nLab signature genus

Idea

The signature of an oriented manifold $X$ of dimension $4 k$, $k \in \mathbb{N}$ is the signature of the quadratic form which is the intersection pairing in integral cohomology

$$H^{2k}(X,\mathbb{Z}) \times H^{2k}(X,\mathbb{Z}) \stackrel{-\cup-}{\to} H^{4k}(X, \mathbb{Z}) \stackrel{\langle -,X\rangle}{\to} \mathbb{Z}$$

If the signature of a manifold with dimension not divisible by 4 is taken to be 0.

(Milnor-Stasheff 74, p. 224)

This defines a genus: the signature genus.

Properties

(…)

• $L_0 = 1$

• $L_1 = p_1 / 3$

• $L_2 = (7 p_2 - (p_1)^2) / 45$

Related concepts

• Hirzebruch signature theorem

manifold dimension	invariant	quadratic form	quadratic refinement
$4k$	signature genus	intersection pairing	integral Wu structure
$4k+2$	Kervaire invariant		framing

References

• John Milnor, James Stasheff, p. 224 of Characteristic classes, Princeton University Press, 1974

See also

• Wikipedia, L genus and the Hirzebruch signature theorem