9.3 The Thom isomorphism theorem

The Thom isomorphism theorem is a key result in vector bundle theory. It connects the cohomology of a vector bundle's base space to the cohomology of its Thom space, providing a powerful tool for calculations.

This theorem has wide-ranging applications in algebraic topology. It's crucial for understanding characteristic classes, cobordism theory, and manifold topology, bridging the gap between vector bundles and cohomology computations.

Thom Isomorphism Theorem

Statement and Isomorphism

• The Thom isomorphism theorem asserts that for an oriented real vector bundle $E$ over a compact base space $B$, there exists an isomorphism between the cohomology of the base space $B$ and the cohomology of the Thom space of $E$, shifted by the rank of the bundle
• Precisely, if $E$ is an oriented real vector bundle of rank $k$ over a compact base space $B$, then there is an isomorphism: $H^i(B; R) \cong H^{i+k}(Th(E); R)$ for all $i$, where $Th(E)$ represents the Thom space of $E$ and $R$ is the coefficient ring
  • For instance, if $E$ is a rank 2 bundle over a sphere $S^2$, then $H^i(S^2; R) \cong H^{i+2}(Th(E); R)$ for all $i$
• The isomorphism is given by the cup product with the Thom class of the vector bundle $E$, which is a cohomology class in $H^k(Th(E); R)$

Significance and Applications

• The Thom isomorphism theorem serves as a powerful tool in algebraic topology, establishing a relationship between the cohomology of the base space and the cohomology of the Thom space, which often possesses a simpler structure
• It enables the computation of cohomology groups of Thom spaces by leveraging the knowledge of the cohomology groups of the base space
  • For example, if the cohomology groups of a base manifold are known, the Thom isomorphism can be employed to determine the cohomology groups of the associated Thom space
• The theorem finds applications in various areas of topology, such as characteristic classes, cobordism theory, and the study of manifolds

Thom Spaces and Classes

Thom Space Construction

• The Thom space $Th(E)$ of a vector bundle $E$ over a base space $B$ is obtained by collapsing the complement of the zero section in the total space of $E$ to a single point
• Intuitively, the Thom space is formed by adding a "point at infinity" to each fiber of the vector bundle, effectively compactifying the fibers
  • For a line bundle over a circle, the Thom space resembles a torus with a disk attached at each point of the circle
• The construction of the Thom space provides a way to study the topology of vector bundles by considering the cohomology of the associated Thom space

Thom Class and its Properties

• The Thom class is a cohomology class $u \in H^k(Th(E); R)$, where $k$ is the rank of the vector bundle $E$ and $R$ is the coefficient ring
• It is characterized by the property that its restriction to each fiber of the vector bundle is a generator of the cohomology of the fiber (which is isomorphic to $R$)
  • In other words, the Thom class evaluates to a generator of $R$ on each fiber of the bundle
• For an oriented vector bundle, the Thom class is unique and generates the cohomology of the Thom space as a free module over the cohomology of the base space
• The Thom class plays a crucial role in the statement and proof of the Thom isomorphism theorem, acting as the bridge between the cohomology of the base space and the cohomology of the Thom space

Cohomology Computations with Thom Isomorphism

Computing Cohomology of Thom Spaces

• The Thom isomorphism allows for the computation of the cohomology groups of the Thom space $Th(E)$ in terms of the cohomology groups of the base space $B$
• Given the cohomology groups of the base space $B$, one can apply the Thom isomorphism to determine the cohomology groups of the Thom space by shifting the degrees by the rank of the vector bundle
  • For instance, if $E$ is an oriented rank $k$ bundle over $B$, and the cohomology groups of $B$ are known, then $H^{i+k}(Th(E); R) \cong H^i(B; R)$ for all $i$
• This computational technique simplifies the process of determining the cohomology of Thom spaces, as it reduces the problem to understanding the cohomology of the base space

Integration with Other Tools

• In practice, the Thom isomorphism is frequently used in combination with other tools, such as the Serre spectral sequence or the Gysin sequence, to compute the cohomology of more complex spaces
• The Serre spectral sequence is a powerful tool for computing the cohomology of fiber bundles, and it can be applied in conjunction with the Thom isomorphism to study the cohomology of Thom spaces associated with vector bundles
  • For example, the Serre spectral sequence can be used to relate the cohomology of the base space, fiber, and total space of a fiber bundle, and the Thom isomorphism can then be applied to compute the cohomology of the associated Thom space
• The Gysin sequence, on the other hand, is a long exact sequence that relates the cohomology of a manifold to the cohomology of a submanifold and its normal bundle, and it can be used in tandem with the Thom isomorphism to study the cohomology of submanifolds and their neighborhoods

Thom Isomorphism vs Poincaré Duality

Poincaré Duality

• Poincaré duality is a fundamental result in algebraic topology that establishes a relationship between the homology and cohomology of a closed, oriented manifold
• For a closed, oriented manifold $M$ of dimension $n$, Poincaré duality states that there is an isomorphism between the homology and cohomology groups: $H_i(M; R) \cong H^{n-i}(M; R)$ for all $i$, where $R$ is the coefficient ring
  • For example, for a closed, oriented surface of genus $g$, Poincaré duality implies that $H_1(M; R) \cong H^1(M; R)$
• Poincaré duality provides a deep connection between the homology and cohomology of a manifold, allowing for the computation of one from the other

Relationship between Thom Isomorphism and Poincaré Duality

• The Thom isomorphism theorem can be viewed as a generalization of Poincaré duality to the setting of vector bundles
• In the case of the tangent bundle of a closed, oriented manifold $M$, the Thom space of the tangent bundle is homotopy equivalent to the suspension of $M$, and the Thom class corresponds to the fundamental class of $M$ under this equivalence
  • The suspension of a manifold $M$ is obtained by taking the product of $M$ with an interval and collapsing the ends to points
• By applying the Thom isomorphism to the tangent bundle of a closed, oriented manifold, one can recover the Poincaré duality isomorphism between the homology and cohomology of the manifold
• The Thom isomorphism and Poincaré duality are closely related and provide a powerful framework for studying the topology of manifolds and vector bundles
  • While Poincaré duality relates the homology and cohomology of a manifold, the Thom isomorphism extends this relationship to the setting of vector bundles and their associated Thom spaces