It's embarrassing to say this, but I wasn't be able to be convinced that that's the only relation. We have relations up to degree $2n$, but the dimension of Grassmannian is $2k(n−k)$. Where do relations of higher degrees come from? For instance, for $G(2,5)$, I couldn't get the obvious relation $c_1^7=0$ from $(1+c_1 +c_2)(1+c_1 ' +c_2 ' +c_3 ')=1$. What am I missing?

I understood. Thank you for your answers and patience.

A diffeomorphism $\psi$ on $M$ induces an isomorphism on the tangent space at each point. So two vector fields $X$ and $(\psi^{-1})_{*} X$ have the same number of zeroes on $M$. What am I thinking wrong?

I am sorry I am confused. What do you mean by $\psi^{*}X$? Is it a pushforward of a vector field $X$ by a diffeomorphism $\psi^{-1}$? But then the resulting vector field still has zeroes.

I couldn't understand Step 1. It seems that there is a nowhere vanishing vector field on an open manifold, but I have no idea how to prove it. Could you explain what it means "moving zero outside the manifold", or what reference I should read?

I thought I knew K\"unneth theorem but it was not true. Thank you.