# 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 (i.e. up to suitable cobordism) 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.

## Contents

## Definition[edit]

A **genus** φ assigns a number φ(*X*) to each manifold *X* such that

- φ(
*X*∪*Y*) = φ(*X*) + φ(*Y*) (where ∪ is the disjoint union) - φ(
*X*×*Y*) = φ(*X*)φ(*Y*) - φ(
*X*) = 0 if*X*is the boundary of a manifold with boundary.

The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value φ(*X*) is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms.

The conditions on φ can be rephrased as saying that φ is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.

Example: If φ(*X*) is the signature of the oriented manifold *X*, then φ is a genus from oriented manifolds to the ring of integers.

## The genus associated to a formal power series[edit]

A sequence of polynomials *K*_{1}, *K*_{2}, ... in variables *p*_{1}, *p*_{2}, ... is called **multiplicative** if

implies that

If *Q*(*z*) is a formal power series in *z* with constant term 1, we can define a multiplicative sequence

by

where *p _{k}* is the

*k*th elementary symmetric function of the indeterminates

*z*

_{i}. (The variables

*p*will often in practice be Pontryagin classes.)

_{k}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[edit]

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

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

(for further *L*-polynomials see ^{[1]} or OEIS: A237111). 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* (i.e. the signature of the intersection form on the 2*n*th cohomology group of *M*):

This is now known as the **Hirzebruch signature theorem** (or sometimes the **Hirzebruch index 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.

## Todd genus[edit]

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 (i.e. ), 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[edit]

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

(There is also an Â genus which is less commonly used, associated to the characteristic 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 (which in dimension 4 implies Rochlin's theorem) – 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[edit]

A genus is called an **elliptic genus** if the power series *Q*(*z*) = *z*/*f*(*z*) satisfies the condition

for constants δ and ε. (As usual, *Q* is the characteristic power series of the genus.)

One explicit expression for *f*(*z*) 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.

The first few values of such genera are:

Example (Elliptic genus for quaternionic projective plane) :

Example (Elliptic genus for octonionic projective plane (Cayley plane)):

## Witten genus[edit]

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.

## See also[edit]

## Notes[edit]

**^**McTague, Carl (2014) "Computing Hirzebruch L-Polynomials".

## References[edit]

- Friedrich Hirzebruch
*Topological Methods in Algebraic Geometry*ISBN 3-540-58663-6 Text of the original German version: http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf - Friedrich Hirzebruch, Thomas Berger, Rainer Jung
*Manifolds and Modular Forms*ISBN 3-528-06414-5 - Milnor, Stasheff,
*Characteristic classes*, ISBN 0-691-08122-0 - A.F. Kharshiladze (2001) [1994], "Pontryagin class", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Hazewinkel, Michiel, ed. (2001) [1994], "Elliptic genera",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4