I’m reading Manifolds and Modular Forms.
According to its introduction chapter, the definition of genus is a ring homomorphism $\varphi:\Omega\otimes\mathbb{Q}\rightarrow R$ with $R$ an integral domain over $\mathbb{Q}$, and $\Omega$ the cobordism ring.
But on Wikipedia, the definition seems to be $\varphi:\Omega\rightarrow R$, so what role does ‘$\otimes\mathbb{Q}$’ plays in the prior def?
The Wikipedia article describes the general notion of a genus, allowing for homomorphisms to general rings, for instance, to ${\mathbb Z}/2$. The authors of the book you are reading, prefer to work over ${\mathbb Q}$, i.e. when $R$ is a ${\mathbb Q}$-algebra, eliminating examples such as the ring ${\mathbb Z}/2$. Of course, one can still consider ring-homomorphisms $$ \Omega\to R, $$ but (if one thinks in terms of ${\mathbb Q}$-algebras) it is more natural to consider ${\mathbb Q}$-algebra homomorphisms
$$ \Omega \otimes {\mathbb Q}\to R. $$ This is what your book does.