# Genus of a multiplicative sequence

A cobordism (W; M, N).

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.

## Definition

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

1. φ(XY) = φ(X) + φ(Y) (where ∪ is the disjoint union)
2. φ(X×Y) = φ(X)φ(Y)
3. 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ 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

A sequence of polynomials K1, K2, ... in variables p1, p2, ... is called multiplicative if

${\displaystyle 1+p_{1}z+p_{2}z^{2}+\dots =(1+q_{1}z+q_{2}z^{2}+\cdots )(1+r_{1}z+r_{2}z^{2}+\cdots )}$

implies that

${\displaystyle \sum _{j}K_{j}(p_{1},p_{2},\ldots )z^{j}=\sum _{j}K_{j}(q_{1},q_{2},\ldots )z^{j}\sum _{k}K_{k}(r_{1},r_{2},\ldots )z^{k}}$

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

${\displaystyle K=1+K_{1}+K_{2}+\cdots }$

by

${\displaystyle K(p_{1},p_{2},p_{3},\ldots )=Q(z_{1})Q(z_{2})Q(z_{3})\cdots }$

where pk is the kth elementary symmetric function of the indeterminates zi. (The variables pk will often in practice be Pontryagin classes.)

The genus φ of oriented manifolds corresponding to Q is given by

${\displaystyle \Phi (X)=K(p_{1},p_{2},p_{3},\ldots )}$

where the pk 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 4k 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

${\displaystyle {{\sqrt {z}} \over \tanh({\sqrt {z}})}=\sum _{k\geq 0}{\frac {2^{2k}B_{2k}z^{k}}{(2k)!}}=1+{z \over 3}-{z^{2} \over 45}+\cdots }$

where the numbers ${\displaystyle B_{2k}}$ are the Bernoulli numbers. The first few values are:

${\displaystyle L_{0}=1}$
${\displaystyle L_{1}={\tfrac {1}{3}}p_{1}}$
${\displaystyle L_{2}={\tfrac {1}{45}}(7p_{2}-p_{1}^{2})}$
${\displaystyle L_{3}={\tfrac {1}{945}}(62p_{3}-13p_{1}p_{2}+2p_{1}^{3})}$
${\displaystyle L_{4}={\tfrac {1}{14175}}(381p_{4}-71p_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{1}^{4})}$

(for further L-polynomials see [1] or ). Now let M be a closed smooth oriented manifold of dimension 4n with Pontrjagin classes ${\displaystyle p_{i}=p_{i}(M).}$ Friedrich Hirzebruch showed that the L genus of M in dimension 4n evaluated on the fundamental class of ${\displaystyle M,[M],}$ is equal to ${\displaystyle \sigma (M),}$ the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M):

${\displaystyle \sigma (M)=\langle L_{n}(p_{1}(M),\ldots ,p_{n}(M)),[M]\rangle .}$

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

The fact that L2 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 p2, and so was not smoothable.

## Todd genus

The Todd genus is the genus of the formal power series

${\displaystyle {\frac {z}{1-\exp(-z)}}=1+{\frac {1}{2}}z+\sum _{i=1}^{\infty }(-1)^{i+1}{\frac {B_{2i}}{(2i)!}}z^{2i}}$

with ${\displaystyle B_{2k}}$ as before, Bernoulli numbers. The first few values are

${\displaystyle Td_{0}=1}$
${\displaystyle Td_{1}={\frac {1}{2}}c_{1}}$
${\displaystyle Td_{2}={\frac {1}{12}}\left(c_{2}+c_{1}^{2}\right)}$
${\displaystyle Td_{3}={\frac {1}{24}}c_{1}c_{2}}$
${\displaystyle Td_{4}={\frac {1}{720}}\left(-c_{1}^{4}+4c_{2}c_{1}^{2}+3c_{2}^{2}+c_{3}c_{1}-c_{4}\right)}$

The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. ${\displaystyle \mathrm {Td} _{n}(\mathbb {CP} ^{n})=1}$), 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

${\displaystyle Q(z)={{\sqrt {z}}/2 \over \sinh({\sqrt {z}}/2)}=1-{\frac {z}{24}}+{\frac {7z^{2}}{5760}}-\cdots }$

(There is also an Â genus which is less commonly used, associated to the characteristic series ${\displaystyle Q(16z)}$.) The first few values are

${\displaystyle {\hat {A}}_{0}=1}$
${\displaystyle {\hat {A}}_{1}=-{\tfrac {1}{24}}p_{1}}$
${\displaystyle {\hat {A}}_{2}={\tfrac {1}{5760}}(-4p_{2}+7p_{1}^{2})}$
${\displaystyle {\hat {A}}_{3}={\tfrac {1}{967680}}(-16p_{3}+44p_{2}p_{1}-31p_{1}^{3})}$
${\displaystyle {\hat {A}}_{4}={\tfrac {1}{464486400}}(-192p_{4}+512p_{3}p_{1}+208p_{2}^{2}-904p_{2}p_{1}^{2}+381p_{1}^{4})}$

