# nLab signature genus

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## 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, this defines a genus: the signature genus.

## Properties

(…)

• $L_0 = 1$

• $L_1 = p_1 / 3$

• $L_2 = (7 p_2 - (p_1)^2) / 45$

manifold dimensioninvariantquadratic formquadratic refinement
$4k$signature genusintersection pairingintegral Wu structure
$4k+2$Kervaire invariantframing

## References

Revised on June 3, 2012 22:15:23 by Urs Schreiber (131.130.246.204)