- Source: Fibration
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 Postnikov systems or obstruction theory.
In this article, all mappings are continuous mappings between topological spaces.
Formal definitions
= Homotopy lifting property
=A mapping
p
:
E
→
B
{\displaystyle p\colon E\to B}
satisfies the homotopy lifting property for a space
X
{\displaystyle X}
if:
for every homotopy
h
:
X
×
[
0
,
1
]
→
B
{\displaystyle h\colon X\times [0,1]\to B}
and
for every mapping (also called lift)
h
~
0
:
X
→
E
{\displaystyle {\tilde {h}}_{0}\colon X\to E}
lifting
h
|
X
×
0
=
h
0
{\displaystyle h|_{X\times 0}=h_{0}}
(i.e.
h
0
=
p
∘
h
~
0
{\displaystyle h_{0}=p\circ {\tilde {h}}_{0}}
)
there exists a (not necessarily unique) homotopy
h
~
:
X
×
[
0
,
1
]
→
E
{\displaystyle {\tilde {h}}\colon X\times [0,1]\to E}
lifting
h
{\displaystyle h}
(i.e.
h
=
p
∘
h
~
{\displaystyle h=p\circ {\tilde {h}}}
) with
h
~
0
=
h
~
|
X
×
0
.
{\displaystyle {\tilde {h}}_{0}={\tilde {h}}|_{X\times 0}.}
The following commutative diagram shows the situation: : 66
= Fibration
=A fibration (also called Hurewicz fibration) is a mapping
p
:
E
→
B
{\displaystyle p\colon E\to B}
satisfying the homotopy lifting property for all spaces
X
.
{\displaystyle X.}
The space
B
{\displaystyle B}
is called base space and the space
E
{\displaystyle E}
is called total space. The fiber over
b
∈
B
{\displaystyle b\in B}
is the subspace
F
b
=
p
−
1
(
b
)
⊆
E
.
{\displaystyle F_{b}=p^{-1}(b)\subseteq E.}
: 66
= Serre fibration
=A Serre fibration (also called weak fibration) is a mapping
p
:
E
→
B
{\displaystyle p\colon E\to B}
satisfying the homotopy lifting property for all CW-complexes.: 375-376
Every Hurewicz fibration is a Serre fibration.
= Quasifibration
=A mapping
p
:
E
→
B
{\displaystyle p\colon E\to B}
is called quasifibration, if for every
b
∈
B
,
{\displaystyle b\in B,}
e
∈
p
−
1
(
b
)
{\displaystyle e\in p^{-1}(b)}
and
i
≥
0
{\displaystyle i\geq 0}
holds that the induced mapping
p
∗
:
π
i
(
E
,
p
−
1
(
b
)
,
e
)
→
π
i
(
B
,
b
)
{\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(b),e)\to \pi _{i}(B,b)}
is an isomorphism.
Every Serre fibration is a quasifibration.: 241-242
Examples
The projection onto the first factor
p
:
B
×
F
→
B
{\displaystyle p\colon B\times F\to B}
is a fibration. That is, trivial bundles are fibrations.
Every covering
p
:
E
→
B
{\displaystyle p\colon E\to B}
is a fibration. Specifically, for every homotopy
h
:
X
×
[
0
,
1
]
→
B
{\displaystyle h\colon X\times [0,1]\to B}
and every lift
h
~
0
:
X
→
E
{\displaystyle {\tilde {h}}_{0}\colon X\to E}
there exists a uniquely defined lift
h
~
:
X
×
[
0
,
1
]
→
E
{\displaystyle {\tilde {h}}\colon X\times [0,1]\to E}
with
p
∘
h
~
=
h
.
