangle$, where $Sq^i$ is the $i$-th Steenrod square
• Wu's theorem states that the total Stiefel-Whitney class of the tangent bundle $TM$ is related to the total Wu class by $w(TM) = Sq(v)$, where $Sq = \sum_{i \geq 0} Sq^i$ and $v = \sum_{i \geq 0} v_i$

Applications of Stiefel-Whitney classes

• Stiefel-Whitney classes have numerous applications in topology, geometry, and algebraic topology, providing invariants and obstructions for various geometric and topological problems
• These applications demonstrate the power and utility of Stiefel-Whitney classes in studying the properties of vector bundles, manifolds, and their interplay

Obstruction to existence of linearly independent sections

• Stiefel-Whitney classes can be used to determine the maximum number of linearly independent sections of a vector bundle
• For a rank $n$ vector bundle $E$ over a CW-complex $B$, the maximum number $r$ of linearly independent sections of $E$ is related to the vanishing of the Stiefel-Whitney classes $w_i(E)$ for $i > n - r$
• In particular, if $w_n(E) \neq 0$, then $E$ does not admit a nowhere-vanishing section, providing an obstruction to the existence of a frame for the vector bundle

Characteristic numbers and cobordism

• Stiefel-Whitney numbers, obtained by evaluating products of Stiefel-Whitney classes on the fundamental class of a closed manifold, are important invariants in the study of cobordism theory
• Two closed $n$-dimensional manifolds are unoriented cobordant if and only if they have the same Stiefel-Whitney numbers for all partitions of $n$
• The Stiefel-Whitney numbers of a manifold determine its cobordism class in the unoriented cobordism ring, which is isomorphic to a polynomial ring over $\mathbb{Z}/2\mathbb{Z}$ with generators in degrees $2^i - 1$ for $i \geq 1$

Embeddings and immersions of manifolds

• Stiefel-Whitney classes provide obstructions to the existence of embeddings and immersions of manifolds into Euclidean spaces
• For a compact $n$-dimensional manifold $M$, the Whitney embedding theorem states that $M$ can be embedded in $\mathbb{R}^{2n}$, but the Stiefel-Whitney classes can obstruct embeddings into lower-dimensional Euclidean spaces
• Similarly, the Stiefel-Whitney classes of the normal bundle of an immersion can provide information about the codimension and the topology of the immersed manifold

Orientability and spin structures

• The first and second Stiefel-Whitney classes of a vector bundle or a manifold are related to the existence of orientations and spin structures
• A vector bundle $E$ is orientable if and only if its first Stiefel-Whitney class $w_1(E)$ vanishes, and a manifold $M$ is orientable if and only if $w_1(TM) = 0$
• A manifold $M$ admits a spin structure if and only if both $w_1(TM)$ and $w_2(TM)$ vanish, providing a topological obstruction to the existence of a spin structure on the manifold

Relation to other characteristic classes

• Stiefel-Whitney classes are one of several types of characteristic classes associated with vector bundles, each capturing different aspects of the topology of the bundle and the base space
• Understanding the relationships between Stiefel-Whitney classes and other characteristic classes, such as Chern classes, Pontryagin classes, and the Euler class, can provide a more comprehensive view of the topological properties of vector bundles

Chern classes vs Stiefel-Whitney classes

• Chern classes are characteristic classes associated with complex vector bundles, analogous to Stiefel-Whitney classes for real vector bundles
• For a complex vector bundle $E$, the Chern classes $c_i(E)$ live in the cohomology ring $H^{2i}(B; \mathbb{Z})$, while Stiefel-Whitney classes $w_i(E_{\mathbb{R}})$ of the underlying real vector bundle $E_{\mathbb{R}}$ live in $H^i(B; \mathbb{Z}/2\mathbb{Z})$
• The mod 2 reduction of the total Chern class of $E$ is related to the total Stiefel-Whitney class of $E_{\mathbb{R}}$ by the formula $c(E) \equiv w(E_{\mathbb{R}})^2 \pmod{2}$

Pontryagin classes and Stiefel-Whitney classes

• Pontryagin classes are characteristic classes associated with real vector bundles, living in the cohomology ring $H^{4i}(B; \mathbb{Z})$
• For a real vector bundle $E$, the Pontryagin classes $p_i(E)$ are defined as the Chern classes of the complexification $E \otimes \mathbb{C}$, and they are related to the Stiefel-Whitney classes by the formula $p_i(E) \equiv w_{2i}(E)^2 \pmod{2}$
• The vanishing of Pontryagin classes provides information about the stable parallelizability of the vector bundle and the base space

Euler class and top Stiefel-Whitney class

• The Euler class is a characteristic class associated with oriented real vector bundles, living in the top cohomology group $H^n(B; \mathbb{Z})$ for a rank $n$ bundle
• For an oriented real vector bundle $E$, the Euler class $e(E)$ is related to the top Stiefel-Whitney class $w_n(E)$ by the formula $e(E) \equiv w_n(E) \pmod{2}$
• The non-vanishing of the Euler class provides an obstruction to the existence of a nowhere-vanishing section of the vector bundle, similar to the top Stiefel-Whitney class