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Concept
Genus of a multiplicative sequence
Summary
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.
A genus assigns a number to each manifold X such that
(where is the disjoint union);
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 is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms.
The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.
Example: If is the signature of the oriented manifold X, then is a genus from oriented manifolds to the ring of integers.
Multiplicative sequence
A sequence of polynomials in variables is called multiplicative if
implies that
If is a formal power series in z with constant term 1, we can define a multiplicative sequence
by
where is the kth elementary symmetric function of the indeterminates . (The variables will often in practice be Pontryagin classes.)
The genus of compact, connected, smooth, oriented manifolds corresponding to Q is given by
where the are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus . 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.
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