User:Julian Nill/Fiber bundle

In algebraic topology, a branch of mathematics, a fiber bundle is a topological space that can be represented locally as a Cartesian product of two topological spaces, together with a mapping that reflects this similarity.

Fiber bundles play an important role in homotopy theory, differential geometry, and differential topology.

History
The concept of a fiber bundle first appeared in connection with the topology and geometry of manifolds.$$^{[1]}$$ Herbert Seifert introduced the terms fiber and fiber space in 1933.$$^{[2]}$$

The first definition of a fiber bundle was given by Hassler Whitney in 1935 under the name sphere space. In the years from 1935 to 1940, fiber bundles became a seperate field of research in mathematics. The work of Whitney, Heinz Hopf, and Eduard Stiefel provided perspectives on the importance of fiber bundles in topology and differential geometry.$$^{[3] Preface}$$

By 1950, the definition of a fiber bundle was clearly noted and the theory on homotopy classification and characteristic classes of fiber bundles was advanced by several mathematicians, including Shiing-Shen Chern, Lev Pontryagin, Stiefel, and Whitney. In the years from 1950 to 1955, Friedrich Hirzebruch was able to prove the Hirzebruch-Riemann-Roch theorem using the characteristic classes of fiber bundles. John Milnor gave a construction of a universal fiber bundle for arbitrary topological groups in 1955. In the early 1960s, Alexander Groethendieck, Michael Atiyah, and Hirzebruch developed a generalized cohomology theory, K-theory, using stability classes of vector bundles.$$^{[4] Preface}$$

Formal definition
A fiber bundle is a quadrupel $$(E, B, \pi, F)$$ consisting of topological spaces $$E,$$ $$B$$ and $$F$$ and a continuous surjective mapping $$\pi \colon E \to B,$$ where for every $$x \in B$$ there exists an open neighbourhood $$U \subseteq B$$ of $$x$$ and a homeomorphism $$\varphi \colon \pi^{-1}(U) \to U \times F,$$ so that the following diagram commutes: Here the mapping $$proj_1 \colon U \times F \to U$$ is the natural projection. Such a homeomorphism $$\varphi$$ is called local trivialization of the bundle and the mapping $$\pi$$ is called projection. The space $$B$$ is called base space of the bundle, $$E$$ the total space and $$F$$ the fiber.

The space $$U \times F$$ is provided with the product topology and $$\pi^{-1}(U)$$ with the subspace topology.

In order to additionally mention the fiber of the bundle, it is common to use the notation $$F \hookrightarrow E \to B$$ for the fiber bundle. Here the mapping $$F \hookrightarrow E$$ is the inclusion and $$F$$ is identified with $$F_b = \pi^{-1}(b),$$ the fiber over a point $$b \in B.$$$$^{[5] p. 376-377}$$

Each fiber bundle is a Serre fibration.$$^{[5] p. 379}$$

Tivial bundle
Let $$E = B \times F$$ and $$\pi \colon E \to B$$ be the projection onto the first factor, then the total space $$E$$ is not only locally a product, but also globally. Such a fiber bundle is called a trivial bundle or product bundle.$$^{[3] p.3}$$

Covering
A fiber bundle with discrete fiber is a covering. Similarly, any covering whose fibers all have the same cardinality is a fiber bundle with discrete fiber. In particular, a covering over a connected base space is a fiber bundle.$$^{[5] p.377}$$

Möbius band
The Möbius band is an illustrative example of a nontrivial fiber bundle. The base space is the circular line $$S^1$$ which runs through the center of the band. The fiber is given by a closed interval, e.g. $$[-1, 1].$$