{\displaystyle p\circ {\tilde {h}}=h.}
: 159 : 50
Every fiber bundle
p
:
E
→
B
{\displaystyle p\colon E\to B}
satisfies the homotopy lifting property for every CW-complex.: 379
A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.: 379
An example of a fibration which is not a fiber bundle is given by the mapping
i
∗
:
X
I
k
→
X
∂
I
k
{\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}}}
induced by the inclusion
i
:
∂
I
k
→
I
k
{\displaystyle i\colon \partial I^{k}\to I^{k}}
where
k
∈
N
,
{\displaystyle k\in \mathbb {N} ,}
X
{\displaystyle X}
a topological space and
X
A
=
{
f
:
A
→
X
}
{\displaystyle X^{A}=\{f\colon A\to X\}}
is the space of all continuous mappings with the compact-open topology.: 198
The Hopf fibration
S
1
→
S
3
→
S
2
{\displaystyle 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
:
E
1
→
E
2
{\displaystyle f\colon E_{1}\to E_{2}}
between total spaces of two fibrations
p
1
:
E
1
→
B
{\displaystyle p_{1}\colon E_{1}\to B}
and
p
2
:
E
2
→
B
{\displaystyle p_{2}\colon E_{2}\to B}
with the same base space is a fibration homomorphism if the following diagram commutes:
The mapping
f
{\displaystyle f}
is a fiber homotopy equivalence if in addition a fibration homomorphism
g
:
E
2
→
E
1
{\displaystyle g\colon E_{2}\to E_{1}}
exists, such that the mappings
f
∘
g
{\displaystyle f\circ g}
and
g
∘
f
{\displaystyle g\circ f}
are homotopic, by fibration homomorphisms, to the identities
Id
E
2
{\displaystyle \operatorname {Id} _{E_{2}}}
and
Id
E
1
.
{\displaystyle \operatorname {Id} _{E_{1}}.}
: 405-406
= Pullback fibration
=Given a fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
and a mapping
f
:
A
→
B
{\displaystyle f\colon A\to B}
, the mapping
p
f
:
f
∗
(
E
)
→
A
{\displaystyle p_{f}\colon f^{*}(E)\to A}
is a fibration, where
f
∗
(
E
)
=
{
(
a
,
e
)
∈
A
×
E
|
f
(
a
)
=
p
(
e
)
}
{\displaystyle f^{*}(E)=\{(a,e)\in A\times E|f(a)=p(e)\}}
is the pullback and the projections of
f
∗
(
E
)
{\displaystyle f^{*}(E)}
onto
A
{\displaystyle A}
and
E
{\displaystyle E}
yield the following commutative diagram:
The fibration
p
f
{\displaystyle p_{f}}
is called the pullback fibration or induced fibration.: 405-406
= 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
{\displaystyle E_{f}}
of the pathspace fibration for a continuous mapping
f
:
A
→
B
{\displaystyle f\colon A\to B}
between topological spaces consists of pairs
(
a
,
γ
)
{\displaystyle (a,\gamma )}
with
a
∈
A
{\displaystyle a\in A}
and paths
γ
:
I
→
B
{\displaystyle \gamma \colon I\to B}
with starting point
γ
(
0
)
=
f
(
a
)
,
{\displaystyle \gamma (0)=f(a),}
where
I
=
[
0
,
1
]
{\displaystyle I=[0,1]}
is the unit interval. The space
E
f
=
{
(
a
,
γ
)
∈
A
×
B
I
|
γ
(
0
)
=
f
(
a
)
}
{\displaystyle E_{f}=\{(a,\gamma )\in A\times B^{I}|\gamma (0)=f(a)\}}
carries the subspace topology of
A
×
B
I
,
{\displaystyle A\times B^{I},}
where
B
I
{\displaystyle B^{I}}
describes the space of all mappings
I
→
B
{\displaystyle I\to B}
and carries the compact-open topology.
The pathspace fibration is given by the mapping
p
:
E
f
→
B
{\displaystyle p\colon E_{f}\to B}
with
p
(
a
,
γ
)
=
γ
(
1
)
.
{\displaystyle p(a,\gamma )=\gamma (1).}
The fiber
F
f
{\displaystyle F_{f}}
is also called the homotopy fiber of
f
{\displaystyle f}
and consists of the pairs
(
a
,
γ
)
{\displaystyle (a,\gamma )}
with
a
∈
A
{\displaystyle a\in A}
and paths
γ
:
[
0
,
1
]
→
B
,
{\displaystyle \gamma \colon [0,1]\to B,}
where
γ
(
0
)
=
f
(
a
)
{\displaystyle \gamma (0)=f(a)}
and
γ
(
1
)
=
b
0
∈
B
{\displaystyle \gamma (1)=b_{0}\in B}
holds.
