nLab
signature genus

Contents

Idea

The signature of an oriented manifold XX of dimension 4k4 k, kk \in \mathbb{N} is the signature of the quadratic form which is the intersection pairing in integral cohomology

H 2k(X,)×H 2k(X,)H 4k(X,),X 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=1L_0 = 1

  • L 1=p 1/3L_1 = p_1 / 3

  • L 2=(7p 2(p 1) 2)/45L_2 = (7 p_2 - (p_1)^2) / 45

manifold dimensioninvariantquadratic formquadratic refinement
4k4ksignature genusintersection pairingintegral Wu structure
4k+24k+2Kervaire invariantframing

References

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