Genus of a multiplicative sequence

In mathematics, the genus of a sequence is a ring homomorphism, from the ring of smooth compact manifolds to another ring, usually the ring of rational numbers.

Definition

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 a boundary.

The manifolds may have some extra structure; for example, they might be oriented, or spin, 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 Z/2Z 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 given 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 of a formal power series

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

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

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

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

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

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

\Phi (X)=K(p_{1},p_{2},p_{3},\cdots )

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 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\geq 0}}{2^{{2k}}B_{{2k}}z^{k} \over (2k)!}=1+{z \over 3}-{z^{2} \over 45}+\cdots

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

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

(for further L-polynomials see ^[1] or A237111). 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, $[M]$ , 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),\dots ,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

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

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

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

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

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

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

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 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, 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 Hitchin later discovered an analogous ${{\mathbb Z}}_{2}$ -valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Gromov, Lawson, and Stolz later proved that the Â genus and Hitchin's ${{\mathbb Z}}_{2}$ -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

{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

f \left( z \right) =\frac{{\it JacobiSN} \left( az,{\frac {\sqrt {\epsilon}} {{a}^{2}}} \right)} {a}

where

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

and JacobiSN is the Jacobi elliptic function sn.

Examples:

$\delta =\epsilon =1,f(z)=\tanh(z)$ . This is the L-genus.
$\delta =-1/8,\epsilon =0,f(z)=2\sinh(z/2)$ . 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:

${\tfrac 13}\delta p_{1}$
${\tfrac 1{90}}{\big [}(-4\delta ^{2}+18\epsilon )p_{2}+(7\delta ^{2}-9\epsilon )p_{1}^{2}{\big ]}$
${\tfrac 1{1890}}{\big [}(16\delta ^{3}+108\delta \epsilon )p_{3}+(-44\delta ^{3}+18\delta \epsilon )p_{2}p_{1}+(31\delta ^{3}-27\delta \epsilon )p_{1}^{3}{\big ]}$
${\tfrac 1{113400}}{\big [}(-192\delta ^{4}+1728\delta ^{2}\epsilon +1512\epsilon ^{2})p_{4}+(512\delta ^{4}+432\delta ^{2}\epsilon -1512\epsilon ^{2})p_{3}p_{1}+(208\delta ^{4}-1872\delta ^{2}\epsilon +1512\epsilon ^{2})p_{2}^{2}+(-904\delta ^{4}+1836\delta ^{2}\epsilon -756\epsilon ^{2})p_{2}p_{1}^{2}+(381\delta ^{4}-594\delta ^{2}\epsilon +189\epsilon ^{2})p_{1}^{4}{\big ]}$
$\left( -{\frac {53}{41580}}\,{\delta}^{3}\epsilon+{\frac {17}{27720}}\,\delta\,{\epsilon}^{2}+{\frac {73}{106920}}\,{\delta}^{5} \right) {p_{{1}}}^{5}+ \left( {\frac {481}{103950}}\,{\delta}^{3} \epsilon-{\frac {29}{11550}}\,\delta\,{\epsilon}^{2}-{\frac {1073}{ 467775}}\,{\delta}^{5} \right) p_{{2}}{p_{{1}}}^{3}+ \left( {\frac { 244}{155925}}\,{\delta}^{5}-{\frac {37}{51975}}\,{\delta}^{3}\epsilon- {\frac {2}{5775}}\,\delta\,{\epsilon}^{2} \right) p_{{3}}{p_{{1}}}^{2} + \left( \left( {\frac {622}{467775}}\,{\delta}^{5}-{\frac {202}{ 51975}}\,{\delta}^{3}\epsilon+{\frac {7}{2475}}\,\delta\,{\epsilon}^{2 } \right) {p_{{2}}}^{2}+ \left( -{\frac {31}{5775}}\,\delta\,{\epsilon }^{2}-{\frac {424}{467775}}\,{\delta}^{5}+{\frac {32}{7425}}\,{\delta} ^{3}\epsilon \right) p_{{4}} \right) p_{{1}}+ \left( -{\frac {4}{2475} }\,{\delta}^{3}\epsilon+{\frac {4}{2475}}\,\delta\,{\epsilon}^{2}-{ \frac {16}{22275}}\,{\delta}^{5} \right) p_{{3}}p_{{2}}+ \left( { \frac {8}{10395}}\,{\delta}^{3}\epsilon+{\frac {32}{93555}}\,{\delta}^ {5}+{\frac {34}{3465}}\,\delta\,{\epsilon}^{2} \right) p_{{5}}$

Witten genus

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

Q(z)=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.

Notes

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

References

Friedrich Hirzebruch Topological Methods in Algebraic Geometry ISBN 3-540-58663-6
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, 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

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