In algebraic topology , a branch of mathematics , the cap product defines a link between the cohomology and homology of a space .
definition
Let be a topological space , let the -th singular chain group, i.e. the free Abelian group over the set of all continuous maps of the standard simplex after and . Use or to denote the inclusions of the standard or simplex as the “front- dimensional side” or “rear- dimensional side” in the standard simplex.
X
{\ displaystyle X}
C.
n
(
X
)
{\ displaystyle C_ {n} (X)}
n
{\ displaystyle n}
n
{\ displaystyle n}
Δ
n
{\ displaystyle \ Delta ^ {n}}
X
{\ displaystyle X}
C.
n
(
X
)
=
H
O
m
(
C.
n
(
X
)
,
Z
)
{\ displaystyle C ^ {n} (X) = Hom (C_ {n} (X), \ mathbb {Z})}
ι
0
...
p
:
Δ
p
→
Δ
p
+
q
{\ displaystyle \ iota _ {0 \ ldots p} \ colon \ Delta ^ {p} \ rightarrow \ Delta ^ {p + q}}
ι
p
...
p
+
q
:
Δ
q
→
Δ
p
+
q
{\ displaystyle \ iota _ {p \ ldots p + q} \ colon \ Delta ^ {q} \ rightarrow \ Delta ^ {p + q}}
p
{\ displaystyle p}
q
{\ displaystyle q}
p
{\ displaystyle p}
q
{\ displaystyle q}
(
p
+
q
)
{\ displaystyle (p + q)}
For and a singular simplex (with ) is defined
ψ
∈
C.
q
(
X
)
{\ displaystyle \ psi \ in C ^ {q} (X)}
σ
:
Δ
p
→
X
{\ displaystyle \ sigma: \ Delta ^ {p} \ rightarrow X}
p
≥
q
{\ displaystyle p \ geq q}
σ
⌢
ψ
: =
(
-
1
)
p
q
ψ
(
σ
∘
ι
0
...
q
)
σ
∘
ι
q
...
p
{\ displaystyle \ sigma \ frown \ psi: = (- 1) ^ {pq} \ psi (\ sigma \ circ \ iota _ {0 \ ldots q}) \ sigma \ circ \ iota _ {q \ ldots p}}
and sets this linearly to a mapping
C.
q
(
X
)
×
C.
p
(
X
)
→
C.
p
-
q
(
X
)
{\ displaystyle C ^ {q} (X) \ times C_ {p} (X) \ rightarrow C_ {pq} (X)}
away.
More generally be a ring and be . Then you get a picture
R.
{\ displaystyle R}
C.
n
(
X
;
R.
)
=
C.
n
(
X
)
⊗
Z
R.
,
C.
n
(
X
)
=
H
O
m
(
C.
n
(
X
)
,
R.
)
{\ displaystyle C_ {n} (X; R) = C_ {n} (X) \ otimes _ {\ mathbb {Z}} R, C ^ {n} (X) = Hom (C_ {n} (X) , R)}
C.
q
(
X
;
R.
)
×
C.
p
(
X
;
R.
)
→
C.
p
-
q
(
X
;
R.
)
{\ displaystyle C ^ {q} (X; R) \ times C_ {p} (X; R) \ rightarrow C_ {pq} (X; R)}
.
From the relation
∂
(
σ
⌢
ψ
)
=
(
-
1
)
q
(
∂
σ
⌢
ψ
-
σ
⌢
δ
ψ
)
{\ displaystyle \ partial (\ sigma \ frown \ psi) = (- 1) ^ {q} (\ partial \ sigma \ frown \ psi - \ sigma \ frown \ delta \ psi)}
it follows that the cap product is a well-defined map
H
q
(
X
;
R.
)
×
H
p
(
X
;
R.
)
→
H
p
-
q
(
X
;
R.
)
{\ displaystyle H ^ {q} (X; R) \ times H_ {p} (X; R) \ rightarrow H_ {pq} (X; R)}
Are defined.
properties
The following applies to
continuous images
f
:
X
→
Y
{\ displaystyle f: X \ rightarrow Y}
f
∗
(
c
)
⌢
ψ
=
f
∗
(
c
⌢
f
∗
(
ψ
)
)
{\ displaystyle f _ {*} (c) \ frown \ psi = f _ {*} (c \ frown f ^ {*} (\ psi))}
with , .
c
∈
C.
p
(
X
;
R.
)
{\ displaystyle c \ in C_ {p} (X; R)}
ψ
∈
C.
q
(
Y
;
R.
)
{\ displaystyle \ psi \ in C ^ {q} (Y; R)}
The cap product is related to the cup product by the following equation:
ψ
(
c
⌢
φ
)
=
(
φ
⌣
ψ
)
(
c
)
{\ displaystyle \ psi (c \ frown \ varphi) = (\ varphi \ smile \ psi) (c)}
for , ,
c
∈
C.
p
(
X
;
R.
)
{\ displaystyle c \ in C_ {p} (X; R)}
ψ
∈
C.
q
(
X
;
R.
)
{\ displaystyle \ psi \ in C ^ {q} (X; R)}
φ
∈
C.
p
-
q
(
X
;
R.
)
.
{\ displaystyle \ varphi \ in C ^ {pq} (X; R).}
Application: Poincaré duality
Be a closed, orientable -manifold and
M.
{\ displaystyle M}
n
{\ displaystyle n}
[
M.
]
∈
H
n
(
M.
;
Z
)
{\ displaystyle \ left [M \ right] \ in H_ {n} (M; \ mathbb {Z})}
the fundamental class . Then the Cap product realizes with an isomorphism
[
M.
]
{\ displaystyle \ left [M \ right]}
H
k
(
M.
;
Z
)
→
H
n
-
k
(
M.
;
Z
)
{\ displaystyle H ^ {k} (M; \ mathbb {Z}) \ rightarrow H_ {nk} (M; \ mathbb {Z})}
for .
k
=
0
,
...
,
n
{\ displaystyle k = 0, \ ldots, n}
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">