9.5 Euler class

The Euler class is a key concept in cohomology theory, providing insights into the topology of oriented vector bundles. It measures the twisting of a bundle and connects to other important ideas like Chern classes and the Euler characteristic.

This cohomological invariant has wide-ranging applications in algebraic and differential topology. The Euler class plays a crucial role in intersection theory, obstruction theory, and the study of sphere bundles, bridging geometric and topological properties of manifolds and vector bundles.

Definition of Euler class

• The Euler class is a characteristic class associated to oriented vector bundles, providing a cohomological invariant that captures topological properties of the bundle
• It is an element of the cohomology ring of the base space, lying in the twice the rank of the bundle degree cohomology group
• The Euler class generalizes the concept of the Euler characteristic of a manifold to the setting of vector bundles

Euler class for oriented vector bundles

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• Given an oriented rank $n$ vector bundle $E \rightarrow B$ over a topological space $B$, the Euler class $e(E)$ is a cohomology class in $H^n(B; \mathbb{Z})$
• The Euler class measures the twisting or non-triviality of the vector bundle
• It is defined using the Thom class of the bundle, which is a cohomology class in the compactly supported cohomology of the total space $H^n_c(E; \mathbb{Z})$
  • The Thom class is obtained by extending a chosen orientation class of a fiber by zero to the total space
  • The Euler class is the pullback of the Thom class along the zero section $s: B \rightarrow E$, i.e., $e(E) = s^*(\text{Thom class})$

Euler class in cohomology ring

• The Euler class is an element of the cohomology ring $H^*(B; \mathbb{Z})$ of the base space $B$
• It satisfies the property that its cup product with any class in $H^{n-1}(B; \mathbb{Z})$ vanishes
• The Euler class is natural with respect to pullbacks of vector bundles
  • Given a map $f: X \rightarrow B$ and the pullback bundle $f^*E$ over $X$, the Euler class satisfies $e(f^*E) = f^*(e(E))$

Properties of Euler class

• The Euler class is invariant under bundle isomorphisms
  • Isomorphic vector bundles have the same Euler class
• For a trivial vector bundle, the Euler class is zero
• The Euler class is multiplicative under Whitney sum of vector bundles
  • Given vector bundles $E$ and $F$, $e(E \oplus F) = e(E) \smile e(F)$, where $\smile$ denotes the cup product
• The Euler class satisfies the product formula for fibrations
  • Given a fibration $F \rightarrow E \rightarrow B$ with oriented vector bundles over $F$ and $B$, the Euler class of the bundle over $E$ is the product of the pullbacks of the Euler classes from $F$ and $B$

Euler class and Chern classes

• The Euler class is closely related to the Chern classes, which are characteristic classes associated to complex vector bundles
• While the Euler class is defined for oriented real vector bundles, Chern classes are defined for complex vector bundles
• The Euler class and Chern classes provide important invariants for studying the topology of vector bundles

Relationship between Euler and Chern classes

• For a complex vector bundle of rank $n$, the Euler class is related to the top Chern class
• The Euler class $e(E)$ is the Poincaré dual of the top Chern class $c_n(E)$
  • $e(E) = PD(c_n(E))$, where $PD$ denotes the Poincaré duality isomorphism $H^{2n}(B; \mathbb{Z}) \rightarrow H_0(B; \mathbb{Z})$
• The relationship between Euler and Chern classes allows for computations and comparisons between real and complex vector bundles

Euler class as top Chern class

• For a complex vector bundle $E$ of rank $n$, the Euler class $e(E_{\mathbb{R}})$ of the underlying real vector bundle $E_{\mathbb{R}}$ is equal to the top Chern class $c_n(E)$
• This relationship provides a way to compute the Euler class of a complex vector bundle using Chern classes
• The top Chern class $c_n(E)$ can be expressed as the Euler class of the underlying real bundle

Computations using Chern classes

• Chern classes provide a powerful tool for computing characteristic classes of complex vector bundles
• The Chern character, defined in terms of Chern classes, allows for computations in rational cohomology
  • The Chern character is a ring homomorphism $ch: K(B) \rightarrow H^{**}(B; \mathbb{Q})$ from the K-theory of vector bundles to the cohomology ring with rational coefficients
• Chern classes satisfy the Whitney product formula, which enables computations for direct sums of vector bundles
  • For vector bundles $E$ and $F$, the total Chern class satisfies $c(E \oplus F) = c(E) \smile c(F)$

