I met the following definition on a text I'm reading.
Let $f : M \rightarrow N$ be a smooth map between two closed oriented manifolds of dimension $d$. Then $f$ defines a pullback on de Rham cohomology $f^* : H^d_{dR}(N,\mathbb{R}) \rightarrow H^d_{dR}(M,\mathbb{R})$. Let $\omega$ be the generator of $H^d_{dR}(N,\mathbb{R})$, then the mapping degree of $f$ is defined to be $\int_M f^* \omega$.
Can someone explain the answers to the following questions?
- How do you define the generator of $H^d_{dR}(N,\mathbb{R})$? Can we take any $d-form$ as a generator?
- If $\omega$ is such a generator does it satisfy the condition $\int_N \omega=1$?
Thank you.