9.4 Applications to manifolds

Characteristic classes are powerful tools in algebraic topology, helping us understand vector bundles over manifolds. They provide crucial information about a manifold's topology, including orientability and the existence of non-vanishing vector fields.

This section explores applications of characteristic classes to manifolds, focusing on Stiefel-Whitney, Chern, and Pontryagin classes. We'll see how these classes relate to important theorems like the hairy ball theorem and Poincaré-Hopf theorem.

Characteristic Classes for Manifold Topology

Cohomology Classes and Vector Bundles

• Characteristic classes are cohomology classes associated to vector bundles over a manifold that provide information about the topology of the manifold
• The Stiefel-Whitney classes are characteristic classes associated to real vector bundles
  • Stiefel-Whitney classes are elements of the cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients
  • The $i$-th Stiefel-Whitney class $w_i(E)$ of a real vector bundle $E$ is an element of $H^i(M; \mathbb{Z}/2\mathbb{Z})$, where $M$ is the base manifold
• The Chern classes are characteristic classes associated to complex vector bundles
  • Chern classes are elements of the cohomology ring with integer coefficients
  • The $i$-th Chern class $c_i(E)$ of a complex vector bundle $E$ is an element of $H^{2i}(M; \mathbb{Z})$, where $M$ is the base manifold
• The Pontryagin classes are characteristic classes associated to real vector bundles that are derived from the Chern classes of the complexification of the bundle
  • Pontryagin classes are elements of the cohomology ring with integer coefficients
  • The $i$-th Pontryagin class $p_i(E)$ of a real vector bundle $E$ is an element of $H^{4i}(M; \mathbb{Z})$, where $M$ is the base manifold

Euler Class and Orientability

• The Euler class is a characteristic class associated to oriented real vector bundles that measures the twisting of the bundle
  • The Euler class $e(E)$ of an oriented real vector bundle $E$ is an element of the top cohomology group $H^n(M; \mathbb{Z})$, where $n$ is the dimension of the base manifold $M$
  • The Euler class is related to the obstruction to the existence of a non-vanishing section of the vector bundle
• The top Stiefel-Whitney class of the tangent bundle of a closed manifold vanishes if and only if the manifold is orientable
  • A manifold is orientable if its tangent bundle admits a consistent choice of orientation
  • The vanishing of the top Stiefel-Whitney class $w_n(TM)$ is equivalent to the existence of an orientation on the manifold $M$
• The Euler characteristic of a closed manifold can be computed as the integral of the Euler class of its tangent bundle
  • The Euler characteristic $\chi(M)$ of a closed manifold $M$ is a topological invariant that measures the alternating sum of the dimensions of the cohomology groups
  • The Gauss-Bonnet theorem states that $\chi(M) = \int_M e(TM)$, where $e(TM)$ is the Euler class of the tangent bundle of $M$

Vector Fields on Spheres

Hairy Ball Theorem

• The hairy ball theorem states that there is no non-vanishing continuous tangent vector field on an even-dimensional sphere
  • A non-vanishing vector field is a continuous assignment of a non-zero tangent vector at each point of the manifold
  • The theorem implies that it is impossible to comb the hair on a spherical object (like a tennis ball) without creating a cowlick or bald spot
• The proof of the hairy ball theorem relies on the Euler characteristic and the Poincaré-Hopf theorem
  • The Poincaré-Hopf theorem relates the Euler characteristic to the sum of the indices of the zeros of a vector field
  • The index of a zero of a vector field measures the local behavior of the vector field around the zero (winding number)
• The Euler characteristic of an even-dimensional sphere is 2, while the Euler characteristic of an odd-dimensional sphere is 0
  • The Euler characteristic of the $n$-sphere $S^n$ is given by $\chi(S^n) = 1 + (-1)^n$
  • For even $n$, $\chi(S^n) = 2$, and for odd $n$, $\chi(S^n) = 0$

