Signature (topology)

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

${\displaystyle H^{2k}(M,\mathbf {R} )}$.

The basic identity for the cup product

${\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}$

shows that with p = q = 2k the product is symmetric. It takes values in

${\displaystyle H^{4k}(M,\mathbf {R} )}$.

If we assume also that M is compact, Poincaré duality identifies this with

${\displaystyle H^{0}(M,\mathbf {R} )}$

which can be identified with ${\displaystyle \mathbf {R} }$. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H^2k(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 ${\displaystyle \sigma (M)}$ of M is by definition the signature of Q, that is, ${\displaystyle \sigma (M)=n_{+}-n_{-}}$ where any diagonal matrix defining Q has ${\displaystyle n_{+}}$ positive entries and ${\displaystyle n_{-}}$ negative entries.[2] If M is not connected, its signature is defined to be the sum of the signatures of its connected components.

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 ${\displaystyle L^{4k},}$ or as the 4k-dimensional quadratic L-group ${\displaystyle L_{4k},}$ and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of ${\displaystyle \mathbf {Z} /2}$) for framed manifolds of dimension 4k+2 (the quadratic L-group ${\displaystyle L_{4k+2}}$), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group ${\displaystyle L^{4k+1}}$); the other dimensional L-groups vanish.

• Compact oriented manifolds M and N satisfy ${\displaystyle \sigma (M\sqcup N)=\sigma (M)+\sigma (N)}$ by definition, and satisfy ${\displaystyle \sigma (M\times N)=\sigma (M)\sigma (N)}$ by a Künneth formula.

• If M is an oriented boundary, then ${\displaystyle \sigma (M)=0}$.

• 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.[3] For example, in four dimensions, it is given by ${\displaystyle {\frac {p_{1}}{3}}}$. 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.

• Hirzebruch signature theorem
• Genus of a multiplicative sequence
• Rokhlin's theorem