The total space is given by the quotient space $$E = ([0, 1] \times [-1, 1]) / \sim$$ with the equivalence relation $$\sim ,$$ given by $$(0, a) \sim (1, -a).$$ The bundle projection $$\pi \colon E \to S^1$$ is given by the mapping, induced by the projection $$proj \colon [0, 1] \times [-1, 1] \to [0, 1],$$ i.e. an equivalence class $$[(x, y)] \in E$$ is mapped under the bundle projection onto the equivalence class $$[x]$$, where the equivalence relation on $$S^1$$ is given by $$(0 \sim 1).$$

The corresponding trivial bundle $$S^1 \times [-1, 1]$$ is a cylinder. The Möbius band and the cylinder differ by a twist of the fiber. This twist is only globally visible, locally the Möbius band and the cylinder are identical.$$^{[5] p.377}$$

Klein bottle
Another nontrivial fiber bundle is the Klein bottle. The base space and the fiber are given by $$S^1$$ and the total space is given by the quotient space $$E = ([0, 1] \times [0, 1]) / \sim ,$$ where the equivalence relation $$\sim$$ is given by $$(0, y) \sim (1, y)$$ and $$(x, 0) \sim (1 - x, 1).$$ The bundle projection $$\pi \colon E \to S^1$$ maps an element $$[(a, b)] \in E$$ onto $$\pi([(a, b)]) = [b] $$ with the equivalence relation $$(0 \sim 1) $$ on $$S^1. $$

The corresponding trivial bundle $$S^1 \times S^1 $$ is a torus. This is locally indistinguishable from the Klein bottle.$$^{[3] p.4}$$

Hopf bundle
The Hopf bundle $$S^1 \hookrightarrow S^3 \to S^2$$ has spheres as fiber, total space and base space and is one of the first discovered nontrivial fiber bundles. It is a special case for $$n=1$$ of the fiber bundle $$S^1 \to S^{2n +1} \to \Complex P^n$$ over the $$n$$-dimensional complex projective space.

Further Hopf bundles, also called generalized Hopf bundles, can be derived by replacing the complex numbers by the real numbers, the quaternions and the octonions:


 * The covering $$S^0 \to S^n \to \R P^n$$ over the $$n$$-dimensional projective space yields for $$n=1$$ the real Hopf bundle $$S^0 \to S^1 \to S^1.$$
 * The quaternions yield a Hopf bundle given by $$S^3 \to S^7 \to S^4 \cong \mathbb{H} P^1.$$
 * The octonions yield a Hopf bundle given by $$S^7 \to S^{15} \to S^8.$$

Further fiber bundles whose fiber, total space and base space are spheres do not exist. This is a consequence of Adam's theorem which solves H. Hopf's problem on the number of mappings between spheres with Hopf-Invariant 1.$$^{[5] p. 377-379}$$

Cross sections
The cross section of a fiber bundle $$(E, B, \pi, F)$$ is a continuous mapping $$s \colon B \to E$$ which is right inverse to the projection $$\pi .$$ For every $$b \in B$$ is the link of projection and section equal to the identity: $$(\pi \circ s)(b) = b.$$ In other words for every $$b \in B$$ the image of the section lies in the fiber over $$b :$$ $$s(b) \in \pi^{-1}(b).$$

A local section of a fiber bundle is a continuous mapping $$s \colon V \to E,$$ where $$V \subseteq B$$ is an open subset and $$(\pi \circ s) (b) = b$$ holds for all $$b \in V.$$$$^{[4] p.11}$$

Bundle morphism
A bundle morphism (also called bundle mapping) between two fiber bundles $$(E_1, B_1, \pi_1, F_1)$$ and $$(E_2, B_2, \pi_2, F_2)$$ is a mapping, that preserves the bundle structure; in some sense, it is a fiber-preserving mapping. More precisely, a bundle morphism is given by a tupel $$(u, f)$$ of two mappings $$u \colon E_1 \to E_2$$ and $$f \colon B_1 \to B_2,$$ such that $$\pi_2 \circ u = f \circ \pi_1$$ holds. The situation is illustrated by the following commutative diagram: A fiber over $$b \in B_1$$ is mapped under $$u$$ onto a fiber over $$f(b);$$ this is given by the relation $$u(\pi_1^{-1}(b)) \subseteq \pi_2^{-1}(f(b)).$$