For the special case of the inclusion of the base point
i
:
b
0
→
B
{\displaystyle i\colon b_{0}\to B}
, an important example of the pathspace fibration emerges. The total space
E
i
{\displaystyle E_{i}}
consists of all paths in
B
{\displaystyle B}
which starts at
b
0
.
{\displaystyle b_{0}.}
This space is denoted by
P
B
{\displaystyle PB}
and is called path space. The pathspace fibration
p
:
P
B
→
B
{\displaystyle p\colon PB\to B}
maps each path to its endpoint, hence the fiber
p
−
1
(
b
0
)
{\displaystyle p^{-1}(b_{0})}
consists of all closed paths. The fiber is denoted by
Ω
B
{\displaystyle \Omega B}
and is called loop space.: 407-408
Properties
The fibers
p
−
1
(
b
)
{\displaystyle p^{-1}(b)}
over
b
∈
B
{\displaystyle b\in B}
are homotopy equivalent for each path component of
B
.
{\displaystyle B.}
: 405
For a homotopy
f
:
[
0
,
1
]
×
A
→
B
{\displaystyle f\colon [0,1]\times A\to B}
the pullback fibrations
f
0
∗
(
E
)
→
A
{\displaystyle f_{0}^{*}(E)\to A}
and
f
1
∗
(
E
)
→
A
{\displaystyle f_{1}^{*}(E)\to A}
are fiber homotopy equivalent.: 406
If the base space
B
{\displaystyle B}
is contractible, then the fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
is fiber homotopy equivalent to the product fibration
B
×
F
→
B
.
{\displaystyle B\times F\to B.}
: 406
The pathspace fibration of a fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
is very similar to itself. More precisely, the inclusion
E
↪
E
p
{\displaystyle E\hookrightarrow E_{p}}
is a fiber homotopy equivalence.: 408
For a fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
{\displaystyle F}
and contractible total space, there is a weak homotopy equivalence
F
→
Ω
B
.
{\displaystyle F\to \Omega B.}
: 408
Puppe sequence
For a fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
{\displaystyle F}
and base point
b
0
∈
B
{\displaystyle b_{0}\in B}
the inclusion
F
↪
F
p
{\displaystyle F\hookrightarrow F_{p}}
of the fiber into the homotopy fiber is a homotopy equivalence. The mapping
i
:
F
p
→
E
{\displaystyle i\colon F_{p}\to E}
with
i
(
e
,
γ
)
=
e
{\displaystyle i(e,\gamma )=e}
, where
e
∈
E
{\displaystyle e\in E}
and
γ
:
I
→
B
{\displaystyle \gamma \colon I\to B}
is a path from
p
(
e
)
{\displaystyle p(e)}
to
b
0
{\displaystyle b_{0}}
in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration
P
B
→
B
{\displaystyle PB\to B}
along
p
{\displaystyle p}
. This procedure can now be applied again to the fibration
i
{\displaystyle i}
and so on. This leads to a long sequence:
⋯
→
F
j
→
F
i
→
j
F
p
→
i
E
→
p
B
.
{\displaystyle \cdots \to F_{j}\to F_{i}\xrightarrow {j} F_{p}\xrightarrow {i} E\xrightarrow {p} B.}
The fiber of
i
{\displaystyle i}
over a point
e
0
∈
p
−
1
(
b
0
)
{\displaystyle e_{0}\in p^{-1}(b_{0})}
consists of the pairs
(
e
0
,
γ
)
{\displaystyle (e_{0},\gamma )}
where
γ
{\displaystyle \gamma }
is a path from
p
(
e
0
)
=
b
0
{\displaystyle p(e_{0})=b_{0}}
to
b
0
{\displaystyle b_{0}}
, i.e. the loop space
Ω
B
{\displaystyle \Omega B}
. The inclusion
Ω
B
↪
F
i
{\displaystyle \Omega B\hookrightarrow F_{i}}
of the fiber of
i
{\displaystyle i}
into the homotopy fiber of
i
{\displaystyle i}
is again a homotopy equivalence and iteration yields the sequence:
⋯
Ω
2
B
→
Ω
F
→
Ω
E
→
Ω
B
→
F
→
E
→
B
.
