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Genus of a multiplicative sequence

## Summary

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 ${\displaystyle \varphi }$  assigns a number ${\displaystyle \Phi (X)}$  to each manifold X such that

1. ${\displaystyle \Phi (X\sqcup Y)=\Phi (X)+\Phi (Y)}$  (where ${\displaystyle \sqcup }$  is the disjoint union);
2. ${\displaystyle \Phi (X\times Y)=\Phi (X)\Phi (Y)}$ ;
3. ${\displaystyle \Phi (X)=0}$  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 ${\displaystyle \Phi (X)}$  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 ${\displaystyle \Phi }$  can be rephrased as saying that ${\displaystyle \varphi }$  is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.

Example: If ${\displaystyle \Phi (X)}$  is the signature of the oriented manifold X, then ${\displaystyle \Phi }$  is a genus from oriented manifolds to the ring of integers.

## The genus associated to a formal power series

A sequence of polynomials ${\displaystyle K_{1},K_{2},\ldots }$  in variables ${\displaystyle p_{1},p_{2},\ldots }$  is called multiplicative if

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

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 ${\displaystyle Q(z)}$  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 ${\displaystyle p_{k}}$  is the kth elementary symmetric function of the indeterminates ${\displaystyle z_{i}}$ . (The variables ${\displaystyle p_{k}}$  will often in practice be Pontryagin classes.)

The genus ${\displaystyle \Phi }$  of compact, connected, smooth, oriented manifolds corresponding to Q is given by

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

where the ${\displaystyle p_{k}}$  are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus ${\displaystyle \Phi }$ . A theorem of René Thom, 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 {\begin{aligned}L_{0}&=1\\L_{1}&={\tfrac {1}{3}}p_{1}\\L_{2}&={\tfrac {1}{45}}\left(7p_{2}-p_{1}^{2}\right)\\L_{3}&={\tfrac {1}{945}}\left(62p_{3}-13p_{1}p_{2}+2p_{1}^{3}\right)\\L_{4}&={\tfrac {1}{14175}}\left(381p_{4}-71p_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{1}^{4}\right)\end{aligned}}}

(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 ${\displaystyle M}$ , denoted ${\displaystyle [M]}$ , 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 ${\displaystyle L_{2}}$  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 ${\displaystyle p_{2}}$ , 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 ${\displaystyle p_{1}}$  in ${\displaystyle H^{4}(X)}$ . It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since ${\displaystyle L_{1}=-16}$ , we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has ${\displaystyle \operatorname {dim} \left(H^{2}(X)\right)=22}$ , and using the classification of unimodular lattices.[2]

## Todd genus

The Todd genus is the genus of the formal power series

${\displaystyle {\frac {z}{1-\exp(-z)}}=\sum _{i=0}^{\infty }{\frac {B_{i}}{i!}}z^{i}}$

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

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

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

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

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

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

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 ${\displaystyle Q(z)=z/f(z)}$  satisfies the condition

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

for constants ${\displaystyle \delta }$  and ${\displaystyle \epsilon }$ . (As usual, Q is the characteristic power series of the genus.)

One explicit expression for f(z) is

${\displaystyle f(z)={\frac {1}{a}}\operatorname {sn} \left(az,{\frac {\sqrt {\epsilon }}{a^{2}}}\right)}$

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.
• ${\displaystyle \delta =-{\frac {1}{8}},\epsilon =0,f(z)=2\sinh \left({\frac {1}{2}}z\right)}$ . 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 {\begin{aligned}\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 ]}\\&=\int _{HP^{2}}{\tfrac {1}{90}}{\big [}(-4\delta ^{2}+18\epsilon )(7u^{2})+(7\delta ^{2}-9\epsilon )(2u)^{2}{\big ]}\\&=\int _{HP^{2}}[u^{2}\epsilon ]\\&=\epsilon \int _{HP^{2}}[u^{2}]\\&=\epsilon *1=\epsilon \end{aligned}}}

Example (elliptic genus for octonionic projective plane, or Cayley plane):

{\displaystyle {\begin{aligned}\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]\\&=\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 ]}\\&=\int _{OP^{2}}{\big [}\epsilon ^{2}u^{2}{\big ]}\\&=\epsilon ^{2}\int _{OP^{2}}{\big [}u^{2}{\big ]}\\&=\epsilon ^{2}*1=\epsilon ^{2}\\&=\Phi _{ell}(HP^{2})^{2}\end{aligned}}}

## 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.