🔢Algebraic Topology Unit 9 – Vector Bundles & Characteristic Classes

Definition and Basic Properties

• Vector bundle $(E,B,\pi)$ consists of total space $E$, base space $B$, and projection map $\pi:E\to B$
• Each fiber $\pi^{-1}(b)$ over a point $b\in B$ is a vector space (real or complex) of fixed dimension $n$
• Locally trivial condition: each point $b\in B$ has a neighborhood $U$ such that $\pi^{-1}(U)$ is isomorphic to $U\times \mathbb{R}^n$ (or $U\times \mathbb{C}^n$)
  • Isomorphism preserves the vector space structure of the fibers
• Rank of a vector bundle is the dimension of its fibers
• Vector bundles are classified up to isomorphism by their base space, rank, and the field of their fibers (real or complex)
• Morphisms between vector bundles $(E,B,\pi)$ and $(E',B',\pi')$ are continuous maps $f:E\to E'$ that respect the vector space structure of the fibers and commute with the projection maps

Types of Vector Bundles

• Trivial vector bundle: $B\times \mathbb{R}^n$ (or $B\times \mathbb{C}^n$) with projection onto the first factor
• Tangent bundle $TM$ of a smooth manifold $M$: fibers are tangent spaces $T_pM$ at each point $p\in M$
• Normal bundle of a submanifold $N\subset M$: fibers are normal spaces to $N$ in $M$ at each point of $N$
• Cotangent bundle $T^*M$ of a smooth manifold $M$: fibers are dual spaces $(T_pM)^*$ of the tangent spaces
• Line bundle: rank 1 vector bundle
  • Möbius band is a non-trivial real line bundle over the circle $S^1$
• Principal $G$-bundle: fibers are copies of a Lie group $G$ acting freely and transitively on each fiber
• Associated vector bundle: constructed from a principal $G$-bundle using a representation of $G$ on a vector space

Sections and Transitions

• Section of a vector bundle $(E,B,\pi)$ is a continuous map $s:B\to E$ such that $\pi\circ s=\text{id}_B$
  • Assigns a vector in the fiber to each point of the base space
• Space of sections $\Gamma(E)$ is a vector space (or module) over the ring of continuous functions on $B$
• Transition functions describe how the local trivializations of a vector bundle are related on overlaps
  • Cocycle condition ensures consistency of the transition functions
• Transition functions determine the isomorphism class of a vector bundle
• Frames: ordered bases for the fibers that vary continuously over the base space
  • Local frames correspond to local trivializations
• Connection on a vector bundle: a way to parallel transport vectors along paths in the base space
  • Defined using a covariant derivative operator satisfying certain properties

Operations on Vector Bundles

• Whitney sum (direct sum) of vector bundles $(E,B,\pi)$ and $(E',B,\pi')$ over the same base space $B$: fiberwise direct sum of the vector spaces
• Tensor product of vector bundles $(E,B,\pi)$ and $(E',B,\pi')$ over the same base space $B$: fiberwise tensor product of the vector spaces
• Dual bundle $E^*$ of a vector bundle $(E,B,\pi)$: fibers are dual spaces of the original fibers
• Pullback of a vector bundle $(E,B,\pi)$ along a continuous map $f:B'\to B$: fibers over $b'\in B'$ are the same as fibers over $f(b')\in B$
• Restriction of a vector bundle to a subspace of the base space
• Induced bundle construction: creates a vector bundle over a quotient space using a given vector bundle and a group action compatible with the projection map
• Exterior powers and symmetric powers of vector bundles: fiberwise constructions

Characteristic Classes

• Characteristic classes are invariants associated to vector bundles, measuring their non-triviality
• Stiefel-Whitney classes $w_i(E)\in H^i(B;\mathbb{Z}/2\mathbb{Z})$ for real vector bundles
  • $w_1(E)$ measures orientability: vanishes if and only if $E$ is orientable
  • Whitney sum formula: $w(E\oplus E')=w(E)\smile w(E')$, where $w(E)=1+w_1(E)+w_2(E)+\cdots$
• Chern classes $c_i(E)\in H^{2i}(B;\mathbb{Z})$ for complex vector bundles
  • $c_1(E)$ is the first obstruction to the existence of a nowhere-zero section
  • Whitney sum formula: $c(E\oplus E')=c(E)\smile c(E')$, where $c(E)=1+c_1(E)+c_2(E)+\cdots$
• Pontryagin classes $p_i(E)\in H^{4i}(B;\mathbb{Z})$ for real vector bundles
  • Defined using Chern classes of the complexification: $p_i(E)=(-1)^ic_{2i}(E\otimes\mathbb{C})$
• Euler class $e(E)\in H^n(B;\mathbb{Z})$ for oriented real vector bundles of rank $n$
  • Measures the obstruction to the existence of a nowhere-zero section
  • Related to the Euler characteristic of the base space through the Poincaré-Hopf theorem

Examples and Applications

• Möbius band as a non-trivial line bundle over the circle
• Tangent bundle of the 2-sphere $S^2$: non-trivial, as $\chi(S^2)=2$ by the Poincaré-Hopf theorem
• Hopf fibration: non-trivial circle bundle over the 2-sphere, with total space $S^3$
• Classifying spaces $BO(n)$, $BU(n)$, $BSO(n)$ for real, complex, and oriented real vector bundles of rank $n$
  • Characteristic classes are pulled back from universal classes on these spaces
• Gauss-Bonnet theorem: relates the Euler characteristic of a compact oriented even-dimensional Riemannian manifold to the integral of the Pfaffian of the curvature form
• Chern-Weil theory: expresses characteristic classes in terms of the curvature of a connection on the vector bundle
• Index theory: relates analytic properties of elliptic operators on a manifold to topological invariants of the manifold and its vector bundles
  • Atiyah-Singer index theorem, Riemann-Roch theorem for complex manifolds

Connections to Other Areas

• Differential geometry: vector bundles as a generalization of tangent bundles, connections, curvature
• Algebraic topology: classifying spaces, characteristic classes as cohomology classes, K-theory
• Algebraic geometry: algebraic vector bundles, Chern classes in Chow rings, Grothendieck's theory of schemes
• Mathematical physics: gauge theory, Berry phase, Dirac monopole, instantons
• Representation theory: vector bundles associated to principal bundles via representations of the structure group
• Index theory: Atiyah-Singer index theorem relates topology of vector bundles to analysis of elliptic operators
• Symplectic and contact geometry: prequantization line bundles, Maslov index
• Topological quantum field theory (TQFT): vector bundles as a source of examples and inspiration for the axioms of TQFT

Problem-Solving Techniques

• Compute characteristic classes using the axioms and properties they satisfy
  • Stiefel-Whitney classes: axioms of normalization, naturality, Whitney sum formula, and vanishing on trivial bundles
  • Chern classes: axioms of normalization, naturality, Whitney sum formula, and vanishing on trivial bundles
• Use the splitting principle to reduce computations for higher-rank bundles to line bundles
• Apply the Gysin sequence or Leray-Hirsch theorem to compute cohomology of the total space of a vector bundle
• Utilize classifying spaces and the universal bundle to pullback characteristic classes
• Employ obstruction theory to determine the (non-)existence of certain sections or structures on a vector bundle
• Relate characteristic classes to other invariants, such as the Euler characteristic or the index of an elliptic operator
• Use the Chern-Weil theory to express characteristic classes in terms of the curvature of a connection
• Apply the Atiyah-Singer index theorem to relate the index of an elliptic operator to topological invariants of the underlying manifold and its vector bundles