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Fibration

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Colon Fiber Fibration Homotopy

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The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

Fibrations are used, for example, in or obstruction theory.

In this article, all mappings are continuous mappings between topological spaces.

Formal definitions

Homotopy lifting property

A mapping

p \colon E \to B

satisfies the homotopy lifting property for a space

X

if:

for every homotopy $h \colon X \times 0, \to B$ and
for every mapping (also called lift) $\tilde h_0 \colon X \to E$ lifting $h|_{X \times 0} = h_0$ (i.e. $h_0 = p \circ \tilde h_0$ )

there exists a (not necessarily unique) homotopy $\tilde h \colon X \times 0, \to E$ lifting $h$ (i.e. $h = p \circ \tilde h$ ) with $\tilde h_0 = \tilde h|_{X \times 0}.$

The following commutative diagram shows the situation:

Fibration

A fibration (also called Hurewicz fibration) is a mapping

p \colon E \to B

satisfying the homotopy lifting property for all spaces

X.

The space

B

is called base space and the space

E

is called total space. The fiber over

b \in B

is the subspace

F_b = p^{-1}(b) \subseteq E.

Serre fibration

A Serre fibration (also called weak fibration) is a mapping

p \colon E \to B

satisfying the homotopy lifting property for all CW complex.

Every Hurewicz fibration is a Serre fibration.

Quasifibration

A mapping

p \colon E \to B

is called quasifibration, if for every

b \in B,

e \in p^{-1}(b)

and

i \geq 0

holds that the induced mapping

p_* \colon \pi_i(E, p^{-1}(b), e) \to \pi_i(B, b)

is an isomorphism.

Every Serre fibration is a quasifibration.

Examples

The projection onto the first factor $p \colon B \times F \to B$ is a fibration. That is, trivial bundles are fibrations.
Every Covering space $p \colon E \to B$ is a fibration. Specifically, for every homotopy $h \colon X \times 0, \to B$ and every lift $\tilde h_0 \colon X \to E$ there exists a uniquely defined lift $\tilde h \colon X \times 0,1 \to E$ with $p \circ \tilde h = h.$
Every fiber bundle $p \colon E \to B$ satisfies the homotopy lifting property for every CW-complex.
A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.
An example of a fibration which is not a fiber bundle is given by the mapping $i^* \colon X^{I^k} \to X^{\partial I^k}$ induced by the inclusion $i \colon \partial I^k \to I^k$ where $k \in \N,$ $X$ a topological space and $X^{A} = \{f \colon A \to X\}$ is the space of all continuous mappings with the compact-open topology.
The Hopf fibration $S^1 \to S^3 \to S^2$ is a non-trivial fiber bundle and, specifically, a Serre fibration.

Basic concepts

Fiber homotopy equivalence

A mapping

f \colon E_1 \to E_2

between total spaces of two fibrations

p_1 \colon E_1 \to B

and

p_2 \colon E_2 \to B

with the same base space is a fibration homomorphism if the following diagram commutes: The mapping

f

is a fiber homotopy equivalence if in addition a fibration homomorphism

g \colon E_2 \to E_1

exists, such that the mappings

f \circ g

and

g \circ f

are homotopic, by fibration homomorphisms, to the identities

\operatorname{Id}_{E_2}

and

\operatorname{Id}_{E_1}.

Pullback fibration

Given a fibration

p \colon E \to B

and a mapping

f \colon A \to B

, the mapping

p_f \colon f^*(E) \to A

is a fibration, where

f^*(E) = \{(a, e) \in A \times E\ |\ f(a) = p(e)\}

is the Pullback bundle and the projections of

f^*(E)

onto

A

and

E

yield the following commutative diagram: The fibration

p_f

is called the pullback fibration or induced fibration.

Pathspace fibration

With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration.

The total space $E_f$ of the pathspace fibration for a continuous mapping $f \colon A \to B$ between topological spaces consists of pairs $(a, \gamma)$ with $a \in A$ and paths $\gamma \colon I \to B$ with starting point $\gamma (0) = f(a),$ where $I = 0,$ is the unit interval. The space $E_f = \{ (a, \gamma) \in A \times B^I | \gamma (0) = f(a) \}$ carries the subspace topology of $A \times B^I,$ where $B^I$ describes the space of all mappings $I \to B$ and carries the compact-open topology.

