- Source: Signature (topology)
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem.
Definition
Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group
H
2
k
(
M
,
R
)
{\displaystyle H^{2k}(M,\mathbf {R} )}
.
The basic identity for the cup product
α
p
⌣
β
q
=
(
−
1
)
p
q
(
β
q
⌣
α
p
)
{\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}
shows that with p = q = 2k the product is symmetric. It takes values in
H
4
k
(
M
,
R
)
{\displaystyle H^{4k}(M,\mathbf {R} )}
.
If we assume also that M is compact, Poincaré duality identifies this with
H
0
(
M
,
R
)
{\displaystyle H^{0}(M,\mathbf {R} )}
which can be identified with
R
{\displaystyle \mathbf {R} }
. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature
σ
(
M
)
{\displaystyle \sigma (M)}
of M is by definition the signature of Q, that is,
σ
(
M
)
=
n
+
−
n
−
{\displaystyle \sigma (M)=n_{+}-n_{-}}
where any diagonal matrix defining Q has
n
+
{\displaystyle n_{+}}
positive entries and
n
−
{\displaystyle n_{-}}
negative entries. If M is not connected, its signature is defined to be the sum of the signatures of its connected components.
Other dimensions
If M has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in L-theory: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group
L
4
k
,
{\displaystyle L^{4k},}
or as the 4k-dimensional quadratic L-group
L
4
k
,
{\displaystyle L_{4k},}
and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of
Z
/
2
{\displaystyle \mathbf {Z} /2}
) for framed manifolds of dimension 4k+2 (the quadratic L-group
L
4
k
+
2
{\displaystyle L_{4k+2}}
), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group
L
4
k
+
1
{\displaystyle L^{4k+1}}
); the other dimensional L-groups vanish.
= Kervaire invariant
=When
d
=
4
k
+
2
=
2
(
2
k
+
1
)
{\displaystyle d=4k+2=2(2k+1)}
is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.
Properties
Compact oriented manifolds M and N satisfy
σ
(
M
⊔
N
)
=
σ
(
M
)
+
σ
(
N
)
{\displaystyle \sigma (M\sqcup N)=\sigma (M)+\sigma (N)}
by definition, and satisfy
σ
(
M
×
N
)
=
σ
(
M
)
σ
(
N
)
{\displaystyle \sigma (M\times N)=\sigma (M)\sigma (N)}
by a Künneth formula.
If M is an oriented boundary, then
σ
(
M
)
=
0
{\displaystyle \sigma (M)=0}
.
René Thom (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers. For example, in four dimensions, it is given by
p
1
3
{\displaystyle {\frac {p_{1}}{3}}}
. Friedrich Hirzebruch (1954) found an explicit expression for this linear combination as the L genus of the manifold.
William Browder (1962) proved that a simply connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.
Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.
See also
Hirzebruch signature theorem
Genus of a multiplicative sequence
Rokhlin's theorem
References
Kata Kunci Pencarian:
- Keamanan komputer
- Modul Clifford
- Stephen Hawking
- Henri Poincaré
- Signature (topology)
- Signature (disambiguation)
- List of geometric topology topics
- William Browder (mathematician)
- Interior algebra
- Hirzebruch signature theorem
- Pseudometric space
- Quadratic form
- Signature of a knot
- Circuit topology