If the base spaces are identical, the bundle morphism is given by $$(u, 1_B)$$ and one speaks of a $$B$$-morphism or a bundle morphism over $$B,$$ where $$B = B_1 = B_2.$$ The relation $$\pi_1 = \pi_2 \circ u$$ is given by the following diagram: For every $$b \in B$$ the condition $$u( \pi_1^{-1}(b)) \subseteq \pi_2^{-1}(b)$$ holds, which is why $$u$$ is also called fiber-preserving.$$^{[4] p.14}$$

Coordinate bundle
For every base space of a fiber bundle there exists an atlas $$\{ (U_i, h_{U_i}^{-1}) | i \in I \}$$ of charts where $$U_i \subseteq B$$ are open subsets and $$h_{U_i}$$ are the local trivializations of the fiber bundle. Two charts $$(U_i, h_{U_i}^{-1})$$ and $$(U_j, h_{U_j}^{-1})$$ can be compared by using continuous transition functions $$\Phi_{i,j} \colon (U_i \cap U_j) \to Aut(F).$$ The transition functions provide information about which symmetries of the fibers are used in the transition. For a point $$b \in U_i \cap U_j$$ the transition function is given by the expression $$\Phi_{i, j}(b)= (pr_F \circ h_{U_i} \circ h_{U_j}^{-1})(b, \cdot ) \colon F \to F.$$ The following diagram illustrates the situation: In the first line, the first component is given by the identity and the second component is given by the transition function.$$^{[6] p. 184}$$

A topological transformation group $$G$$ of a topological space $$F$$ relative to a map $$\eta \colon G \times F \to F$$ is a topological group $$G$$ such that:


 * $$\eta$$ is continuous
 * $$\eta (e, f) = f$$ where $$e$$ is the identity of $$G$$ and
 * $$\eta(g_1 g_2, f) = \eta(g_1, \eta(g_2, f))$$ for all $$g_1, g_2 \in G$$ and $$f \in F.$$

Often one considers more than one such map $$\eta$$ and therefore abbreviate $$\eta(g, f)$$ by $$g \cdot f.$$$$^{[3] p. 7}$$

A coordinate bundle is a fiber bundle together with an effective topological transformation group $$G$$ of $$F,$$ such that the following two conditions hold:


 * For every $$b \in U_i \cap U_j$$ and $$i, j \in I$$ the homeomorphism $$(h_{U_j} \circ h_{U_i}^{-1}) (b, \cdot ) \colon F \to F$$ corresponds to the operation of a group element in $$G$$ and
 * for every $$i, j \in I$$ the mapping $$\tau_{j, i} \colon U_i \cap U_j \to G$$ with $$\tau_{j, i}(b) = (h_{U_j} \circ h_{U_i}^{-1}) (b, \cdot )$$ is continuous.

The mappings $$\tau_{j, i}$$ are called coordinate transformations (sometimes called transition functions$$^{[7] p.77 - 80}$$) and $$G$$ is called the structure group of the bundle. The coordinate transformations have the following three properties:


 * $$\tau_{k, j}(b) \tau_{j, i}(b) = \tau_{k, i}(b)$$ for every $$i, j, k \in I$$ and every $$b \in U_i \cap U_j \cap U_k.$$
 * $$\tau_{i, i}(b) = id_G$$ for every $$b \in U_i.$$
 * $$\tau_{j, k}(b) = (\tau_{k, j}(b))^{-1}$$ for every $$b \in U_j \cap U_k.$$

Two coordinate bundles with the same base space and total space, the same fiber, projection and structure group are called equivalent if the atlases $$\{(U_i, h_{U_i}) | i \in I \}$$ and $$\{(U_j^\prime, h^\prime_{U_j^\prime}) |j \in J \}$$ for two index sets $$I$$ and $$J$$ satisfy the following two conditions:


 * For every $$b \in U_i \cap U_k^\prime$$ the expression $$\tilde \tau_{k, i}(b) = (h_{U_k^\prime}^\prime \circ h_{U_i}^{-1}) (b, \cdot ) $$ coincides with the operation of a group element and
 * the coordinate transformations $$\tilde \tau_{k, i} \colon U_i \cap U_k^\prime \to G$$ are continuous.

