are key tools in algebraic topology, providing insights into over topological spaces. These measure bundle twisting and non-orientability, serving as characteristic classes analogous to for complex .
Defined by four axioms, Stiefel-Whitney classes capture essential topological properties of real vector bundles. They're used to study , , and , 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 wi(E)∈Hi(B;Z/2Z), 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:
w0(E)=1∈H0(B;Z/2Z) for any real vector bundle E
wi(E)=0 for i>rank(E)
Naturality: for any continuous map f:B′→B, f∗(wi(E))=wi(f∗E)
: for vector bundles E and F over B, w(E⊕F)=w(E)⌣w(F), where ⌣ 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
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→BO(n) such that E≅f∗γn, where γn 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: wi(E)=f∗(wi(γn))
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 : for a continuous map f:B′→B and a real vector bundle E over B, the Stiefel-Whitney classes of the pullback bundle f∗E over B′ are given by wi(f∗E)=f∗(wi(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⊕F are given by the cup product of the Stiefel-Whitney classes of E and F: w(E⊕F)=w(E)⌣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
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=(i1,…,ik) of n, the Stiefel-Whitney number ⟨wi1(TM)⌣⋯⌣wik(TM),[M]⟩∈Z/2Z 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 of certain Stiefel-Whitney classes can provide information about the of a vector bundle and the topology of the base space
For example, if w1(E)=0, then the vector bundle E is not orientable, and if w2(TM)=0 for a manifold M, then M does not admit a spin structure
The top Stiefel-Whitney class wn(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 wi(γn)
For a real vector bundle E of rank n over a space B, the classifying map f:B→BO(n) induces a pullback of the universal Stiefel-Whitney classes, giving wi(E)=f∗(wi(γn))
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 RPn and their associated tautological line bundles γn1 provide a rich source of examples for computing Stiefel-Whitney classes
The of the tautological line bundle over RPn is given by w(γn1)=1+a, where a∈H1(RPn;Z/2Z) 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
relates Stiefel-Whitney classes of the tangent bundle of a smooth manifold to the action of on the cohomology of the manifold
For a smooth compact n-dimensional manifold M, there exist unique cohomology classes vi∈Hi(M;Z/2Z), called Wu classes, such that for any x∈Hn−i(M;Z/2Z), ⟨vi⌣x,[M]⟩=⟨Sqi(x),[M]⟩, where Sqi 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=∑i≥0Sqi and v=∑i≥0vi
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 wi(E) for i>n−r
In particular, if wn(E)=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 Z/2Z with generators in degrees 2i−1 for i≥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 R2n, 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 w1(E) vanishes, and a manifold M is orientable if and only if w1(TM)=0
A manifold M admits a spin structure if and only if both w1(TM) and w2(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 ci(E) live in the cohomology ring H2i(B;Z), while Stiefel-Whitney classes wi(ER) of the underlying real vector bundle ER live in Hi(B;Z/2Z)
The mod 2 reduction of the total Chern class of E is related to the total Stiefel-Whitney class of ER by the formula c(E)≡w(ER)2(mod2)
Pontryagin classes and Stiefel-Whitney classes
Pontryagin classes are characteristic classes associated with real vector bundles, living in the cohomology ring H4i(B;Z)
For a real vector bundle E, the Pontryagin classes pi(E) are defined as the Chern classes of the complexification E⊗C, and they are related to the Stiefel-Whitney classes by the formula pi(E)≡w2i(E)2(mod2)
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 Hn(B;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 wn(E) by the formula e(E)≡wn(E)(mod2)
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
Key Terms to Review (24)
Characteristic classes in physics: Characteristic classes in physics refer to a set of invariants that describe the geometry and topology of vector bundles. These classes capture essential information about the structure of the bundles, such as curvature and how they twist and turn over a base space. They play a crucial role in understanding phenomena such as gauge theory, general relativity, and other areas of theoretical physics where the geometry of fields is significant.
