1.3 Operations on vector bundles

Vector bundle operations are crucial tools in K-Theory. Direct sum and tensor product allow us to combine bundles, while pullback lets us transfer them between spaces. These operations help us build complex structures from simpler ones.

Understanding these operations is key to grasping how vector bundles behave. They're essential for computing characteristic classes, studying the Picard group, and applying the Atiyah-Singer index theorem. These concepts form the backbone of advanced K-Theory applications.

Direct Sum and Tensor Product of Vector Bundles

Definition and Properties

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• The direct sum of two vector bundles $E$ and $F$ over the same base space $B$, denoted $E \oplus F$, is a vector bundle over $B$ whose fiber at each point $b \in B$ is the direct sum of the fibers $E_b$ and $F_b$
  • Example: If $E$ is a rank 2 vector bundle and $F$ is a rank 3 vector bundle, then $E \oplus F$ is a rank 5 vector bundle
• The tensor product of two vector bundles $E$ and $F$ over the same base space $B$, denoted $E \otimes F$, is a vector bundle over $B$ whose fiber at each point $b \in B$ is the tensor product of the fibers $E_b$ and $F_b$
  • Example: If $E$ and $F$ are both rank 2 vector bundles, then $E \otimes F$ is a rank 4 vector bundle
• The direct sum and tensor product of vector bundles satisfy the usual properties of direct sums and tensor products, such as associativity, commutativity (for the direct sum), and distributivity of the tensor product over the direct sum
• The rank of the direct sum bundle $E \oplus F$ is the sum of the ranks of $E$ and $F$, while the rank of the tensor product bundle $E \otimes F$ is the product of the ranks of $E$ and $F$

Applications and Examples

• The Whitney sum formula expresses the total Stiefel-Whitney class of a direct sum bundle in terms of the Stiefel-Whitney classes of the summands: $w(E \oplus F) = w(E) \smile w(F)$, where $\smile$ denotes the cup product
  • This formula allows for the computation of characteristic classes of direct sum bundles
• The tensor product of line bundles can be used to study the Picard group of a manifold, which classifies isomorphism classes of line bundles
  • Example: On a complex projective space $\mathbb{CP}^n$, the Picard group is isomorphic to $\mathbb{Z}$, generated by the tautological line bundle $\mathcal{O}(-1)$
• Operations on vector bundles play a crucial role in the Atiyah-Singer index theorem, which relates the index of an elliptic operator on a manifold to topological invariants of the manifold and its vector bundles
  • The index theorem has applications in geometry, topology, and mathematical physics

Dual Vector Bundles and their Properties

Definition and Basic Properties

• The dual vector bundle of a vector bundle $E$ over a base space $B$, denoted $E^*$, is a vector bundle over $B$ whose fiber at each point $b \in B$ is the dual vector space $(E_b)^*$ of the fiber $E_b$
• The dual vector bundle $E^*$ has the same rank as the original vector bundle $E$
  • Example: If $E$ is a rank 3 vector bundle, then $E^*$ is also a rank 3 vector bundle
• There is a natural pairing between a vector bundle $E$ and its dual $E^*$, given by the evaluation map $ev: E \otimes E^* \to B \times \mathbb{R}$, where $ev(v \otimes \phi) = (\pi(v), \phi(v))$ for $v \in E$ and $\phi \in E^*$
  • This pairing generalizes the notion of the inner product between a vector space and its dual

Double Dual and Isomorphisms

• The double dual of a vector bundle $E$, denoted $(E^*)^*$, is naturally isomorphic to the original vector bundle $E$
  • This isomorphism is analogous to the double dual isomorphism for finite-dimensional vector spaces
• The natural pairing between $E$ and $E^*$ induces an isomorphism between $E$ and $(E^*)^*$
  • Example: For a line bundle $L$, the double dual $L^{**}$ is isomorphic to $L$ itself

Pullback Operation on Vector Bundles

Definition and Functoriality

• Given a vector bundle $E$ over a base space $B$ and a continuous map $f: B' \to B$, the pullback of $E$ along $f$, denoted $f^*E$, is a vector bundle over $B'$ whose fiber at each point $b' \in B'$ is the fiber $E_{f(b')}$ of $E$ over $f(b')$
  • The pullback operation allows for the "transfer" of vector bundles from one base space to another via a continuous map
• The pullback operation is functorial, meaning that it respects composition of maps: if $g: B'' \to B'$ and $f: B' \to B$ are continuous maps, then $(f \circ g)^*E \cong g^*(f^*E)$
  • This property ensures that the pullback operation behaves well under composition of maps

Compatibility with Operations and Triviality

• The pullback operation is compatible with the direct sum and tensor product of vector bundles: $f^*(E \oplus F) \cong f^*E \oplus f^*F$ and $f^*(E \otimes F) \cong f^*E \otimes f^*F$
  • These isomorphisms allow for the computation of pullbacks of direct sums and tensor products in terms of the pullbacks of their components
• The pullback of the trivial bundle $B \times \mathbb{R}^n$ along a map $f: B' \to B$ is isomorphic to the trivial bundle $B' \times \mathbb{R}^n$
  • This property shows that the pullback of a trivial bundle remains trivial

Applications of Vector Bundle Operations

Characteristic Classes

• Pullback bundles can be used to study the behavior of vector bundles under maps between base spaces and to define characteristic classes of vector bundles
  • Example: The Chern classes of a complex vector bundle $E$ can be defined using the pullback of $E$ along the classifying map of $E$
• Operations on vector bundles, such as direct sum, tensor product, and dualization, can be used to construct new vector bundles from existing ones and to study their properties
  • Example: The Chern character of a complex vector bundle $E$ can be expressed in terms of the Chern classes of the exterior powers of $E$

Atiyah-Singer Index Theorem

• The Atiyah-Singer index theorem relates the index of an elliptic operator on a manifold to topological invariants of the manifold and its vector bundles
  • The index theorem has applications in geometry, topology, and mathematical physics
• The proof of the index theorem relies heavily on the use of vector bundle operations, such as the pullback, tensor product, and direct sum
  • Example: The symbol of an elliptic operator can be viewed as a section of a vector bundle constructed using the pullback and tensor product operations