nLab
genus

this entry is about the notion of genus in cohomology. For classication of surfaces see genus of a surface.

Definition

For $R$ a ring, an $R$-valued genus is a ring homomorphism

$$\sigma :{\Omega }_n\to R$$\sigma : \Omega_n \to R

from the bordism ring.

The cobordism ring here may be replaced by rings of cobordisms with extra structure.

Higher category theory interpretation

The notion of genus finds its natural interpretation in higher category theory, where it is refined to a morphism of symmetric monoidal ∞-groupoids

$$\sigma :{\mathrm{Bord}}_{(\mathrm{\infty },\mathrm{\infty })}\to S$$\sigma : Bord_{(\infty,\infty)} \to S

from the (∞,n)-category of cobordisms for $n\to \mathrm{\infty }$ to a ring spectrum $S$.

See cobordism ring for more.

Examples

• The Euler characteristic $X\mapsto \chi (X)$ is close to being a genus, but is not cobordism invariant

  (this is the index of the Dirac operator $D=d+d^{†}$)

• The signature genus?;

• The A-hat genus? is the index of a Dirac operator? coming from a spin bundle?;

• The elliptic genus? or Witten genus may be interpreted as the index of a Dirac operator on loop space.