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.

Definition
A genus $$\varphi$$ assigns a number $$\Phi(X)$$ to each manifold X such that
 * 1) $$\Phi(X \sqcup Y) = \Phi(X) + \Phi(Y)$$ (where $$\sqcup$$ is the disjoint union);
 * 2) $$\Phi(X \times Y) = \Phi(X)\Phi(Y)$$;
 * 3) $$\Phi(X) = 0$$  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 $$\Phi(X)$$ is in some ring, often the ring of rational numbers, though it can be other rings such as $$\Z/2\Z$$ or the ring of modular forms.

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

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

The genus associated to a formal power series
A sequence of polynomials $$K_1, K_2,\ldots$$ in variables $$p_1, p_2,\ldots$$ is called multiplicative if


 * $$1 + p_1z + p_2z^2 + \cdots = (1 + q_1z + q_2z^2 + \cdots) (1 + r_1z + r_2z^2 + \cdots)$$

implies that


 * $$\sum_j K_j(p_1, p_2, \ldots)z^j = \sum_j K_j (q_1, q_2, \ldots) z^j\sum_k K_k (r_1, r_2, \ldots)z^k$$

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


 * $$K = 1+ K_1 + K_2 + \cdots$$

by


 * $$K(p_1, p_2, p_3, \ldots) = Q(z_1)Q(z_2)Q(z_3)\cdots$$,

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

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


 * $$\Phi(X) = K(p_1, p_2, p_3, \ldots)$$

where the $$p_k$$ are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus $$\Phi$$. 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


 * $${\sqrt{z}\over \tanh(\sqrt z)} = \sum_{k\ge 0} \frac{2^{2k}B_{2k}z^k}{(2k)!} = 1 + {z \over 3} - {z^2 \over 45} + \cdots$$

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


 * $$\begin{align}

L_0 &= 1 \\ L_1 &= \tfrac13 p_1 \\ L_2 &= \tfrac1{45}\left(7p_2 - p_1^2\right) \\ L_3 &= \tfrac1{945}\left(62 p_3 - 13 p_1 p_2 + 2 p_1^3\right) \\ L_4 &= \tfrac1{14175}\left(381 p_4 - 71 p_1 p_3 - 19 p_2^2 + 22 p_1^2 p_2 - 3 p_1^4\right) \end{align}$$

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


 * $$\sigma(M) = \langle L_n(p_1(M), \ldots, p_n(M)), [M]\rangle$$.

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

The fact that $$L_2$$ 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 $$p_1$$ in $$H^4(X)$$. It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since $$L_1 = -16$$, we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has $$\operatorname{dim}\left(H^2(X)\right) = 22 $$, and using the classification of unimodular lattices.

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


 * $$\frac{z}{1 - \exp(-z)} = \sum_{i=0}^\infty\frac{B_i}{i!}z^i$$

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


 * $$\begin{align}

Td_0 &= 1 \\ Td_1 &= \frac1{2} c_1 \\ Td_2 &= \frac1{12} \left (c_2 + c_1^2 \right ) \\ Td_3 &= \frac1{24} c_1 c_2 \\ Td_4 &= \frac1{720} \left(-c_1^4 + 4 c_2 c_1^2 + 3c_2^2 + c_3 c_1 - c_4\right) \end{align}$$

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



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 $$Q(16z)$$.) The first few values are


 * $$\begin{align}

\hat{A}_0 &= 1 \\ \hat{A}_1 &= -\tfrac1{24}p_1 \\ \hat{A}_2 &= \tfrac1{5760}\left(-4p_2 + 7 p_1^2\right) \\ \hat{A}_3 &= \tfrac1{967680}\left(-16p_3 + 44p_2 p_1 - 31 p_1^3\right) \\ \hat{A}_4 &= \tfrac1{464486400}\left(-192p_4 + 512 p_3 p_1 + 208p_2^2 - 904p_2 p_1^2 + 381p_1^4\right) \end{align}$$

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 $$\Z_2$$-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 $$\Z_2$$-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


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

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

One explicit expression for f(z) is


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

where


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

and sn is the Jacobi elliptic function.

Examples:


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

The first few values of such genera are:
 * $$\frac{1}{3}\delta p_1$$
 * $$\frac{1}{90} \left [ \left (-4\delta^2 +18\epsilon \right )p_2+ \left (7\delta^2-9\epsilon \right )p_1^2\right ]$$
 * $$\frac{1}{1890} \left [ \left (16\delta^3 + 108\delta \epsilon \right )p_3 + \left (-44\delta^3 +18\delta \epsilon \right )p_2p_1 + \left (31\delta^3 -27\delta \epsilon \right )p_1^3\right ]$$

Example (elliptic genus for quaternionic projective plane) :


 * $$\begin{align}

\Phi_{ell}(HP^2) &= \int_{HP^2}\tfrac1{90}\big[(-4\delta^2 +18\epsilon )p_2+(7\delta^2-9\epsilon )p_1^2\big] \\ &= \int_{HP^2}\tfrac1{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{align}$$

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


 * $$\begin{align}

\Phi_{ell}(OP^2) &= \int_{OP^2}\tfrac1{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}\tfrac1{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{align}$$

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


 * $$Q(z) = \frac{z}{\sigma_L(z)} = \exp\left(\sum_{k\ge 2} {2G_{2k}(\tau)z^{2k}\over(2k)!}\right)$$

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.