Euler class and characteristic classes

• The Euler class is an example of a characteristic class, which is a cohomology class associated to vector bundles that satisfies certain axioms
• Characteristic classes provide topological invariants that capture important properties of vector bundles
• The study of characteristic classes is central to the classification and understanding of vector bundles

Euler class as a characteristic class

• The Euler class satisfies the axioms of a characteristic class
  • Naturality: For a map $f: X \rightarrow B$ and a vector bundle $E$ over $B$, $e(f^*E) = f^*(e(E))$
  • Normalization: For the tautological line bundle $\gamma^1$ over the projective space $\mathbb{RP}^1$, $e(\gamma^1)$ is the generator of $H^1(\mathbb{RP}^1; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$
  • Multiplicativity: For vector bundles $E$ and $F$, $e(E \oplus F) = e(E) \smile e(F)$
• The Euler class is a primary example of a characteristic class, alongside Stiefel-Whitney classes, Chern classes, and Pontryagin classes

Naturality of Euler class

• The Euler class is natural with respect to pullbacks of vector bundles
• Given a map $f: X \rightarrow B$ and a vector bundle $E$ over $B$, the Euler class satisfies $e(f^*E) = f^*(e(E))$
  • This means that the Euler class of the pullback bundle $f^*E$ over $X$ is the pullback of the Euler class of $E$ under the map $f$
• Naturality allows for the comparison of Euler classes under maps between spaces and provides functorial properties

Axioms for characteristic classes

• Characteristic classes, including the Euler class, satisfy certain axioms that uniquely determine them
• The axioms for a characteristic class $c$ of rank $n$ vector bundles include:
  • Naturality: For a map $f: X \rightarrow B$ and a vector bundle $E$ over $B$, $c(f^*E) = f^*(c(E))$
  • Normalization: The value of $c$ on a specific bundle (e.g., the tautological line bundle over projective space) is prescribed
  • Multiplicativity: For vector bundles $E$ and $F$, $c(E \oplus F) = c(E) \smile c(F)$
• The axioms provide a framework for studying and classifying characteristic classes

Applications of Euler class

• The Euler class has numerous applications in algebraic and differential topology
• It provides a bridge between the topological properties of vector bundles and the cohomology of the base space
• The Euler class is used in the study of characteristic classes, obstruction theory, and intersection theory

Euler class and Euler characteristic

• The Euler class is related to the Euler characteristic of a manifold
• For a closed oriented manifold $M$ of even dimension $n$, the Euler characteristic $\chi(M)$ is equal to the evaluation of the Euler class of the tangent bundle $e(TM)$ on the fundamental class $[M]$
  • $\chi(M) = \langle e(TM), [M] \rangle$
• This relationship provides a connection between the Euler class and the topological invariant of Euler characteristic

Euler class and intersection theory

• The Euler class plays a role in intersection theory, which studies the intersection of submanifolds and the resulting cohomology classes
• In the context of oriented vector bundles, the Euler class can be interpreted as the self-intersection of the zero section
  • The zero section $s: B \rightarrow E$ of a rank $n$ vector bundle $E$ over a closed oriented manifold $B$ of dimension $n$ represents a homology class $[s(B)] \in H_n(E; \mathbb{Z})$
  • The self-intersection of the zero section is given by the evaluation of the Euler class on the fundamental class: $\langle e(E), [B] \rangle = [s(B)] \cdot [s(B)]$
• The Euler class provides information about the intersection properties of vector bundles

Euler class in obstruction theory

• The Euler class appears in obstruction theory, which studies the existence and extension of cross-sections of vector bundles
• The vanishing of the Euler class is a necessary condition for the existence of a non-vanishing section of an oriented vector bundle
  • If a rank $n$ vector bundle $E$ over a CW complex $B$ admits a non-vanishing section, then the Euler class $e(E)$ must be zero
• The Euler class can be used to construct obstruction classes that measure the failure of extending a section from the skeleton of a CW complex to the entire space

Euler class and sphere bundles

• The Euler class is particularly relevant in the study of sphere bundles, which are fiber bundles with spheres as fibers
• The Euler class of the tangent bundle of a sphere provides important geometric and topological information
• Sphere bundles and their Euler classes are connected to the Hopf invariant and the Gysin sequence in cohomology

