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, 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.
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