🪡K-Theory Unit 6 – Thom Isomorphism: Theory and Applications

Definition and Basics

• Thom Isomorphism establishes a connection between the cohomology of a vector bundle and the cohomology of its base space
• Relates the cohomology of a space $X$ to the cohomology of the Thom space of a vector bundle over $X$
• Given a vector bundle $\xi$ over a space $X$, the Thom space $T(\xi)$ is the one-point compactification of the total space of $\xi$
• The Thom class $u_\xi \in H^n(T(\xi); R)$ is a unique cohomology class that restricts to the orientation class of each fiber
• The Thom Isomorphism states that the map $\phi: H^*(X; R) \to H^{*+n}(T(\xi); R)$ given by $\phi(a) = \pi^*(a) \cup u_\xi$ is an isomorphism
  • $\pi: T(\xi) \to X$ is the projection map
  • $R$ is a ring and $n$ is the rank of the vector bundle $\xi$
• Provides a powerful tool for computing cohomology groups of vector bundles and their associated spaces

Historical Context

• The Thom Isomorphism was introduced by René Thom in the 1950s
• Thom's work was influenced by earlier results in algebraic topology, such as the Gysin sequence and the Pontryagin-Thom construction
• The Thom Isomorphism played a crucial role in the development of cobordism theory, which studies manifolds up to cobordism equivalence
• Thom's ideas were further generalized by Atiyah and Hirzebruch in the context of K-theory, leading to the Atiyah-Hirzebruch spectral sequence
• The Thom Isomorphism has since found numerous applications in various areas of mathematics, including algebraic topology, differential geometry, and mathematical physics
• Thom's work on the Thom Isomorphism and cobordism theory earned him the Fields Medal in 1958, recognizing the significance of his contributions

Key Concepts and Components

• Vector bundles: A vector bundle $\xi$ over a space $X$ consists of a total space $E(\xi)$, a projection map $\pi: E(\xi) \to X$, and a vector space structure on each fiber $\pi^{-1}(x)$
• Thom space: The Thom space $T(\xi)$ of a vector bundle $\xi$ is obtained by taking the one-point compactification of the total space $E(\xi)$
  • Intuitively, it is constructed by adding a point at infinity to each fiber of the vector bundle
• Thom class: The Thom class $u_\xi \in H^n(T(\xi); R)$ is a unique cohomology class that restricts to the orientation class of each fiber
  • It serves as a fundamental ingredient in the construction of the Thom Isomorphism
• Orientation: A vector bundle $\xi$ is orientable if there exists a consistent choice of orientation for each fiber
  • Orientability is a necessary condition for the existence of the Thom class and the Thom Isomorphism
• Cohomology: Cohomology groups $H^*(X; R)$ are algebraic invariants associated with a space $X$ and a coefficient ring $R$
  • The Thom Isomorphism relates the cohomology of the base space $X$ to the cohomology of the Thom space $T(\xi)$
• Cup product: The cup product $\cup$ is a binary operation on cohomology classes that allows for the multiplication of cohomology classes
  • It is used in the construction of the Thom Isomorphism map $\phi(a) = \pi^*(a) \cup u_\xi$

Proof Outline

• The proof of the Thom Isomorphism typically involves several key steps:
  1. Construct the Thom class $u_\xi \in H^n(T(\xi); R)$ using the orientation of the vector bundle $\xi$
  2. Define the Thom Isomorphism map $\phi: H^*(X; R) \to H^{*+n}(T(\xi); R)$ by $\phi(a) = \pi^*(a) \cup u_\xi$
  3. Show that $\phi$ is a homomorphism of cohomology groups
  4. Prove that $\phi$ is an isomorphism by constructing an inverse map or using other algebraic techniques
• The construction of the Thom class relies on the orientability of the vector bundle and the existence of a consistent choice of orientation for each fiber
• The proof often makes use of the Mayer-Vietoris sequence, a long exact sequence relating the cohomology of a space to the cohomology of its subspaces
• The naturality of the Thom Isomorphism with respect to bundle maps and pullbacks is also an important aspect of the proof
• Various algebraic and topological techniques, such as spectral sequences and the Leray-Hirsch theorem, may be employed in the proof depending on the specific context and generalization of the Thom Isomorphism being considered

