Signature (topology)
Template:Short description 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.
DefinitionEdit
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
- <math>H^{2k}(M,\mathbf{R})</math>.
The basic identity for the cup product
- <math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)</math>
shows that with p = q = 2k the product is symmetric. It takes values in
- <math>H^{4k}(M,\mathbf{R})</math>.
If we assume also that M is compact, Poincaré duality identifies this with
- <math>H_{0}(M,\mathbf{R})</math>
which can be identified with <math>\mathbf{R}</math>. 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.<ref>Template:Cite book</ref> More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature <math>\sigma(M)</math> of M is by definition the signature of Q, that is, <math>\sigma(M) = n_+ - n_-</math> where any diagonal matrix defining Q has <math>n_+</math> positive entries and <math>n_-</math> negative entries.<ref>Template:Cite book</ref> If M is not connected, its signature is defined to be the sum of the signatures of its connected components.
Other dimensionsEdit
Template:Details 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 <math>L^{4k},</math> or as the 4k-dimensional quadratic L-group <math>L_{4k},</math> and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of <math>\mathbf{Z}/2</math>) for framed manifolds of dimension 4k+2 (the quadratic L-group <math>L_{4k+2}</math>), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group <math>L^{4k+1}</math>); the other dimensional L-groups vanish.
Kervaire invariantEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} When <math>d=4k+2=2(2k+1)</math> 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.
PropertiesEdit
- Compact oriented manifolds M and N satisfy <math>\sigma(M \sqcup N) = \sigma(M) + \sigma(N)</math> by definition, and satisfy <math>\sigma(M\times N) = \sigma(M)\sigma(N)</math> by a Künneth formula.
- If M is an oriented boundary, then <math>\sigma(M)=0</math>.
- 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.<ref>Template:Cite news</ref> For example, in four dimensions, it is given by <math>\frac{p_1}{3}</math>. 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.