Euler class of tangent bundle

• For the tangent bundle $TS^n$ of the $n$-dimensional sphere $S^n$, the Euler class $e(TS^n)$ is a generator of the cohomology group $H^n(S^n; \mathbb{Z}) \cong \mathbb{Z}$
• The Euler class of the tangent bundle of a sphere is closely related to the Euler characteristic
  • For even-dimensional spheres, $\chi(S^{2n}) = \langle e(TS^{2n}), [S^{2n}] \rangle = 2$
  • For odd-dimensional spheres, $\chi(S^{2n+1}) = 0$ and the Euler class $e(TS^{2n+1})$ is trivial

Euler class and Hopf invariant

• The Euler class is related to the Hopf invariant, which is an invariant of maps between spheres
• For a map $f: S^{2n-1} \rightarrow S^n$, the Hopf invariant $H(f)$ is defined as the linking number of the preimages of two regular values
• The Hopf invariant can be expressed in terms of the Euler class of the pullback of the tangent bundle of $S^n$ under the map $f$
  • $H(f) = \langle f^*e(TS^n), [S^{2n-1}] \rangle$
• The Euler class provides a cohomological interpretation of the Hopf invariant

Euler class and Gysin sequence

• The Euler class appears in the Gysin sequence, which is a long exact sequence in cohomology associated to sphere bundles
• For an oriented sphere bundle $S^{n-1} \rightarrow E \rightarrow B$ with associated disk bundle $D^n \rightarrow \bar{E} \rightarrow B$, the Gysin sequence relates the cohomology of the base space $B$ to the cohomology of the total space $E$:
  • $\cdots \rightarrow H^{i-n}(B) \xrightarrow{\smile e} H^i(B) \xrightarrow{\pi^*} H^i(E) \xrightarrow{\phi} H^{i+1-n}(B) \rightarrow \cdots$
  • The map $\smile e$ is the cup product with the Euler class $e \in H^n(B)$
• The Gysin sequence provides a tool for computing the cohomology of sphere bundles using the Euler class and the cohomology of the base space

Generalizations of Euler class

• The concept of Euler class can be generalized and extended to various settings beyond oriented vector bundles
• These generalizations include the Euler class for non-oriented bundles, the Euler class in K-theory, and the equivariant Euler class
• The generalized Euler classes provide invariants and tools for studying more general types of bundles and cohomology theories

Euler class for non-oriented bundles

• The Euler class can be defined for non-oriented vector bundles by considering the orientation bundle
• For a non-oriented vector bundle $E$, the orientation bundle $\text{Or}(E)$ is a double cover of the base space $B$ that parametrizes the local orientations of the fibers
• The Euler class of a non-oriented vector bundle $E$ is defined as the first Stiefel-Whitney class of the orientation bundle: $e(E) := w_1(\text{Or}(E)) \in H^1(B; \mathbb{Z}/2\mathbb{Z})$
• The Euler class of a non-oriented bundle provides an obstruction to the existence of a global orientation

Euler class in K-theory

• The Euler class can be generalized to the setting of K-theory, which is a generalized cohomology theory that studies vector bundles
• In K-theory, the Euler class is defined as a K-theoretic characteristic class
• For a complex vector bundle $E$ of rank $n$ over a space $B$, the K-theoretic Euler class $\lambda(E)$ is defined as the alternating sum of exterior powers of $E$:
  • $\lambda(E) := \sum_{i=0}^n (-1)^i \Lambda^i E \in K(B)$
• The K-theoretic Euler class satisfies analogues of the properties of the ordinary Euler class, such as naturality and multiplicativity

Euler class in equivariant cohomology

• The Euler class can be extended to the setting of equivariant cohomology, which studies spaces with group actions and equivariant vector bundles
• For a G-equivariant vector bundle $E$ over a G-space $B$, where $G$ is a topological group, the equivariant Euler class $e_G(E)$ is defined in the equivariant cohomology ring $H^*_G(B; \mathbb{Z})$
• The equivariant Euler class captures the equivariant twisting of the vector bundle with respect to the group action
• Equivariant characteristic classes, including the equivariant Euler class, provide invariants for studying the topology of spaces and bundles with group actions