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 formula_1 assigns a number formula_2 to each manifold "X" such that
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 formula_2 is in some ring, often the ring of rational numbers, though it can be other rings such as formula_8 or the ring of modular forms.
The conditions on formula_9 can be rephrased as saying that formula_1 is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring. 
Example: If formula_2 is the signature of the oriented manifold "X", then formula_9 is a genus from oriented manifolds to the ring of integers.
The genus associated to a formal power series.
A sequence of polynomials formula_13 in variables formula_14 is called multiplicative if
implies that
If formula_17 is a formal power series in "z" with constant term 1, we can define a multiplicative sequence 
by 
where formula_20 is the "k"th elementary symmetric function of the indeterminates formula_21. (The variables formula_20 will often in practice be Pontryagin classes.)
The genus formula_9 of compact, connected, smooth, oriented manifolds corresponding to "Q" is given by
where the formula_20 are the Pontryagin classes of "X". The power series "Q" is called the characteristic power series of the genus formula_9. A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4"k" 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 
where the numbers formula_28 are the Bernoulli numbers. The first few values are: 
(for further "L"-polynomials see or ). Now let "M" be a closed smooth oriented manifold of dimension 4"n" with Pontrjagin classes formula_34. Friedrich Hirzebruch showed that the "L" genus of "M" in dimension 4"n" evaluated on the fundamental class of formula_35, denoted formula_36, is equal to formula_37, the signature of "M" (i.e., the signature of the intersection form on the 2"n"th cohomology group of "M"):
This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem). 
The fact that formula_39 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 formula_40, 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 formula_41 in formula_42. It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since formula_43, we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has formula_44, and using the classification of unimodular lattices.
Todd genus.
The Todd genus is the genus of the formal power series 
with formula_28 as before, Bernoulli numbers. The first few values are 
The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. formula_52), 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
(There is also an Â genus which is less commonly used, associated to the characteristic series formula_54.) The first few values are
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 formula_60-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 formula_60-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 formula_62 satisfies the condition 
for constants formula_64 and formula_65. (As usual, "Q" is the characteristic power series of the genus.)
One explicit expression for "f"("z") is
where
and "sn" is the Jacobi elliptic function.
Examples:
The first few values of such genera are:
Example (Elliptic genus for quaternionic projective plane) :
Example (Elliptic genus for octonionic projective plane (Cayley plane)):
Witten genus.
The Witten genus is the genus associated to the characteristic power series
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 4"k" dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2"k", with integral Fourier coefficients.