{\displaystyle \cdots \Omega ^{2}B\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 Puppe sequences or the sequences of fibrations and cofibrations.: 407-409
Principal fibration
A fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
{\displaystyle 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 towers.: 412
Long exact sequence of homotopy groups
For a Serre fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
there exists a long exact sequence of homotopy groups. For base points
b
0
∈
B
{\displaystyle b_{0}\in B}
and
x
0
∈
F
=
p
−
1
(
b
0
)
{\displaystyle x_{0}\in F=p^{-1}(b_{0})}
this is given by:
⋯
→
π
n
(
F
,
x
0
)
→
π
n
(
E
,
x
0
)
→
π
n
(
B
,
b
0
)
→
π
n
−
1
(
F
,
x
0
)
→
{\displaystyle \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 }
⋯
→
π
0
(
F
,
x
0
)
→
π
0
(
E
,
x
0
)
.
{\displaystyle \cdots \rightarrow \pi _{0}(F,x_{0})\rightarrow \pi _{0}(E,x_{0}).}
The homomorphisms
π
n
(
F
,
x
0
)
→
π
n
(
E
,
x
0
)
{\displaystyle \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})}
and
π
n
(
E
,
x
0
)
→
π
n
(
B
,
b
0
)
{\displaystyle \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})}
are the induced homomorphisms of the inclusion
i
:
F
↪
E
{\displaystyle i\colon F\hookrightarrow E}
and the projection
p
:
E
→
B
.
{\displaystyle p\colon E\rightarrow B.}
: 376
= Hopf fibration
=Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:
S
0
↪
S
1
→
S
1
,
{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1},}
S
1
↪
S
3
→
S
2
,
{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2},}
S
3
↪
S
7
→
S
4
,
{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4},}
S
7
↪
S
15
→
S
8
.
{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}.}
The long exact sequence of homotopy groups of the hopf fibration
S
1
↪
S
3
→
S
2
{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}}
yields:
⋯
→
π
n
(
S
1
,
x
0
)
→
π
n
(
S
3
,
x
0
)
→
π
n
(
S
2
,
b
0
)
→
π
n
−
1
(
S
1
,
x
0
)
→
{\displaystyle \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 }
⋯
→
π
1
(
S
1
,
x
0
)
→
π
1
(
S
3
,
x
0
)
→
π
1
(
S
2
,
b
0
)
.
{\displaystyle \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
{\displaystyle S^{1}}
in
S
3
{\displaystyle S^{3}}
is contractible to a point:
0
→
π
i
(
S
3
)
→
π
i
(
S
2
)
→
π
i
−
1
(
S
1
)
→
0.
{\displaystyle 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
ϕ
:
π
i
−
1
(
S
1
)
→
π
i
(
S
2
)
{\displaystyle \phi \colon \pi _{i-1}(S^{1})\to \pi _{i}(S^{2})}
and there are isomorphisms:
π
i
(
S
2
)
≅
π
i
(
S
3
)
⊕
π
i
−
1
(
S
1
)
.
{\displaystyle \pi _{i}(S^{2})\cong \pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}
The homotopy groups
π
i
−
1
(
S
1
)
{\displaystyle \pi _{i-1}(S^{1})}
are trivial for
i
≥
3
,
{\displaystyle i\geq 3,}
so there exist isomorphisms between
π
i
(
S
2
)
{\displaystyle \pi _{i}(S^{2})}
and
π
i
(
S
3
)
{\displaystyle \pi _{i}(S^{3})}
for
i
≥
3.
{\displaystyle i\geq 3.}
Analog the fibers
S
3
{\displaystyle S^{3}}
in
S
7
{\displaystyle S^{7}}
and
S
7
{\displaystyle S^{7}}
in
S
15
{\displaystyle S^{15}}
are contractible to a point. Further the short exact sequences split and there are families of isomorphisms:: 111
π
i
(
S
4
)
≅
π
i
(
S
7
)
⊕
π
i
−
1
(
S
3
)
{\displaystyle \pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}
and
π
i
(
S
8
)
≅
π
i
(
S
15
)
⊕
π
i
−
1
(
S
7
)
.
{\displaystyle \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
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
,
{\displaystyle F,}
where the base space is a path connected CW-complex, and an additive homology theory
G
∗
{\displaystyle G_{*}}
there exists a spectral sequence:: 242
H
k
(
B
;
G
q
(
F
)
)
≅
E
k
,
q
2
⟹
G
k
+
q
(
E
)
.
