Misplaced Pages

This is an old revision of this page, as edited by Dave.Dunford (talk | contribs) at 16:43, 11 September 2022 (no hyphen after -ly). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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

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

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(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 of M is by definition the signature of Q, an ordered triple according to its definition. 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 ${\displaystyle L^{4k},}$ 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.

Kervaire invariant

When ${\displaystyle d=4k+2=2(2k+1)}$ 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

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 ${\displaystyle {\frac {p_{1}}{3}}}$. 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

1. Milnor, John; Stasheff, James (1962). Characteristic classes. Annals of Mathematics Studies 246. p. 224. CiteSeerX 10.1.1.448.869. ISBN 978-0691081229.
2. Hatcher, Allen (2003). Algebraic topology (PDF) (Repr. ed.). Cambridge: Cambridge Univ. Pr. p. 250. ISBN 978-0521795401. Retrieved 8 January 2017.
3. Thom, René. "Quelques proprietes globales des varietes differentiables" (PDF) (in French). Comm. Math. Helvetici 28 (1954), S. 17–86. Retrieved 26 October 2019.

• Geometric topology
• Quadratic forms