The pathspace fibration is given by the mapping $p \colon E_f \to B$ with $p(a, \gamma) = \gamma (1).$ The fiber $F_f$ is also called the homotopy fiber of $f$ and consists of the pairs $(a, \gamma)$ with $a \in A$ and paths $\gamma \colon 0, \to B,$ where $\gamma(0) = f(a)$ and $\gamma(1) = b_0 \in B$ holds.

For the special case of the inclusion of the base point $i \colon b_0 \to B$ , an important example of the pathspace fibration emerges. The total space $E_i$ consists of all paths in $B$ which starts at $b_0.$ This space is denoted by $PB$ and is called path space. The pathspace fibration $p \colon PB \to B$ maps each path to its endpoint, hence the fiber $p^{-1}(b_0)$ consists of all closed paths. The fiber is denoted by $\Omega B$ and is called loop space.

Properties

The fibers $p^{-1}(b)$ over $b \in B$ are homotopy equivalent for each path component of $B.$
For a homotopy $f \colon 0, \times A \to B$ the pullback fibrations $f^*_0(E) \to A$ and $f^*_1(E) \to A$ are fiber homotopy equivalent.
If the base space $B$ is contractible, then the fibration $p \colon E \to B$ is fiber homotopy equivalent to the product fibration $B \times F \to B.$
The pathspace fibration of a fibration $p \colon E \to B$ is very similar to itself. More precisely, the inclusion $E \hookrightarrow E_p$ is a fiber homotopy equivalence.
For a fibration $p \colon E \to B$ with fiber $F$ and contractible total space, there is a weak homotopy equivalence $F \to \Omega B.$

Puppe sequence

For a fibration

p \colon E \to B

with fiber

F

and base point

b_0 \in B

the inclusion

F \hookrightarrow F_p

of the fiber into the homotopy fiber is a homotopy equivalence. The mapping

i \colon F_p \to E

with

i (e, \gamma) = e

, where

e \in E

and

\gamma \colon I \to B

is a path from

p(e)

b_0

in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration

PB \to B

along

p

. This procedure can now be applied again to the fibration

i

and so on. This leads to a long sequence:

$\cdots \to F_j \to F_i \xrightarrow {j} F_p \xrightarrow i E \xrightarrow p B.$

The fiber of

i

over a point

e_0 \in p^{-1}(b_0)

consists of the pairs

(e_0, \gamma)

where

\gamma

is a path from

p(e_0) = b_0

b_0

, i.e. the loop space

\Omega B

. The inclusion

\Omega B \hookrightarrow F_i

of the fiber of

i

into the homotopy fiber of

i

is again a homotopy equivalence and iteration yields the sequence:

$\cdots \Omega^2B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B.$

Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the or the sequences of fibrations and cofibrations.

Principal fibration

A fibration

p \colon E \to B

with fiber

F

is called principal, if there exists a commutative diagram: The bottom row is a sequence of fibrations and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov system.

Long exact sequence of homotopy groups

For a Serre fibration

p \colon E \to B

there exists a long exact sequence of . For base points

b_0 \in B

and

x_0 \in F = p^{-1}(b_0)

this is given by:

$\cdots \rightarrow \pi_n(F,x_0) \rightarrow \pi_n(E, x_0) \rightarrow \pi_n(B, b_0) \rightarrow \pi_{n - 1}(F, x_0) \rightarrow$
$\cdots \rightarrow \pi_0(F, x_0) \rightarrow \pi_0(E, x_0).$

The

\pi_n(F, x_0) \rightarrow \pi_n(E, x_0)

and

\pi_n(E, x_0) \rightarrow \pi_n(B, b_0)

are the induced homomorphisms of the inclusion

i \colon F \hookrightarrow E

and the projection

p \colon E \rightarrow B.

Hopf fibration

are a family of whose fiber, total space and base space are N-sphere:

$S^0 \hookrightarrow S^1 \rightarrow S^1,$
$S^1 \hookrightarrow S^3 \rightarrow S^2,$
$S^3 \hookrightarrow S^7 \rightarrow S^4,$
$S^7 \hookrightarrow S^{15} \rightarrow S^8.$

The long exact sequence of homotopy groups of the hopf fibration

S^1 \hookrightarrow S^3 \rightarrow S^2

yields:

