# Genus of a multiplicative sequence

In mathematics, a

**genus of a multiplicative sequence**is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.

## Definition

A**genus**assigns a number to each manifold

*X*such that

- ;
- ;
- if
*X*is the boundary of a manifold with boundary.

The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds to another ring.

Example: If is the signature of the oriented manifold

*X*, then is a genus from oriented manifolds to the ring of integers.

## The genus associated to a formal power series

A sequence of polynomials*K*

_{1},

*K*

_{2},... in variables

*p*

_{1},

*p*

_{2},... is called

**multiplicative**if

implies that

If

*Q*is a formal power series in

*z*with constant term 1, we can define a multiplicative sequence

by

where

*p*is the

_{k}*k*th elementary symmetric function of the indeterminates

*z*

_{i}.

The genus φ of oriented manifolds corresponding to

*Q*is given by

where the

*p*

_{k}are the Pontryagin classes of

*X*. The power series

*Q*is called the

**characteristic power series**of the genus φ. Thom's theorem, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4

*k*for positive integers

*k*, implies that this gives a bijection between formal power series

*Q*with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.

## L genus

The**L genus**is the genus of the formal power series

where the numbers are the Bernoulli numbers. The first few values are:

. Now let

*M*be a closed smooth oriented manifold of dimension 4

*n*with Pontrjagin classes Friedrich Hirzebruch showed that the

*L*genus of

*M*in dimension 4

*n*evaluated on the fundamental class of is equal to the signature of

*M*:

This is now known as the

**Hirzebruch signature theorem**.

The fact that

*L*

_{2}is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional PL manifold with no smooth structure. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of

*p*

_{2}, and so was not smoothable.

### Application on K3 surfaces

Since projective K3 surfaces are smooth complex manifolds of dimension two, their only non-trivial Pontryagin class is in. It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since, we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has, and using the classification of unimodular lattices.## Todd genus

The**Todd genus**is the genus of the formal power series

with as before, Bernoulli numbers. The first few values are

The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces, and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the Hirzebruch–Riemann–Roch theorem, and in fact is one of the key developments that led to the formulation of that theorem.

## Â genus

The**Â genus**is the genus associated to the characteristic power series

The first few values are

The Â genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 – for general manifolds, the Â genus is not always an integer. This was proven by Hirzebruch and Armand Borel; this result both motivated and was later explained by the Atiyah–Singer index theorem, which showed that the Â genus of a spin manifold is equal to the index of its Dirac operator.

By combining this index result with a Weitzenbock formula for the Dirac Laplacian, André Lichnerowicz deduced that if a compact spin manifold admits a metric with positive scalar curvature, its Â genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but Nigel Hitchin later discovered an analogous -valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Mikhail Gromov, H. Blaine Lawson, and Stephan Stolz later proved that the Â genus and Hitchin's -valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5.

## Elliptic genus

A genus is called an**elliptic genus**if the power series

*Q*=

*z*/

*f*satisfies the condition

for constants δ and ε.

One explicit expression for

*f*is

where

and

*sn*is the Jacobi elliptic function.

Examples:

- . This is the L-genus.
- . This is the Â genus.
- . This is a generalization of the L-genus.

Example :

Example :

## Witten genus

The**Witten genus**is the genus associated to the characteristic power series

where σ

_{L}is the Weierstrass sigma function for the lattice

*L*, and

*G*is a multiple of an Eisenstein series.

The Witten genus of a 4

*k*dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2

*k*, with integral Fourier coefficients.