Differential characters are a term from the mathematical field of differential topology that generalizes the cohomology groups.
Secondary characteristic classes , for example the Cheeger-Chern-Simons classes of vector bundles, are differential characters. In the case of flat bundles , these are then even cohomology classes.
ℤ-valued differential characters
Let be a smooth manifold and an integer. The group of -valent differential characters of degree is
X
{\ displaystyle X}
k
≥
1
{\ displaystyle k \ geq 1}
Z
{\ displaystyle \ mathbb {Z}}
k
{\ displaystyle k}
H
^
k
(
X
;
Z
)
: =
{
H
∈
H
O
m
(
Z
k
-
1
(
X
;
Z
)
,
Z
)
∣
H
∘
∂
∈
Ω
k
(
X
)
}
{\ displaystyle {\ widehat {H}} ^ {k} (X; \ mathbb {Z}): = \ left \ {h \ in \ mathrm {Hom} (Z_ {k-1} (X; \ mathbb { Z}), \ mathbb {Z}) \ mid h \ circ \ partial \ in \ Omega ^ {k} (X) \ right \}}
.
Here the group of - denotes cycles and the notation means that there is a differential form such that
Z
k
-
1
(
X
;
Z
)
{\ displaystyle Z_ {k-1} (X; \ mathbb {Z})}
(
k
-
1
)
{\ displaystyle (k-1)}
H
∘
∂
∈
Ω
k
(
X
)
{\ displaystyle h \ circ \ partial \ in \ Omega ^ {k} (X)}
ω
∈
Ω
k
(
X
)
{\ displaystyle \ omega \ in \ Omega ^ {k} (X)}
H
(
∂
c
)
=
exp
(
2
π
i
∫
c
ω
)
{\ displaystyle h (\ partial c) = \ exp (2 \ pi i \ int _ {c} \ omega)}
holds for every smooth chain .
c
∈
C.
k
(
X
;
Z
)
{\ displaystyle c \ in C_ {k} (X; \ mathbb {Z})}
ℝ / ℤ-valued differential characters
Let be a smooth manifold and an integer. The group of -valent differential characters of degree is
X
{\ displaystyle X}
k
≥
1
{\ displaystyle k \ geq 1}
R.
/
Z
{\ displaystyle \ mathbb {R} / \ mathbb {Z}}
k
{\ displaystyle k}
H
^
k
(
X
;
R.
/
Z
)
: =
{
H
∈
H
O
m
(
Z
k
-
1
(
X
;
Z
)
,
R.
/
Z
)
∣
H
∘
∂
∈
Ω
k
(
X
)
}
{\ displaystyle {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z}): = \ left \ {h \ in \ mathrm {Hom} (Z_ {k-1} (X; \ mathbb {Z}), \ mathbb {R} / \ mathbb {Z}) \ mid h \ circ \ partial \ in \ Omega ^ {k} (X) \ right \}}
.
Here the group of -cycles denotes and the notation means that there is a differential form such that
Z
k
-
1
(
X
;
Z
)
{\ displaystyle Z_ {k-1} (X; \ mathbb {Z})}
(
k
-
1
)
{\ displaystyle (k-1)}
H
∘
∂
∈
Ω
k
(
X
)
{\ displaystyle h \ circ \ partial \ in \ Omega ^ {k} (X)}
ω
∈
Ω
k
(
X
)
{\ displaystyle \ omega \ in \ Omega ^ {k} (X)}
H
(
∂
c
)
=
∫
c
ω
m
O
d
Z
{\ displaystyle h (\ partial c) = \ int _ {c} \ omega \ \ mathrm {mod} \ \ mathbb {Z}}
holds for every smooth chain .
c
∈
C.
k
(
X
;
Z
)
{\ displaystyle c \ in C_ {k} (X; \ mathbb {Z})}
Short exact sequences
Korand illustration
You have a short exact sequence
0
→
H
k
(
X
;
R.
/
Z
)
→
H
^
k
(
X
;
R.
/
Z
)
→
A.
0
k
+
1
(
X
)
→
0
{\ displaystyle 0 \ to H ^ {k} (X; \ mathbb {R} / \ mathbb {Z}) \ to {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z}) \ to A_ {0} ^ {k + 1} (X) \ to 0}
.
Herein refers to the group of closed differential forms integral period and the image
A.
0
k
+
1
(
X
)
{\ displaystyle A_ {0} ^ {k + 1} (X)}
δ
:
H
^
k
(
X
;
R.
/
Z
)
→
A.
0
k
+
1
(
X
)
{\ displaystyle \ delta \ colon {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z}) \ to A_ {0} ^ {k + 1} (X)}
assigns the unique differential form with
too.
H
∈
H
^
k
(
X
;
R.
/
Z
)
{\ displaystyle h \ in {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
ω
∈
Ω
k
(
X
)
{\ displaystyle \ omega \ in \ Omega ^ {k} (X)}
H
(
∂
c
)
=
∫
c
ω
m
O
d
Z
∀
c
∈
C.
k
s
m
O
O
t
H
(
X
;
Z
)
{\ displaystyle h (\ partial c) = \ int _ {c} \ omega \ \ mathrm {mod} \ \ mathbb {Z} \ \ quad \ forall c \ in C_ {k} ^ {smooth} (X; \ mathbb {Z})}
In particular, one can understand as a subgroup of .
H
k
(
X
;
R.
/
Z
)
{\ displaystyle H ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
H
^
k
(
X
;
R.
/
Z
)
{\ displaystyle {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
Secondary characteristic classes of vector bundles give invariants in , which in the case of vanishing curvature even lie in.
H
^
k
(
X
;
R.
/
Z
)
{\ displaystyle {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
H
k
(
X
;
R.
/
Z
)
{\ displaystyle H ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
Bockstein homomorphism
There is a homomorphism
b
:
H
^
k
(
X
;
R.
/
Z
)
→
H
k
+
1
(
X
;
Z
)
{\ displaystyle b \ colon {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z}) \ to H ^ {k + 1} (X; \ mathbb {Z}) }
,
whose restriction is precisely the Bockstein homomorphism . It fits into an exact sequence
H
k
(
X
;
R.
/
Z
)
{\ displaystyle H ^ {k} (X; \ mathbb {R} / \ mathbb {Z})}
0
→
A.
k
(
X
)
/
A.
0
k
(
X
)
→
H
^
k
(
X
;
R.
/
Z
)
→
H
k
+
1
(
X
;
Z
)
→
0
{\ displaystyle 0 \ to A ^ {k} (X) / A_ {0} ^ {k} (X) \ to {\ widehat {H}} ^ {k} (X; \ mathbb {R} / \ mathbb {Z}) \ to H ^ {k + 1} (X; \ mathbb {Z}) \ to 0}
.
literature
Jeff Cheeger, James Simons: Differential characters and geometric invariants. Geometry and topology. In: Lecture Notes in Math. 1167, Springer, Berlin 1985, pp. 50–80.
Christian Bär, Christian Becker: Differential characters. In: Lecture Notes in Mathematics. 2112. Springer, Cham 2014, ISBN 978-3-319-07033-9 .
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