$\cdots \rightarrow \pi_n(S^1,x_0) \rightarrow \pi_n(S^3, x_0) \rightarrow \pi_n(S^2, b_0) \rightarrow \pi_{n - 1}(S^1, x_0) \rightarrow$ $\cdots \rightarrow \pi_1(S^1, x_0) \rightarrow \pi_1(S^3, x_0) \rightarrow \pi_1(S^2, b_0).$

This sequence splits into short exact sequences, as the fiber

S^1

S^3

is contractible to a point:

$0 \rightarrow \pi_i(S^3) \rightarrow \pi_i(S^2) \rightarrow \pi_{i-1}(S^1) \rightarrow 0.$

This short exact sequence splits because of the suspension homomorphism

\phi \colon \pi_{i - 1}(S^1) \to \pi_i(S^2)

and there are :

$\pi_i(S^2) \cong \pi_i(S^3) \oplus \pi_{i - 1}(S^1).$

The homotopy groups

\pi_{i - 1}(S^1)

are trivial for

i \geq 3,

so there exist isomorphisms between

\pi_i(S^2)

and

\pi_i(S^3)

for

i \geq 3.

Analog the fibers $S^3$ in $S^7$ and $S^7$ in $S^{15}$ are contractible to a point. Further the short exact sequences split and there are families of isomorphisms:

$\pi_i(S^4) \cong \pi_i(S^7) \oplus \pi_{i - 1}(S^3)$ and
$\pi_i(S^8) \cong \pi_i(S^{15}) \oplus \pi_{i - 1}(S^7).$

Spectral sequence

Spectral sequences are important tools in algebraic topology for computing (co-)homology groups.

The Leray-Serre spectral sequence connects the (co-)homology of the total space and the fiber with the (co-)homology of the base space of a fibration. For a fibration $p \colon E \to B$ with fiber $F,$ where the base space is a path connected CW-complex, and an additive homology theory $G_*$ there exists a spectral sequence:

H_k (B; G_q(F)) \cong E^2_{k, q} \implies G_{k + q}(E).

Fibrations do not yield long exact sequences in homology, as they do in homotopy. But under certain conditions, fibrations provide exact sequences in homology. For a fibration $p \colon E \to B$ with fiber $F,$ where base space and fiber are path connected, the fundamental group $\pi_1(B)$ acts trivially on $H_*(F)$ and in addition the conditions $H_p(B) = 0$ for $0 and H_q(F) = 0 for 0 hold, an exact sequence exists (also known under the name Serre exact sequence):$

$H_{m+n-1}(F) \xrightarrow {i_*} H_{m+n-1}(E) \xrightarrow {f_*} H_{m+n-1} (B) \xrightarrow \tau H_{m+n-2} (F) \xrightarrow {i^*} \cdots \xrightarrow {f_*} H_1 (B) \to 0.$

This sequence can be used, for example, to prove Hurewicz theorem or to compute the homology of loopspaces of the form

\Omega S^n:

$H_k (\Omega S^n) = \begin{cases} \Z & \exist q \in \Z \colon k = q (n-1)\\
0 & \text{otherwise} \end{cases}.$

For the special case of a fibration

p \colon E \to S^n

where the base space is a

n

-sphere with fiber

F,

there exist exact sequences (also called ) for homology and cohomology:

$\cdots \to H_q(F) \xrightarrow{i_*} H_q(E) \to H_{q-n}(F) \to H_{q-1}(F) \to \cdots$
$\cdots \to H^q(E) \xrightarrow{i^*} H^q(F) \to H^{q-n+1}(F) \to H^{q+1}(E) \to \cdots$

Orientability

For a fibration

p \colon E \to B

with fiber

F

and a fixed commutative ring

R

with a unit, there exists a contravariant functor from the fundamental groupoid of

B

to the category of graded

R

-modules, which assigns to

b \in B

the module

H_*(F_b, R)

and to the path class

\omega

the homomorphism

h\omega_* \colon H_*(F_{\omega (0)}, R) \to H_*(F_{\omega (1)}, R),

where

h\omega

is a homotopy class in

F_{\omega(0)},.

A fibration is called orientable over $R$ if for any closed path $\omega$ in $B$ the following holds: $h\omega_* = 1.$

Euler characteristic

For an orientable fibration

p \colon E \to B

over the field

\mathbb{K}

with fiber

F

and path connected base space, the Euler characteristic of the total space is given by:

$\chi(E) = \chi(B)\chi(F).$

Here the Euler characteristics of the base space and the fiber are defined over the field

\mathbb{K}

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