The Thom isomorphism theorem is a powerful tool in algebraic topology. It links the cohomology of a 's Thom space to the cohomology of its base space, shifted by the bundle's rank.
This theorem has wide-ranging applications, from computing cohomology of various spaces to establishing connections between vector bundle topology and structure. It's a key player in cobordism theory and characteristic classes.
The Thom Isomorphism Theorem
Statement and Significance
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The Thom isomorphism theorem establishes an isomorphism between the cohomology of the Thom space of a real vector bundle E over a compact base space B and the cohomology of B, shifted by the rank of E
The isomorphism is given by the cup product with the of E, a distinguished cohomology class in the cohomology of the Thom space
The theorem provides a powerful tool for computing the cohomology of various spaces (complement of a submanifold, manifold with boundary)
It establishes a deep connection between the topology of vector bundles and the algebraic structure of cohomology rings
The Thom isomorphism generalizes the in cohomology, which relates the cohomology of a space to the cohomology of its suspension (sphere bundle over a space)
Applications and Generalizations
The Thom isomorphism can be applied iteratively for vector bundles constructed as pullbacks or Whitney sums of other vector bundles
When the base space B is a CW-complex, the cellular structure of B can be used to compute its cohomology, and then the Thom isomorphism determines the cohomology of the Thom space
In some cases, the Thom isomorphism can be used with the Serre to compute the cohomology of Thom spaces for vector bundles over more general base spaces
The Thom isomorphism has been generalized to various settings (complex vector bundles, equivariant cohomology, generalized cohomology theories)
It plays a crucial role in the development of cobordism theory and the study of characteristic classes of vector bundles
Thom Class of a Vector Bundle
Construction and Properties
The Thom class uE of a rank-n real vector bundle E over a base space B is a cohomology class in Hn(Th(E);Z), where Th(E) is the Thom space of E
It is constructed using the orientation of the vector bundle and the Thom space's natural structure as a pointed space (collapsing the complement of the unit disk bundle)
The Thom class is natural with respect to bundle maps: if f:E→F is a vector bundle map covering g:B→C, then f∗(uF)=uE, where f∗ is the induced map on cohomology
The restriction of the Thom class to the fiber over a point b∈B is a generator of the cohomology of the fiber, isomorphic to Hn(Rn,Rn−{0};Z) (orientation class)
The Thom class is functorial, meaning it is compatible with the pullback operation on vector bundles (naturality with respect to base change)
Uniqueness and Orientation
The Thom class is unique up to sign, depending on the choice of orientation for the vector bundle
For an oriented vector bundle, the Thom class is the unique class in Hn(Th(E);Z) that restricts to the orientation class on each fiber
Changing the orientation of the vector bundle results in the negative of the original Thom class
The existence of a Thom class is equivalent to the orientability of the vector bundle
Non-orientable vector bundles (Möbius strip) do not admit a Thom class in integer coefficients, but may have a Thom class with twisted coefficients
Cohomology of Thom Spaces
Thom Isomorphism and Computations
The Thom isomorphism states that for a rank-n real vector bundle E over a base space B, there is an isomorphism ϕ:Hi(B;R)→Hi+n(Th(E);R) given by ϕ(x)=π∗(x)∪uE, where π:Th(E)→B is the projection map and uE is the Thom class
To compute the cohomology of a Thom space Th(E), one can use the Thom isomorphism to relate it to the cohomology of the base space B, which may be easier to compute (known cohomology of Grassmannians, projective spaces)
The Thom isomorphism is an H∗(B)-module isomorphism, where the module structure on H∗(Th(E)) is given by the cup product with the pullback of classes from B
For a trivial vector bundle E=B×Rn, the Thom space is homotopy equivalent to the n-fold suspension of B, and the Thom isomorphism recovers the suspension isomorphism in cohomology
The Thom isomorphism can be used to compute the cohomology of vector bundles over spheres (tangent bundle of spheres), yielding important examples in homotopy theory (stunted projective spaces)
Gysin Sequence and Euler Class
The Gysin sequence is a long exact sequence in cohomology associated with a sphere bundle or a vector bundle, relating the cohomology of the base space, total space, and Thom space
For an oriented rank-n vector bundle E over a base space B, the Gysin sequence takes the form:
⋯→Hi(B)∪e(E)Hi+n(B)π∗Hi+n(E)i∗Hi+n(B)→⋯
where e(E) is the Euler class, π∗ is the pullback map, and i∗ is the pushforward map
The Thom isomorphism can be used to derive the Gysin sequence by considering the long exact sequence of the pair (Th(E),B) and identifying the connecting homomorphism with the cup product with the Euler class
The Gysin sequence provides a powerful tool for computing the cohomology of the total space of a vector bundle, given the cohomology of the base space and the Euler class
In the case of the tangent bundle of a compact, connected, oriented manifold M, the Gysin sequence relates the cohomology of M to the cohomology of its unit tangent bundle and the Euler characteristic of M
Thom Isomorphism vs Euler Class
Relationship between Thom Class and Euler Class
The Euler class e(E) of an oriented real vector bundle E over a base space B is a characteristic class in Hn(B;Z), where n is the rank of E
The Euler class can be defined as the pullback of the Thom class uE along the zero section s:B→E, i.e., e(E)=s∗(uE)
The Thom isomorphism theorem implies that the Euler class is the unique class in Hn(B;Z) that maps to the Thom class under the Thom isomorphism
The non-vanishing of the Euler class is an obstruction to the existence of a nowhere-zero section of the vector bundle (hairy ball theorem for even-dimensional spheres)
For a compact, connected, oriented manifold M, the Euler characteristic χ(M) can be computed as the evaluation of the Euler class of the tangent bundle on the fundamental class [M], i.e., χ(M)=⟨e(TM),[M]⟩
Obstruction Theory and Applications
The Euler class is an obstruction to the existence of a nowhere-zero section of a vector bundle, and its vanishing is necessary for the vector bundle to admit a non-vanishing section
The Euler class can be used to study the existence of non-vanishing vector fields on manifolds (Poincaré-Hopf theorem) and the index of a vector field with isolated zeros
The Thom isomorphism and the Euler class provide a link between the global topology of a vector bundle and the local geometry of its base space
In obstruction theory, the Euler class is the primary obstruction to the existence of a section of a spherical fibration, and the Thom class is related to the higher obstructions
The Thom isomorphism and the Euler class have applications in various areas of topology, including characteristic classes, , and cobordism theory (Pontryagin-Thom construction)
Key Terms to Review (14)
Atiyah-Hirzebruch Spectral Sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a way to compute the K-theory of a space from its cohomology. It connects the geometry of vector bundles to topological invariants, allowing for the classification of vector bundles through the lens of K-theory and characteristic classes.
