nLab
genus

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

Idea

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.

In homotopy theory / higher category theory

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

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

from the (∞,n)-category of cobordisms for $n\to \mathrm{\infty }$ to a ring spectrum $S$. This is equivalently the Thom spectrum (see at cobordism ring for more) and so a genus may be thought of as a morphism of spectra

$$\sigma :MO\to R.$$\sigma : M O \to R \,.

In terms of quantum field theory

At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For $G$ some structure, the Thom spectrum $MG$ is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma :MG\to R$ can be thought of as assigning to any target space the corresponding sigma-model.

This is for instance the case at least over the point for the A-hat genus $M\mathrm{Spin}\to KO$, which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus $M\mathrm{String}\to \mathrm{tmf}$, which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).

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.