A $$G$$-fiber bundle is an equivalence class of coordinate bundles. Often the $$G$$-fiber bundle is defined as maximal coordinate bundle.$$^{[3] p. 6-9}$$

The fiber bundle construction theorem provides conditions under which the existence of a coordinate bundle is guaranteed:

For topological transformation group $$G$$ of a space $$F$$ and a system of coordinate transformations of a space $$B,$$ that means a cover $$\{U_i | i \in I \}$$ and a set $$\{ \tau_{j, i} | i,j \in I\}$$ of continuous mappings with the above three properties for coordinate transformations, there exists a coordinate bundle with base space $$B,$$ fiber $$F,$$ structure group $$G$$ and coordinate transformations $$\tau_{j,i}.$$$$^{[3] p. 14}$$

Principal bundle
A principal $$G$$-bundle is a fiber bundle $$\pi \colon E \to B$$ with a fiber $$F = G$$ and structure group $$G$$ acting on the fiber by left translation. The structure group acts freely on the total space by right translation with orbit space $$B.$$$$^{[7] p. 84}$$

A open cover $$(U_i)_{i \in I}$$ of $$B$$ is called numerable, if there exists a locally finite partition of unity: "$1 = \sum_{i \in I} u_i$ with $supp(u_i) \subseteq U_i$ for every $i \in I.$"A principal $$G$$-bundle $$\pi \colon E \to B$$ is called numerable, if there is a numerable cover $$(U_i)_{i \in I}$$ of $$B$$ such that the restricted bundles $$p_i = p |_{p^{-1}(U_i)} \colon p^{-1}(U_i) \to U_i$$ are trivial bundles for all $$i \in I.$$ A numerable principal $$G$$-bundle is called universal bundle, if for every space $$X$$ the map $$\rho \colon [X, B] \to Prin_G(X)$$ from the set of homotopy classes of maps from $$X$$ to $$B$$ to the set of isomorphism classes of principal $$G$$-bundles is a bijection. In the case of a universal bundle $$\pi \colon E \to B$$ the base space is called classifying space of $$G.$$$$^{[4] p.48-50}$$

Principal bundles play an important role in the classification of bundles. Moreover each $$G$$-fiber bundle can be associated with a principal bundle and conversely each principal bundle can be associated with a $$G$$-fiber bundle.

Associated principal bundle
For a given $$G$$-fiber bundle a $$G$$-principal bundle can be constructed. The existence is given by the fiber bundle construction theorem, where the fiber is represented by $$G$$ and in addition $$G$$ acts on itself by left multiplication. The base space and the system of coordinate transformations will be choosen identical to those of the $$G$$-fiber bundle.$$^{[3] p. 36}$$

Associated G-fiber bundle
For a given $$G$$-principal bundle $$\pi \colon E \to B$$ and a left $$G$$-space $$F$$ a $$G$$-fiber bundle can be constructed:

On the product space $$E \times F$$ there is a right $$G$$-space structure defined by $$((x, b), g) = (gx, g^{-1}b). $$ The $$G $$-fiber bundle is given by the mapping $$\pi_F \colon (E \times F) / G \to B $$ with $$\pi_F((x, b)G) = \pi(x) $$ and the fiber $$F. $$$$^{[4] p. 43-44}$$

