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
If the signature of a manifold with dimension not divisible by 4 is taken to be 0, this defines a genus: the signature genus.
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$L_0 = 1$
$L_1 = p_1 / 3$
$L_2 = (7 p_2 - (p_1)^2) / 45$
manifold dimension | invariant | quadratic form | quadratic refinement |
---|---|---|---|
$4k$ | signature genus | intersection pairing | integral Wu structure |
$4k+2$ | Kervaire invariant | framing |