Definition
A genus φ assigns a number φ(X) to each manifold X such that
- φ(X∪Y) = φ(X) + φ(Y) (where ∪ is the disjoint union)
- φ(X×Y) = φ(X)φ(Y)
- 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.
Read more about this topic: Genus Of A Multiplicative Sequence
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