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The signature of an algebraic system is the collection of relations and operations on the basic set of the given algebraic system together with an indication of their arity. An algebraic system (a universal algebra) with signature is also called an -system (respectively, -algebra).

The signature of a quadratic, or symmetric bilinear, form over an ordered field is a pair of non-negative integers , where is the positive and the negative index of inertia of the given form (see Law of inertia; Quadratic form). Sometimes the number is called the signature of the form.

O.A. Ivanova

The signature of a manifold is the signature of the quadratic form

where is the cohomology cup-product and is the fundamental class. The manifold is assumed to be compact, orientable and of dimension . The signature is denoted by .

If , one sets . The signature has the following properties:

a) ;

b) ;

c) .

The signature of a manifold can be represented as a linear function of its Pontryagin numbers (cf. Pontryagin number; [2]). For the representation of the signature as the index of a differential operator see Index formulas.

References

[1] A. Dold, "Lectures on algebraic topology" , Springer (1980)
[2] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)

M.I. Voitsekhovskii

Comments

Let be a commutative graded algebra over a commutative ring with unit. Let denote the group of all elements , , under the obvious multiplication of such expressions:

A sequence of polynomials , with coefficients in is called a multiplicative sequence of polynomials if each is homogeneous of degree and if for each the mapping defines a group homomorphism from to . Given a power series with constant term , there is precisely one multiplicative sequence over such that . This multiplicative sequence is called the multiplicative sequence defined by the power series .

Now, let be the multiplicative sequence defined by the power series

where is the -th Bernoulli number (cf. Bernoulli numbers). The -genus of a manifold of dimension is defined by

where is the fundamental homology class of and is the -th Pontryagin class. One sets if the dimension of is not a multiple of . The Hirzebruch signature theorem now says that the -genus of a manifold is equal to its signature [2], §19.

In some of the older literature the signature of a manifold is referred to as the index of a manifold.

How to Cite This Entry:
Signature. O.A. Ivanova, M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Signature&oldid=16291
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098