Cohomology ring: The cohomology ring is an algebraic structure that encodes the topological properties of a space through cohomology groups, allowing us to perform operations like addition and multiplication on these groups. It consists of the direct sum of the cohomology groups with a product defined by the cup product, making it a key tool for understanding how spaces can be constructed and related. This structure is crucial in applications such as the Thom isomorphism theorem, where it helps relate the cohomology of a manifold to its associated bundle.
Homotopy equivalence: Homotopy equivalence is a concept in topology that describes a relationship between two topological spaces, where they can be continuously deformed into each other through a series of transformations. This relationship is crucial for understanding the fundamental properties of spaces, as it implies that the spaces share the same homotopy type and thus the same topological invariants, which is significant in various areas such as K-Theory and bordism theory.
Index Theory: Index theory is a branch of mathematics that studies the relationship between the analytical properties of differential operators and topological invariants of manifolds. It provides a powerful tool for understanding how various geometric and topological aspects influence the behavior of solutions to differential equations, linking analysis, topology, and geometry.
K-Theory: K-Theory is a branch of mathematics that studies vector bundles and their generalizations through the construction of K-groups, which provide a way to classify and understand vector bundles up to isomorphism. It connects various areas of mathematics, including topology, algebra, and geometry, offering insights into fixed point theorems, quantum field theory, and even string theory.
M. Atiyah: M. Atiyah, short for Michael Atiyah, is a prominent mathematician known for his contributions to geometry and topology, particularly in the realm of K-Theory. His work laid the foundation for significant developments, including the Thom isomorphism theorem, which connects the K-Theory of vector bundles to their intersection theory, revealing deep relationships between algebraic and geometric properties.
R. Thom: R. Thom, or René Thom, was a French mathematician known for his contributions to topology and the development of catastrophe theory, which analyzes how small changes in circumstances can lead to sudden shifts in behavior. His work laid foundational concepts that connect algebraic topology with differential geometry, providing deep insights into the structure of manifolds and their applications in various fields.
Spectral Sequence: A spectral sequence is a mathematical tool used in algebraic topology and homological algebra that provides a method for computing homology or cohomology groups through a series of steps involving filtrations and differentials. This powerful technique helps bridge complex structures and allows mathematicians to derive results about topological spaces and algebraic objects systematically.
Stable homotopy: Stable homotopy refers to a concept in algebraic topology that studies the properties of spaces and maps that remain invariant under stabilization, typically by adding a dimension. This idea connects to various important results and theories, such as the Thom isomorphism theorem, Bott periodicity, and the relationships between K-theory, bordism, and cobordism theory. It plays a crucial role in understanding algebraic K-theory and its applications to schemes and varieties.
Suspension isomorphism: Suspension isomorphism is a fundamental concept in K-theory that states that for any space X, the suspension of X, denoted as 'SX', is isomorphic to the reduced K-theory of X, which connects the topological properties of X to algebraic structures in K-theory. This idea establishes a crucial relationship between the suspension operation and the properties of spaces within reduced K-theory, offering deep insights into the structure of vector bundles and cohomology theories.
Thom class: The Thom class is an important concept in K-theory that arises from the study of vector bundles and their relations to stable homotopy theory. It serves as a tool for understanding the topological properties of manifolds through the lens of stable bundles, linking various algebraic and geometric structures. The Thom class is particularly crucial in establishing the Thom isomorphism theorem, which connects cohomology theories with vector bundles.
Topological invariants: Topological invariants are properties of a topological space that remain unchanged under homeomorphisms, which are continuous functions that have continuous inverses. These invariants provide crucial information about the space's structure and are essential in classifying spaces and understanding their relationships. They play a pivotal role in various mathematical contexts, helping to establish connections between seemingly disparate areas of study.
Topological Space: A topological space is a set of points equipped with a structure that allows for the definition of concepts such as convergence, continuity, and compactness. This structure is given by a collection of open sets that satisfies specific axioms, enabling various mathematical explorations and connections to other areas like algebraic topology and K-Theory.
Vector Bundle: A vector bundle is a mathematical structure that consists of a base space and a collection of vector spaces associated with each point in the base space. This concept allows for the study of smooth manifolds and serves as a fundamental tool in various areas of mathematics, connecting topology, geometry, and algebra through concepts like classification and characteristic classes.