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Revision as of 06:54, 13 January 2022 editJohn Baez (talk \| contribs)Extended confirmed users2,378 edits →Properties: added mention of Rokhlin's theorem, since it's an interesting property that's quick to state.← Previous edit		Revision as of 16:43, 11 September 2022 edit undoDave.Dunford (talk \| contribs)Extended confirmed users48,567 editsm no hyphen after -lyNext edit →
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	== Other dimensions ==		== Other dimensions ==
	{{details\|L-theory}}		{{details\|L-theory}}
	If ''M'' has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in ]: the signature can be interpreted as the 4''k''-dimensional (simply-connected) symmetric L-group <math>L^{4k},</math> or as the 4''k''-dimensional quadratic L-group <math>L_{4k},</math> and these invariants do not always vanish for other dimensions. The ] is a mod 2 (i.e., an element of <math>\mathbf{Z}/2</math>) for framed manifolds of dimension 4''k''+2 (the quadratic L-group <math>L_{4k+2}</math>), while the ] is a mod 2 invariant of manifolds of dimension 4''k''+1 (the symmetric L-group <math>L^{4k+1}</math>); the other dimensional L-groups vanish.		If ''M'' has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in ]: the signature can be interpreted as the 4''k''-dimensional (simply connected) symmetric L-group <math>L^{4k},</math> or as the 4''k''-dimensional quadratic L-group <math>L_{4k},</math> and these invariants do not always vanish for other dimensions. The ] is a mod 2 (i.e., an element of <math>\mathbf{Z}/2</math>) for framed manifolds of dimension 4''k''+2 (the quadratic L-group <math>L_{4k+2}</math>), while the ] is a mod 2 invariant of manifolds of dimension 4''k''+1 (the symmetric L-group <math>L^{4k+1}</math>); the other dimensional L-groups vanish.

	=== Kervaire invariant ===		=== Kervaire invariant ===
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	== Properties ==		== Properties ==
	] (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] ].<ref>{{cite news\|last1=Thom\|first1=René\|title=Quelques proprietes globales des varietes differentiables\|publisher=Comm. Math. Helvetici 28 (1954), S. 17–86\|url=https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf\|accessdate=26 October 2019\|language=fr}}</ref> For example, in four dimensions, it is given by <math>\frac{p_1}{3}</math>. ] (1954) found an explicit expression for this linear combination as the ] of the manifold. ] (1962) proved that a simply-connected compact ] with 4''n''-dimensional ] is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the ]. ] says that the signature of a 4-dimensional simply-connected manifold with a ] is divisible by 16.		] (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its ] ].<ref>{{cite news\|last1=Thom\|first1=René\|title=Quelques proprietes globales des varietes differentiables\|publisher=Comm. Math. Helvetici 28 (1954), S. 17–86\|url=https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf\|accessdate=26 October 2019\|language=fr}}</ref> For example, in four dimensions, it is given by <math>\frac{p_1}{3}</math>. ] (1954) found an explicit expression for this linear combination as the ] of the manifold. ] (1962) proved that a simply connected compact ] with 4''n''-dimensional ] is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the ]. ] says that the signature of a 4-dimensional simply connected manifold with a ] is divisible by 16.

	==See also==		==See also==

Revision as of 16:43, 11 September 2022

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^{2k}(M,\mathbf {R} )

The basic identity for the cup product

\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^{4k}(M,\mathbf {R} )

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

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

which can be identified with $\mathbf {R}$ . 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

Further information: L-theory

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

Kervaire invariant

Main article: Kervaire invariant

When $d=4k+2=2(2k+1)$ 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 ${\frac {p_{1}}{3}}$ . 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.

References

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

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