nLab
signature genus

Idea

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

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

(…)

• $L_0=1$

• $L_1=p_1/3$

• $L_2=(7p_2-(p_1{)}^2)/45$

Related concepts

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

References

• Wikipedia, L genus and the Hirzebruch signature theorem