Chern classes: Chern classes are topological invariants associated with complex vector bundles that provide crucial information about the geometry and topology of the underlying space. They capture characteristics like curvature and the way bundles twist and turn, connecting deeply with other concepts like cohomology, characteristic classes, and various forms of K-theory.
Classifying Spaces: Classifying spaces are topological spaces that serve as a universal space for a particular type of bundle, particularly in the context of principal bundles and vector bundles. They encapsulate the properties of the associated bundles, allowing mathematicians to study them via cohomological methods and connect various concepts such as homotopy, cohomology of groups, and characteristic classes.
Cobordism theory: Cobordism theory is a branch of topology that studies the relationships between manifolds through the notion of cobordism, where two manifolds are considered equivalent if there exists a manifold whose boundary is formed by those two manifolds. This theory links various topological invariants and plays a crucial role in classifying manifolds, particularly in terms of their dimensions and structures. It connects with important concepts such as characteristic classes and helps in understanding how different mathematical objects can be related or transformed into one another.
Cohomology classes: Cohomology classes are equivalence classes of cochains that provide a way to classify and measure the topological features of a space using algebraic methods. They serve as fundamental tools in algebraic topology, allowing mathematicians to derive important invariants that can be used to distinguish between different topological spaces, especially in the study of vector bundles and characteristic classes.
James Stiefel: James Stiefel was a mathematician known for his contributions to algebraic topology, particularly in the study of Stiefel-Whitney classes. These classes are important invariants used to understand the properties of vector bundles and play a significant role in characterizing the topology of manifolds.
John Milnor: John Milnor is a prominent American mathematician known for his contributions to differential topology, particularly in the development of concepts like exotic spheres and Morse theory. His work has significantly influenced various fields such as topology, geometry, and algebraic topology, connecting foundational ideas to more advanced topics in these areas.
Manifold orientability: Manifold orientability refers to the property of a manifold that allows for a consistent choice of direction across its entire structure. When a manifold is orientable, it is possible to define a continuous, non-vanishing 'orientation' on the manifold, which is crucial for understanding its topological features and their implications, particularly when dealing with vector fields and forms.
Natural under pullbacks: Natural under pullbacks refers to a property of certain mathematical structures, particularly in cohomology, where a morphism behaves well with respect to pullbacks. This means that when you have a diagram of spaces and maps, the cohomological properties are preserved when you 'pull back' along one of the maps. This concept is essential for understanding how cohomology classes can be transformed while maintaining their underlying structure, especially in the context of Stiefel-Whitney classes and their behavior under continuous mappings.
Non-triviality: Non-triviality refers to the existence of meaningful or significant elements within a mathematical structure that cannot be reduced to simpler or more basic components. In the context of certain concepts, non-triviality indicates that there are interesting properties, classes, or examples that exhibit deeper insights, as opposed to degenerate cases which are often considered trivial. Understanding non-triviality helps in identifying key characteristics and distinctions within complex systems.
Non-vanishing: In mathematical contexts, particularly in cohomology theory, the term non-vanishing refers to the property of a cohomology class or a topological invariant that does not equal zero. This concept is crucial because it often signifies the existence of certain geometric or topological features, such as non-triviality in vector bundles or characteristic classes, which can imply the presence of significant structures within a space.
Real Projective Space: Real projective space, denoted as $$\mathbb{RP}^n$$, is a topological space that represents the set of lines through the origin in $$\mathbb{R}^{n+1}$$. This space can be thought of as the set of all possible directions in $$\mathbb{R}^{n+1}$$, with each point corresponding to a line, and is crucial for understanding many concepts in topology, including duality and characteristic classes.
Real vector bundles: Real vector bundles are mathematical structures that consist of a continuous collection of real vector spaces parameterized by a topological space. They play a crucial role in the study of differential geometry and algebraic topology, providing a framework for understanding how vector spaces can vary smoothly over a base space. These bundles help in defining various topological invariants, including Stiefel-Whitney classes, which capture information about the orientability and other properties of the underlying space.
