Relates the homology of a fiber bundle with the homologies of its base and fiber
In mathematics , the Leray–Hirsch theorem [1] is a basic result on the algebraic topology of fiber bundles . It is named after Jean Leray and Guy Hirsch , who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula , which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the Leray spectral sequence .
Statement [ ]
Setup [ ]
Let
π
:
E
⟶
B
{\displaystyle \pi \colon E\longrightarrow B}
be a fibre bundle with fibre
F
{\displaystyle F}
. Assume that for each degree
p
{\displaystyle p}
, the singular cohomology rational vector space
H
p
(
F
)
=
H
p
(
F
;
Q
)
{\displaystyle H^{p}(F)=H^{p}(F;\mathbb {Q} )}
is finite-dimensional, and that the inclusion
ι
:
F
⟶
E
{\displaystyle \iota \colon F\longrightarrow E}
induces a surjection in rational cohomology
ι
∗
:
H
∗
(
E
)
⟶
H
∗
(
F
)
{\displaystyle \iota ^{*}\colon H^{*}(E)\longrightarrow H^{*}(F)}
.
Consider a section of this surjection
s
:
H
∗
(
F
)
⟶
H
∗
(
E
)
{\displaystyle s\colon H^{*}(F)\longrightarrow H^{*}(E)}
,
by definition, this map satisfies
ι
∗
∘
s
=
I
d
{\displaystyle \iota ^{*}\circ s=\mathrm {Id} }
.
The Leray–Hirsch isomorphism [ ]
The Leray–Hirsch theorem states that the linear map
H
∗
(
F
)
⊗
H
∗
(
B
)
⟶
H
∗
(
E
)
α
⊗
β
⟼
s
(
α
)
⌣
π
∗
(
β
)
{\displaystyle {\begin{array}{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow &H^{*}(E)\\\alpha \otimes \beta &\longmapsto &s(\alpha )\smallsmile \pi ^{*}(\beta )\end{array}}}
is an isomorphism of
H
∗
(
B
)
{\displaystyle H^{*}(B)}
-modules.
Statement in coordinates [ ]
In other words, if for every
p
{\displaystyle p}
, there exist classes
c
1
,
p
,
…
,
c
m
p
,
p
∈
H
p
(
E
)
{\displaystyle c_{1,p},\ldots ,c_{m_{p},p}\in H^{p}(E)}
that restrict, on each fiber
F
{\displaystyle F}
, to a basis of the cohomology in degree
p
{\displaystyle p}
, the map given below is then an isomorphism of
H
∗
(
B
)
{\displaystyle H^{*}(B)}
modules .
H
∗
(
F
)
⊗
H
∗
(
B
)
⟶
H
∗
(
E
)
∑
i
,
j
,
k
a
i
,
j
,
k
ι
∗
(
c
i
,
j
)
⊗
b
k
⟼
∑
i
,
j
,
k
a
i
,
j
,
k
c
i
,
j
∧
π
∗
(
b
k
)
{\displaystyle {\begin{array}{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow &H^{*}(E)\\\sum _{i,j,k}a_{i,j,k}\iota ^{*}(c_{i,j})\otimes b_{k}&\longmapsto &\sum _{i,j,k}a_{i,j,k}c_{i,j}\wedge \pi ^{*}(b_{k})\end{array}}}
where
{
b
k
}
{\displaystyle \{b_{k}\}}
is a basis for
H
∗
(
B
)
{\displaystyle H^{*}(B)}
and thus, induces a basis
{
ι
∗
(
c
i
,
j
)
⊗
b
k
}
{\displaystyle \{\iota ^{*}(c_{i,j})\otimes b_{k}\}}
for
H
∗
(
F
)
⊗
H
∗
(
B
)
.
{\displaystyle H^{*}(F)\otimes H^{*}(B).}
Notes [ ]