Applications in K-Theory

• The Thom Isomorphism has significant applications in K-theory, a branch of algebraic topology that studies vector bundles and their generalizations
• In K-theory, the Thom Isomorphism relates the K-theory of a space $X$ to the K-theory of the Thom space of a vector bundle over $X$
• The Atiyah-Hirzebruch spectral sequence, which computes the K-theory of a space using its cohomology, relies on the Thom Isomorphism as a key ingredient
• The Thom Isomorphism allows for the computation of the K-theory of various spaces, such as projective spaces and Grassmannians, by reducing the problem to the computation of cohomology
• In the study of characteristic classes, the Thom Isomorphism provides a means to define and study Thom classes in K-theory, which generalize the notion of Chern classes
• The Thom Isomorphism also plays a role in the construction of the Bott periodicity theorem, a fundamental result in K-theory that establishes a periodic relationship between the K-theory of spaces
• The Thom Isomorphism has been generalized to various settings in K-theory, such as equivariant K-theory and twisted K-theory, providing a powerful tool for studying vector bundles and their invariants in these contexts

Examples and Illustrations

• Consider the tautological line bundle $\gamma^1$ over the projective space $\mathbb{CP}^n$
  • The Thom space of $\gamma^1$ is homotopy equivalent to $\mathbb{CP}^{n+1}$
  • The Thom Isomorphism implies that $H^*(\mathbb{CP}^n; \mathbb{Z}) \cong H^{*+2}(\mathbb{CP}^{n+1}; \mathbb{Z})$
• Let $\xi$ be the Möbius bundle over the circle $S^1$
  • The Thom space of $\xi$ is homeomorphic to the Klein bottle
  • The Thom Isomorphism relates the cohomology of $S^1$ to the cohomology of the Klein bottle
• Consider the tangent bundle $T\mathbb{RP}^n$ of the real projective space $\mathbb{RP}^n$
  • The Thom space of $T\mathbb{RP}^n$ is homotopy equivalent to the stunted projective space $\mathbb{RP}^{2n}/\mathbb{RP}^{n-1}$
  • The Thom Isomorphism provides a means to compute the cohomology of the stunted projective space using the cohomology of $\mathbb{RP}^n$
• In the context of K-theory, the Thom Isomorphism can be used to compute the K-theory of the complex projective space $\mathbb{CP}^n$
  • The Thom Isomorphism relates the K-theory of $\mathbb{CP}^n$ to the K-theory of the Thom space of the tautological line bundle over $\mathbb{CP}^n$
  • This allows for the computation of the K-theory groups of $\mathbb{CP}^n$ using the Atiyah-Hirzebruch spectral sequence

Related Theorems

• The Gysin sequence is a long exact sequence that relates the cohomology of a space to the cohomology of the base space and the fiber of a sphere bundle
  • It can be seen as a precursor to the Thom Isomorphism and is closely related to the Euler class of a vector bundle
• The Pontryagin-Thom construction establishes a correspondence between the cobordism groups of manifolds and the homotopy groups of certain Thom spaces
  • It provides a geometric interpretation of the Thom Isomorphism and highlights its connection to cobordism theory
• The Atiyah-Hirzebruch spectral sequence is a spectral sequence that computes the generalized cohomology theory of a space, such as K-theory, using its ordinary cohomology
  • The Thom Isomorphism plays a crucial role in the construction and convergence of this spectral sequence
• The Bott periodicity theorem is a fundamental result in K-theory that establishes a periodic relationship between the K-theory of spaces
  • The proof of the Bott periodicity theorem often relies on the Thom Isomorphism and its generalizations in K-theory
• The Riemann-Roch theorem for complex manifolds relates the Euler characteristic of a vector bundle to the Chern character of the bundle and the Todd class of the manifold
  • The Thom Isomorphism is used in the proof of the Riemann-Roch theorem to establish certain cohomological identities

Challenges and Open Problems

• Generalizing the Thom Isomorphism to broader classes of spaces and vector bundles, such as infinite-dimensional vector bundles or bundles with singularities, remains an active area of research
• Extending the Thom Isomorphism to more general cohomology theories, beyond ordinary cohomology and K-theory, poses interesting challenges and requires the development of new techniques
• Understanding the relationship between the Thom Isomorphism and other invariants of vector bundles, such as characteristic classes and index theorems, is an ongoing area of investigation
• Applying the Thom Isomorphism to the study of geometric and topological properties of manifolds, such as cobordism groups and surgery theory, leads to various open problems and conjectures
• Exploring the connections between the Thom Isomorphism and other areas of mathematics, such as algebraic geometry, mathematical physics, and representation theory, opens up new avenues for research and interdisciplinary collaboration
• Developing computational tools and algorithms for efficiently calculating the Thom Isomorphism and its related invariants in specific examples and applications is an important practical challenge
• Investigating the role of the Thom Isomorphism in the classification of vector bundles and the study of characteristic classes in generalized cohomology theories remains an active area of research with many open questions