The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.

Let's consider the natural embedding $GL_n(\mathbb C) \to \mathbb C^{n^2} \backslash \{0\}$. As was discussed in this question, cohomology with rational coefficients of $GL_n(\mathbb C)$ is an exterior algebra on generators in degrees 1, 3, ..., 2n-1 (one generator in each degree), whereas $\mathbb C^{n^2} \backslash \{0\}$ is homotopy equivalent to a sphere $S^{n^2-1}$.

I'd like to prove that the map induced on cohomology of degree $n^2 - 1$ is a zero map. Any ideas?

Thanks.