{\displaystyle H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\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
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
,
{\displaystyle F,}
where base space and fiber are path connected, the fundamental group
π
1
(
B
)
{\displaystyle \pi _{1}(B)}
acts trivially on
H
∗
(
F
)
{\displaystyle H_{*}(F)}
and in addition the conditions
H
p
(
B
)
=
0
{\displaystyle H_{p}(B)=0}
for
0
<
p
<
m
{\displaystyle 0
and
H
q
(
F
)
=
0
{\displaystyle H_{q}(F)=0}
for
0
<
q
<
n
{\displaystyle 0
hold, an exact sequence exists (also known under the name Serre exact sequence):
H
m
+
n
−
1
(
F
)
→
i
∗
H
m
+
n
−
1
(
E
)
→
f
∗
H
m
+
n
−
1
(
B
)
→
τ
H
m
+
n
−
2
(
F
)
→
i
∗
⋯
→
f
∗
H
1
(
B
)
→
0.
{\displaystyle 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.}
: 250 This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form
Ω
S
n
:
{\displaystyle \Omega S^{n}:}
: 162
H
k
(
Ω
S
n
)
=
{
Z
∃
q
∈
Z
:
k
=
q
(
n
−
1
)
0
otherwise
.
{\displaystyle H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&{\text{otherwise}}\end{cases}}.}
For the special case of a fibration
p
:
E
→
S
n
{\displaystyle p\colon E\to S^{n}}
where the base space is a
n
{\displaystyle n}
-sphere with fiber
F
,
{\displaystyle F,}
there exist exact sequences (also called Wang sequences) for homology and cohomology:: 456
⋯
→
H
q
(
F
)
→
i
∗
H
q
(
E
)
→
H
q
−
n
(
F
)
→
H
q
−
1
(
F
)
→
⋯
{\displaystyle \cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots }
⋯
→
H
q
(
E
)
→
i
∗
H
q
(
F
)
→
H
q
−
n
+
1
(
F
)
→
H
q
+
1
(
E
)
→
⋯
{\displaystyle \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
:
E
→
B
{\displaystyle p\colon E\to B}
with fiber
F
{\displaystyle F}
and a fixed commutative ring
R
{\displaystyle R}
with a unit, there exists a contravariant functor from the fundamental groupoid of
B
{\displaystyle B}
to the category of graded
R
{\displaystyle R}
-modules, which assigns to
b
∈
B
{\displaystyle b\in B}
the module
H
∗
(
F
b
,
R
)
{\displaystyle H_{*}(F_{b},R)}
and to the path class
[
ω
]
{\displaystyle [\omega ]}
the homomorphism
h
[
ω
]
∗
:
H
∗
(
F
ω
(
0
)
,
R
)
→
H
∗
(
F
ω
(
1
)
,
R
)
,
{\displaystyle h[\omega ]_{*}\colon H_{*}(F_{\omega (0)},R)\to H_{*}(F_{\omega (1)},R),}
where
h
[
ω
]
{\displaystyle h[\omega ]}
is a homotopy class in
[
F
ω
(
0
)
,
F
ω
(
1
)
]
.
{\displaystyle [F_{\omega (0)},F_{\omega (1)}].}
A fibration is called orientable over
R
{\displaystyle R}
if for any closed path
ω
{\displaystyle \omega }
in
B
{\displaystyle B}
the following holds:
h
[
ω
]
∗
=
1.
{\displaystyle h[\omega ]_{*}=1.}
: 476
Euler characteristic
For an orientable fibration
p
:
E
→
B
{\displaystyle p\colon E\to B}
over the field
K
{\displaystyle \mathbb {K} }
with fiber
F
{\displaystyle F}
and path connected base space, the Euler characteristic of the total space is given by:
χ
(
E
)
=
χ
(
B
)
χ
(
F
)
.
{\displaystyle \chi (E)=\chi (B)\chi (F).}
Here the Euler characteristics of the base space and the fiber are defined over the field
K
{\displaystyle \mathbb {K} }
.: 481
See also
Approximate fibration
References
Kata Kunci Pencarian:
- Fibration
- Hopf fibration
- Approximate fibration
- G-fibration
- Sharpey's fibres
- Ehresmann's lemma
- Homotopy fiber
- Optical fiber
- Homotopy group
- Wood fibre
