9.1 Vector bundles and classifying spaces

Vector bundles are like fancy spaces that combine a base space with vector spaces. They're crucial in algebraic topology, letting us study complex structures by breaking them into simpler parts. Think of them as mathematical Lego sets for building intricate geometric objects.

Classifying spaces are the ultimate organizers for vector bundles. They help us sort and understand different types of bundles, much like a library catalog for mathematical structures. This concept is key to grasping how vector bundles fit into the bigger picture of topology.

Vector bundles and their properties

Definition and local structure

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• A vector bundle is a topological space that locally resembles the product of a base space and a vector space, equipped with a continuous projection map onto the base space
• The fibers of a vector bundle are vector spaces of the same dimension, known as the rank of the bundle
• A vector bundle has a local trivialization, meaning it can be locally described as the product of an open set in the base space and a vector space (Euclidean space)

Morphisms and operations

• Vector bundle morphisms are continuous maps between vector bundles that preserve the vector space structure of the fibers and commute with the projection maps
  • Example: A map between the tangent bundles of two smooth manifolds induced by a smooth map between the manifolds
• The Whitney sum of two vector bundles is a new vector bundle whose fibers are the direct sums of the fibers of the original bundles
  • Example: The Whitney sum of the tangent bundle and the normal bundle of a submanifold yields the restriction of the tangent bundle of the ambient manifold

Classifying spaces for vector bundles

Grassmannian manifolds

• Classifying spaces are topological spaces that parametrize isomorphism classes of vector bundles over a given base space
• The classifying space for rank-n vector bundles is the Grassmannian manifold $Gr(n, \infty)$, which represents the space of n-dimensional subspaces of an infinite-dimensional vector space
  • Example: $Gr(1, \infty)$ is the infinite-dimensional projective space $\mathbb{R}P^\infty$

Universal bundles and pullbacks

• There exists a universal vector bundle over the classifying space, such that any rank-n vector bundle over a base space $X$ is the pullback of the universal bundle along a continuous map from $X$ to the classifying space
• The set of isomorphism classes of rank-n vector bundles over $X$ is in one-to-one correspondence with the set of homotopy classes of maps from $X$ to the classifying space $Gr(n, \infty)$
  • Example: The tautological line bundle over $\mathbb{R}P^n$ is the pullback of the universal bundle over $\mathbb{R}P^\infty$ along the inclusion map

Constructing vector bundles with transition functions

Transition functions and cocycle conditions

• Transition functions are a way to construct vector bundles by specifying how the local trivializations are related on the overlaps of their domains
• Given an open cover of the base space and a vector space $V$, a vector bundle can be constructed by assigning a continuous function from each double overlap of the cover to the general linear group $GL(V)$
  • Example: The Möbius band can be constructed using two transition functions on the circle, one of which is the identity and the other is a reflection
• The transition functions must satisfy the cocycle condition on triple overlaps, ensuring the consistency of the vector bundle structure

Isomorphism classes and cohomology

• Isomorphic vector bundles have transition functions that differ by a coboundary, i.e., they can be related by a continuous transformation of the local trivializations
• The set of isomorphism classes of vector bundles over a base space $X$ is in one-to-one correspondence with the first Čech cohomology group of $X$ with coefficients in the sheaf of continuous functions to $GL(n, \mathbb{R})$
  • Example: The isomorphism classes of line bundles over a manifold $M$ are classified by $H^1(M; \mathbb{Z}_2)$, as $GL(1, \mathbb{R})$ is homotopy equivalent to $\mathbb{Z}_2$

Vector bundles vs principal bundles

Principal bundles and group actions

• A principal $G$-bundle is a fiber bundle with a continuous right action of a topological group $G$ that preserves the fibers and acts freely and transitively on each fiber
• Given a vector bundle $E$ with rank $n$, one can construct the frame bundle $F(E)$, which is a principal $GL(n, \mathbb{R})$-bundle whose fibers consist of ordered bases for the fibers of $E$
  • Example: The frame bundle of the tangent bundle of a smooth manifold is a principal $GL(n, \mathbb{R})$-bundle

Associated vector bundles

• The vector bundle $E$ can be recovered from its frame bundle $F(E)$ as the associated bundle $F(E) \times_{GL(n, \mathbb{R})} \mathbb{R}^n$, where $GL(n, \mathbb{R})$ acts on $\mathbb{R}^n$ by matrix multiplication
• The classifying space for principal $G$-bundles, denoted by $BG$, is related to the classifying space for vector bundles by the fact that $Gr(n, \infty)$ is homotopy equivalent to $BGL(n, \mathbb{R})$
  • Example: The classifying space for oriented vector bundles is $BSO(n)$, which is homotopy equivalent to the oriented Grassmannian manifold

Classification of principal bundles

• The set of isomorphism classes of principal $G$-bundles over a base space $X$ is in one-to-one correspondence with the set of homotopy classes of maps from $X$ to the classifying space $BG$
• The classification of principal bundles is closely related to the classification of vector bundles, as every vector bundle is associated with a principal bundle
  • Example: The isomorphism classes of principal $SO(n)$-bundles over a manifold $M$ are classified by $H^1(M; \mathbb{Z}_2)$ for $n=1$ and $H^2(M; \mathbb{Z})$ for $n \geq 3$