Genus of a multiplicative sequence

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.

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