# Signature

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 $\Omega$ is also called an $\Omega$- system (respectively, $\Omega$- algebra).

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

O.A. Ivanova

The signature of a manifold $M ^ {n}$ is the signature of the quadratic form

$$Q _ {M} ( x) = \ ( x \cup x, O),$$

where $\cup$ is the cohomology cup-product and $O \in H _ {n} ( M; Z)$ is the fundamental class. The manifold is assumed to be compact, orientable and of dimension $n = 4m$. The signature is denoted by $\sigma ( M)$.

If $n \not\equiv 0$ $\mathop{\rm mod} 4$, one sets $\sigma ( M) = 0$. The signature has the following properties:

a) $\sigma ( M + M ^ \prime ) = \sigma ( M) + \sigma ( M ^ \prime )$;

b) $\sigma ( M \times M ^ \prime ) = \sigma ( M) \sigma ( M ^ \prime )$;

c) $\sigma ( \partial M) = 0$.

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

How to Cite This Entry:
Signature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Signature&oldid=48695
This article was adapted from an original article by O.A. Ivanova, M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article