Poincaré-Hopf Theorem and Applications

• The Poincaré-Hopf theorem implies that any continuous tangent vector field on an even-dimensional sphere must have at least one zero
  • If a non-vanishing vector field existed, the sum of the indices of its zeros would be zero, contradicting the Euler characteristic
  • The existence of a zero of the vector field proves the hairy ball theorem
• The Poincaré-Hopf theorem has various applications in topology and geometry
  • It can be used to prove the fundamental theorem of algebra (every non-constant polynomial has a root)
  • It is related to the study of singularities of vector fields and the Morse theory of functions on manifolds

The Euler Class and its Applications

Definition and Properties

• The Euler class is a characteristic class associated to oriented real vector bundles that measures the twisting of the bundle
  • The Euler class $e(E)$ of an oriented real vector bundle $E$ is an element of the top cohomology group $H^n(M; \mathbb{Z})$, where $n$ is the dimension of the base manifold $M$
  • The Euler class is a obstruction to the existence of a non-vanishing section of the vector bundle
  • If a vector bundle admits a non-vanishing section, then its Euler class vanishes
• The Euler class of an oriented real vector bundle is an element of the top cohomology group of the base space with integer coefficients
  • The Euler class is a cohomology class with integer coefficients, unlike the Stiefel-Whitney classes which have $\mathbb{Z}/2\mathbb{Z}$ coefficients
  • The Euler class captures more refined information about the twisting of the bundle compared to the Stiefel-Whitney classes

Relation to Characteristic Numbers

• The Euler class of the tangent bundle of an oriented closed manifold is equal to the Euler characteristic of the manifold times the generator of the top cohomology group
  • For an oriented closed $n$-manifold $M$, the Euler class of the tangent bundle $e(TM)$ satisfies $e(TM) = \chi(M) \cdot \mu$, where $\mu$ is the generator of $H^n(M; \mathbb{Z}) \cong \mathbb{Z}$
  • This relation connects the Euler class to the Euler characteristic, a fundamental topological invariant
• The Euler class can be used to prove the Poincaré-Hopf theorem and the hairy ball theorem
  • The vanishing of the Euler class of the tangent bundle of an even-dimensional sphere implies the hairy ball theorem
  • The Poincaré-Hopf theorem can be derived from the properties of the Euler class and the Gauss-Bonnet theorem

Characteristic Classes vs Obstructions

Obstructions and Geometric Structures

• Characteristic classes provide information about the obstructions to the existence of certain geometric structures on a manifold
  • Geometric structures include orientations, non-vanishing vector fields, almost complex structures, symplectic structures, etc.
  • The vanishing of certain characteristic classes is a necessary condition for the existence of these structures
• The vanishing of certain characteristic classes is a necessary condition for the existence of certain geometric structures
  • If a characteristic class (like the Euler class or the top Stiefel-Whitney class) does not vanish, it obstructs the existence of the corresponding geometric structure
  • However, the vanishing of a characteristic class is not always sufficient for the existence of the structure, as there may be higher obstructions

Examples of Obstructions

• The Euler class is related to the obstruction to the existence of a non-vanishing section of an oriented real vector bundle
  • If the Euler class of a vector bundle vanishes, it is a necessary condition for the existence of a non-vanishing section
  • The vanishing of the Euler class is not always sufficient, as there may be higher obstructions (like the higher Stiefel-Whitney classes)
• The top Stiefel-Whitney class is related to the obstruction to the existence of an orientation on a manifold
  • A manifold is orientable if and only if its top Stiefel-Whitney class vanishes
  • The vanishing of the top Stiefel-Whitney class is both necessary and sufficient for orientability
• The Pontryagin classes are related to the obstructions to the existence of certain geometric structures on a manifold
  • The vanishing of certain Pontryagin classes is necessary for the existence of almost complex structures or symplectic structures
  • The Pontryagin classes provide information about the topology of the moduli space of geometric structures on a manifold