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; [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 $ A = \oplus_{n=0}^ \infty A ^ {n} $ be a commutative graded algebra over a commutative ring $ R $ with unit. Let $ \Lambda ( A) $ denote the group of all elements $ 1 + a _ {1} + a _ {2} + \dots $, $ a _ {i} \in A ^ {i} $, under the obvious multiplication of such expressions:

$$ ( 1+ a _ {1} + a _ {2} + \dots )( 1+ b _ {1} + b _ {2} + \dots ) = $$

$$ = \ 1 + ( a _ {1} + b _ {1} ) + ( a _ {2} + a _ {1} b _ {1} + b _ {2} ) + \dots . $$

A sequence $ \{ K _ {n} \} $ of polynomials $ K _ {1} ( x _ {1} ) $, $ K _ {2} ( x _ {1} , x _ {2} ) \dots $ with coefficients in $ R $ is called a multiplicative sequence of polynomials if each $ K _ {i} $ is homogeneous of degree $ i $ and if for each $ A $ the mapping $ K : ( 1+ a _ {1} + a _ {2} + \dots ) \mapsto ( 1+ K _ {1} ( a _ {2} ) + K _ {2} ( a _ {1} , a _ {2} ) + \dots ) $ defines a group homomorphism from $ \Lambda ( A) $ to $ \Lambda ( A) $. Given a power series $ f( t) \in R [[ t]] $ with constant term $ 1 $, there is precisely one multiplicative sequence $ \{ K _ {n} \} $ over $ R $ such that $ K( 1+ t) = f ( t) $. This multiplicative sequence is called the multiplicative sequence defined by the power series $ f( t) $.

Now, let $ \{ L _ {n} \} $ be the multiplicative sequence defined by the power series

$$ \frac{\sqrt t }{ \mathop{\rm tanh} ( \sqrt t ) } = $$

$$ = \ 1 + \frac{1}{3} t - \frac{1}{45} t ^ {2} + \dots + (- 1) ^ {k-1} 2 ^ {2 ^ {k} } B _ {k} \frac{t ^ {k} }{(2k)!} + \dots , $$

where $B_k$ is the $k$-th Bernoulli number (cf. Bernoulli numbers). The $L$-genus of a manifold $M$ of dimension $4m$ is defined by

$$ L ( M ^ {4m} ) = \langle L _ {m} ( p _ {1} \dots p _ {m} ) , [ M] \rangle , $$

where $ [ M] $ is the fundamental homology class of $ M $ and $ p _ {i} $ is the $ i $- th Pontryagin class. One sets $ L( M) = 0 $ if the dimension of $ M $ is not a multiple of $ 4 $. The Hirzebruch signature theorem now says that the $ L $- 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.