Hirzebruch signature theorem

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.

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

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.

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

implies that

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

by

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

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.

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