Genus of a multiplicative sequence

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.

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.

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 OEIS: A237111). 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.

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.

The  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  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  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  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  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  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  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.

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  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):

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

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

• Atiyah–Singer index theorem
• List of cohomology theories

• 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", Encyclopedia of Mathematics, EMS Press
• "Elliptic genera", Encyclopedia of Mathematics, EMS Press, 2001 [1994]