Spin Structures: Spin structures are mathematical objects that allow for the definition of spinor fields on a manifold, which is crucial for understanding the behavior of particles with half-integer spin in quantum mechanics. These structures are closely related to the concept of orientability and have significant implications in the study of characteristic classes, particularly Stiefel-Whitney classes, which help classify different types of vector bundles over manifolds.
Steenrod squares: Steenrod squares are cohomology operations that act on the cohomology groups of topological spaces, providing a way to understand how these groups behave under certain transformations. They extend the concept of cup products in cohomology, allowing mathematicians to study the relationships between different cohomology classes and gain insights into the topology of the underlying spaces. Steenrod squares also connect to other advanced concepts, such as Wu classes and Stiefel-Whitney classes, creating a rich framework for exploring algebraic topology.
Stiefel-Whitney classes: Stiefel-Whitney classes are characteristic classes associated with real vector bundles, providing important topological invariants that help classify these bundles. They play a significant role in the study of manifold properties, particularly in relation to cohomology theories, where they reveal information about the intersection of submanifolds and the topology of vector bundles. These classes are particularly useful for understanding orientability and the existence of certain structures on manifolds.
Stiefel-Whitney numbers: Stiefel-Whitney numbers are numerical invariants derived from the Stiefel-Whitney classes, which are characteristic classes used in topology to study vector bundles. These numbers play a crucial role in understanding the properties of manifolds, particularly in relation to their orientability and the existence of non-vanishing sections. By examining these invariants, one can gain insights into the topological structure and characteristics of the underlying space.
Tangent Bundle of Spheres: The tangent bundle of spheres is a mathematical structure that encapsulates all possible tangent vectors at every point on a sphere, creating a new manifold that represents these vectors. This construction allows for the study of differentiable structures and smooth maps on spheres, linking closely to concepts like Stiefel-Whitney classes, which help classify vector bundles and provide information about their topological properties.
Topological invariants: Topological invariants are properties of a topological space that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing. These invariants help classify spaces and reveal essential features about their structure, playing a crucial role in various mathematical theories and applications.
Total stiefel-whitney class: The total Stiefel-Whitney class is a characteristic class associated with a smooth manifold that encodes information about the manifold's orientability and the properties of its tangent bundle. It serves as a tool to study the manifold's topology, particularly in relation to how vector bundles behave over it. The total Stiefel-Whitney class can be expressed as a formal sum of its individual Stiefel-Whitney classes, which are themselves derived from the manifold's properties.
Universal Stiefel-Whitney Classes: Universal Stiefel-Whitney classes are cohomology classes associated with vector bundles, representing the non-vanishing of sections of these bundles. They provide a way to classify vector bundles over any space in a universal manner, meaning that they can be used to represent the characteristic classes of all vector bundles over any manifold. These classes play a significant role in topology, particularly in the study of characteristic classes and how they relate to the geometry of manifolds.
Vector Bundles: A vector bundle is a mathematical structure that consists of a topological space called the base space, along with a vector space attached to each point of that base space. This concept is vital in understanding how vector spaces can vary smoothly over a manifold, allowing for the examination of geometrical and topological properties. The notion of vector bundles is intricately connected to various theories that assign characteristic classes, providing tools to study the geometric nature of the bundles and their implications on other mathematical structures.
Whitney Product Formula: The Whitney Product Formula is a crucial result in cohomology theory that provides a way to compute the Stiefel-Whitney classes of the Cartesian product of two vector bundles. It establishes a relationship between the Stiefel-Whitney classes of individual bundles and their product, highlighting how these classes behave under certain operations. This formula is particularly significant when analyzing the topology of manifold bundles and their associated characteristic classes.
Wu's Theorem: Wu's Theorem is a fundamental result in algebraic topology that relates the Wu classes, which are characteristic classes associated with smooth manifolds, to the Stiefel-Whitney classes, which arise from the oriented intersection theory of vector bundles. This theorem highlights a deep connection between the topology of manifolds and the algebraic structures that can be assigned to them, establishing a link between different ways of classifying topological spaces through their characteristic classes.