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
- [math]\displaystyle{ H^{2k}(M,\mathbf{R}) }[/math].
The basic identity for the cup product
- [math]\displaystyle{ \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]\displaystyle{ H^{4k}(M,\mathbf{R}) }[/math].
If we assume also that M is compact, Poincaré duality identifies this with
- [math]\displaystyle{ H^{0}(M,\mathbf{R}) }[/math]
which can be identified with [math]\displaystyle{ \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.[1] More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature [math]\displaystyle{ \sigma(M) }[/math] of M is by definition the signature of Q, that is, [math]\displaystyle{ \sigma(M) = n_+ - n_- }[/math] where any diagonal matrix defining Q has [math]\displaystyle{ n_+ }[/math] positive entries and [math]\displaystyle{ n_- }[/math] negative entries.[2] 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 [math]\displaystyle{ L^{4k}, }[/math] or as the 4k-dimensional quadratic L-group [math]\displaystyle{ 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]\displaystyle{ \mathbf{Z}/2 }[/math]) for framed manifolds of dimension 4k+2 (the quadratic L-group [math]\displaystyle{ 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]\displaystyle{ L^{4k+1} }[/math]); the other dimensional L-groups vanish.
Kervaire invariant
When [math]\displaystyle{ 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.
Properties
- Compact oriented manifolds M and N satisfy [math]\displaystyle{ \sigma(M \sqcup N) = \sigma(M) + \sigma(N) }[/math] by definition, and satisfy [math]\displaystyle{ \sigma(M\times N) = \sigma(M)\sigma(N) }[/math] by a Künneth formula.
- If M is an oriented boundary, then [math]\displaystyle{ \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.[3] For example, in four dimensions, it is given by [math]\displaystyle{ \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.
See also
References
- ↑ Hatcher, Allen (2003) (in en). Algebraic topology (Repr. ed.). Cambridge: Cambridge Univ. Pr.. p. 250. ISBN 978-0521795401. https://www.math.cornell.edu/~hatcher/AT/AT.pdf. Retrieved 8 January 2017.
- ↑ Milnor, John; Stasheff, James (1962) (in en). Characteristic classes. Annals of Mathematics Studies 246. p. 224. ISBN 978-0691081229.
- ↑ Thom, René. "Quelques proprietes globales des varietes differentiables" (in fr). Comm. Math. Helvetici 28 (1954), S. 17–86. https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf. Retrieved 26 October 2019.
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