Vector bundle
A vector bundle of rank $$n$$ over a field $$\mathbb{K}$$ is a fiber bundle $$V \to E \xrightarrow{\pi} B,$$ whose fibers have the structure of a $$n$$-dimensional $$\mathbb{K}$$-vector space and, in addition, any local trivialization $$\varphi \colon \pi^{-1}(U) \to U \times V $$ for a $$U \subseteq B $$ induces a $$\mathbb{K}$$-linear isomorphism on the individual fibers. This means that the mapping $$\varphi $$ restricted to a $$x \in U $$ is an isomorphism and thus $$\pi^{-1}(x) \cong \{ x \} \times V $$ holds. Often one considers real or complex vector bundles where the field $$\mathbb{K} $$ is given by the real numbers $$\R$$ or by the complex numbers $$\Complex .$$

There is a natural bijection between the isomorphism classes of vector bundles with rank $$k $$ of paracompact spaces $$B $$ and the set of homotopy classes of mappings from $$B $$ into the Grassmann manifold of $$k $$-dimensional subspaces in $$\R^\infty : $$

$$[B, G_n(\R^\infty)] \cong Vect^n(B).$$$$^{[4] p. 23}$$

Examples

 * The tangent bundle of $$S^n \subseteq \R^{n+1} $$ with total space $$E = \{ (x, v) \in S^n \times \R^{n+1} | x \perp v = 0\} $$ and projection $$p \colon E \to S^n $$ is a vector bundle with fibers $$p^{-1}(b) \cong \R^n $$ for every $$b \in S^n. $$
 * The canonical vector bundle $$\gamma_k^n $$ with rank $$k $$ of the Grassmann manifold $$G_k(\R^n) $$ is given by the total space $$E = \{ (V, x) \in G_k(\R^n) \times \R^n | x \in V \} $$ and the projection $$\gamma_k^n \colon E \to G_k(\R^n). $$$$^{[4] p.12-13}$$

Sphere bundle
A $$n$$-sphere bundle is a fiber bundle $$\pi \colon E \to B$$ whose fiber is $$S^n.$$ Often the sphere bundle is given in addition with the orthogonal group $$O (n+1)$$ as structure group.$$^{[8] p. 91}$$

The sphere bundle is called orientable, if the structure group is the rotation group.$$^{[3] p. 34}$$

The cohomology of sphere bundles can be computet using the Gysin sequence.

Cohomology of fiber bundles
The calculation of the cohomology groups of fiber bundles is much more difficult than the calculation of the homotopy groups. The homotopy groups are given by a long exact sequence, whereas the cohomology groups have a long exact sequence only under certain conditions.

For a trivial bundle, the relation of the cohomology groups is given by the Künneth formula. For arbitrary fiber bundles, tools such as spectral sequences are needed.

The Leray-Hirsch theorem provides sufficient conditions on a fiber bundle so that the structure of the cohomology groups is very similar to that of a trivial bundle.

For $$(n-1)$$-sphere bundles $$p \colon E \to B,$$ which additionally satisfy an orientability hypothesis, a long exact sequence of cohomology groups exists. The sequence is known as the Gysin sequence:"$\cdots \to H^{i - n}(B;R) \xrightarrow[]{\smile e} H^i(B;R) \xrightarrow[]{p^*} H^i(E;R) \to H^{i - n + 1}(B;R) \to \cdots .$"Here $$e$$ is a particular Euler class in $$H^n(B;R).$$$$^{[5] p. 438}$$

Examples

 * The Hopf bundle $$S^1 \to S^3 \to S^2$$ does not have the cohomology structure of a trivial bundle, since $$H^*(S^3) \not \approx H^*(S^2) \otimes H^*(S^1)$$ holds.$$^{[5] p. 432}$$
 * For the fiber bundle $$U(n-1) \hookrightarrow U(n) \to S^{2n-1}$$ holds: $$H^*(U(n);\Z) \approx \Lambda_\Z[x_1, x_3, \cdots, x_{2n-1}].$$$$^{[5] p.434}$$