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 ${\displaystyle \mathbb {Z} _{2}}$-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 ${\displaystyle \mathbb {Z} _{2}}$-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) = z/f(z) satisfies the condition

${\displaystyle {f'}^{2}=1-2\delta f^{2}+\epsilon f^{4}}$

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

One explicit expression for f(z) is

${\displaystyle f(z)={\frac {{\rm {sn}}\left(az,{\frac {\sqrt {\epsilon }}{{a}^{2}}}\right)}{a}}}$

where

${\displaystyle a={\sqrt {\delta +{\sqrt {{\delta }^{2}-\epsilon }}}}}$

and sn is the Jacobi elliptic function.

Examples:

• ${\displaystyle \delta =\epsilon =1,f(z)=\tanh(z)}$. This is the L-genus.
• ${\displaystyle \delta =-1/8,\epsilon =0,f(z)=2\sinh(z/2)}$. This is the Â genus.
• ${\displaystyle \epsilon =\delta ^{2},f(z)={\frac {\tanh({\sqrt {\delta }}z)}{\sqrt {\delta }}}}$. This is a generalization of the L-genus.

The first few values of such genera are:

${\displaystyle {\frac {1}{3}}\delta p_{1}}$
${\displaystyle {\frac {1}{90}}\left[\left(-4\delta ^{2}+18\epsilon \right)p_{2}+\left(7\delta ^{2}-9\epsilon \right)p_{1}^{2}\right]}$
${\displaystyle {\frac {1}{1890}}\left[\left(16\delta ^{3}+108\delta \epsilon \right)p_{3}+\left(-44\delta ^{3}+18\delta \epsilon \right)p_{2}p_{1}+\left(31\delta ^{3}-27\delta \epsilon \right)p_{1}^{3}\right]}$

Example (Elliptic genus for quaternionic projective plane) :

${\displaystyle \Phi _{ell}(HP^{2})=\int _{HP^{2}}{\tfrac {1}{90}}{\big [}(-4\delta ^{2}+18\epsilon )p_{2}+(7\delta ^{2}-9\epsilon )p_{1}^{2}{\big ]}}$
${\displaystyle \Phi _{ell}(HP^{2})=\int _{HP^{2}}{\tfrac {1}{90}}{\big [}(-4\delta ^{2}+18\epsilon )(7u^{2})+(7\delta ^{2}-9\epsilon )(2u)^{2}{\big ]}}$
${\displaystyle \Phi _{ell}(HP^{2})=\int _{HP^{2}}[u^{2}\epsilon ]}$
${\displaystyle \Phi _{ell}(HP^{2})=\epsilon \int _{HP^{2}}[u^{2}]}$
${\displaystyle \Phi _{ell}(HP^{2})=\epsilon *1=\epsilon }$

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

${\displaystyle \Phi _{ell}(OP^{2})=\int _{OP^{2}}{\tfrac {1}{113400}}\left[(-192\delta ^{4}+1728\delta ^{2}\epsilon +1512\epsilon ^{2})p_{4}+(208\delta ^{4}-1872\delta ^{2}\epsilon +1512\epsilon ^{2})p_{2}^{2}\right]}$
${\displaystyle \Phi _{ell}(OP^{2})=\int _{OP^{2}}{\tfrac {1}{113400}}{\big [}(-192\delta ^{4}+1728\delta ^{2}\epsilon +1512\epsilon ^{2})(39u^{2})+(208\delta ^{4}-1872\delta ^{2}\epsilon +1512\epsilon ^{2})(6u)^{2}{\big ]}}$
${\displaystyle \Phi _{ell}(OP^{2})=\int _{OP^{2}}{\big [}\epsilon ^{2}u^{2}{\big ]}}$
${\displaystyle \Phi _{ell}(OP^{2})=\epsilon ^{2}\int _{OP^{2}}{\big [}u^{2}{\big ]}}$
${\displaystyle \Phi _{ell}(OP^{2})=\epsilon ^{2}*1=\epsilon ^{2}}$
${\displaystyle \Phi _{ell}(OP^{2})=\Phi _{ell}(HP^{2})^{2}}$

## Witten genus

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

${\displaystyle Q(z)={\frac {z}{\sigma _{L}(z)}}=\exp \left(\sum _{k\geq 2}{2G_{2k}(\tau )z^{2k} \over (2k)!}\right)}$

where σL is the Weierstrass sigma function for the lattice L, and G is a multiple of an Eisenstein series.

The Witten genus of a 4k dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2k, with integral Fourier coefficients.