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 φ {\displaystyle \varphi } assigns a number Φ ( X ) {\displaystyle \Phi (X)} to each manifold X such that
Φ ( X ⊔ Y ) = Φ ( X ) + Φ ( Y ) {\displaystyle \Phi (X\sqcup Y)=\Phi (X)+\Phi (Y)} (where ⊔ {\displaystyle \sqcup } is the disjoint union);
Φ ( X × Y ) = Φ ( X ) Φ ( Y ) {\displaystyle \Phi (X\times Y)=\Phi (X)\Phi (Y)} ;
Φ ( X ) = 0 {\displaystyle \Phi (X)=0} 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 Φ ( X ) {\displaystyle \Phi (X)} is in some ring, often the ring of rational numbers, though it can be other rings such as Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } or the ring of modular forms. The conditions on Φ {\displaystyle \Phi } can be rephrased as saying that Φ {\displaystyle \Phi } is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring. Example: If Φ ( X ) {\displaystyle \Phi (X)} is the signature of the oriented manifold X, then Φ {\displaystyle \Phi } is a genus from oriented manifolds to the ring of integers.
The genus associated to a formal power series
A sequence of polynomials K 1 , K 2 , … {\displaystyle K_{1},K_{2},\ldots } in variables p 1 , p 2 , … {\displaystyle p_{1},p_{2},\ldots } is called multiplicative if
1 + p 1 z + p 2 z 2 + ⋯ = ( 1 + q 1 z + q 2 z 2 + ⋯ ) ( 1 + r 1 z + r 2 z 2 + ⋯ ) {\displaystyle 1+p_{1}z+p_{2}z^{2}+\cdots =(1+q_{1}z+q_{2}z^{2}+\cdots )(1+r_{1}z+r_{2}z^{2}+\cdots )}
implies that
∑ j K j ( p 1 , p 2 , … ) z j = ∑ j K j ( q 1 , q 2 , … ) z j ∑ k K k ( r 1 , r 2 , … ) z k {\displaystyle \sum _{j}K_{j}(p_{1},p_{2},\ldots )z^{j}=\sum _{j}K_{j}(q_{1},q_{2},\ldots )z^{j}\sum _{k}K_{k}(r_{1},r_{2},\ldots )z^{k}}
If Q ( z ) {\displaystyle Q(z)} is a formal power series in z with constant term 1, we can define a multiplicative sequence
K = 1 + K 1 + K 2 + ⋯ {\displaystyle K=1+K_{1}+K_{2}+\cdots }
by
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