Genus of a multiplicative sequence

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

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[edit] Definition

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

  1. φ(X∪Y) = φ(X) + φ(Y) (where ∪ is the disjoint union)
  2. φ(X×Y) = φ(X)φ(Y)
  3. 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.

[edit] The genus of a formal power series

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

1 + p1z + p2z2 + ... = (1 + q1z + q2z2 + ...) (1 + r1z + r2z2 + ...)

implies that

ΣKj(p1,p2,...)zj = ΣKj(q1,q2,...)zjΣKk(r1,r2,...)zk

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

K = 1+ K1 + K2 + ...

by

K(p1,p2,p3,...) = Q(z1)Q(z2)Q(z3)...

where pk is the k'th 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

φ(X) = K(p1,p2,p3,...)

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 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 genuses from oriented manifolds to the rational numbers.

[edit] L genus and the Hirzebruch signature theorem

The L genus is the genus of the formal power series

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

where the numbers B2k are the Bernoulli numbers. The first few values are

  • L0 = 1
  • L1 = p1/3
  • L2 = (7p2 − p12)/45

Friedrich Hirzebruch showed that the L genus of a manifold of dimension 4n is equal to the signature (of the 2nth cohomology group). This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem. René Thom had earlier proved that the signature was given by some linear combination of Pontryagin numbers, and Hirzebruch found the exact formula for this linear combination given above.

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.

[edit] Todd genus

[edit] Â genus

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

  • Â0 = 1
  • Â1 = −p1/24
  • Â2 = (−4p2 + 7 p12)/5760

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). This was explained by Michael Atiyah and Isadore Singer, who showed that the  genus of a spin manifold is equal to the index of its Dirac operator. For general manifolds, the  genus is not always an integer.

Atiyah, Hitchin, Lichnerowicz, and Singer proved that if a compact spin manifold has a metric with positive scalar curvature, then its  genus is 0.

Atiyah and Hirzebruch proved that if a compact spin manifold has a non-trivial circle action on it, then its  genus is 0.

[edit] Elliptic genus

A genus is called an elliptic genus if the power series f(z) = z/Q(z) satisfies the condition

f ′2 = 1 − 2δf2 + εf4

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

Examples:

  • δ = ε = 1, f(z) = tanh(z). This is the L-genus.
  • δ = −1/8, ε = 0, f(z) = 2sinh(z/2). This is the  genus.

[edit] Witten genus

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

Q(z) = 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.

[edit] See also

[edit] References

  • Friedrich Hirzebruch Topological Methods in Algebraic Geometry ISBN 3-540-58663-6
  • Friedrich Hirzebruch, Thomas Berger, Rainer Jung Manifolds and Modular Forms ISBN 3528064145
  • Milnor, Stasheff, Characteristic classes, ISBN 0691081220
  • A.F. Kharshiladze, "Pontryagin class" SpringerLink Encyclopaedia of Mathematics (2001)
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