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.
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