Fundamental group of real $3\times 3$ matrices with rank $1$

Question

In "Manetti - Topologia" there is the following exercise:

Compute the fundamental group of real $3\times 3$ matrices with rank $1$.

He suggests to show that there is a covering map of degree $2$ between the total space $E=S^{2} \times (\mathbb{R}^{3}\smallsetminus \{0\})$ and the base space $X=\mathcal{M}(3,\mathbb{R})$ with rank $1$.

I think $p: E \mapsto X$ sends a vector and its unit vector into a matrix which columns are multiples of this vector, but I can't show that this map has degree $2$.

How can I solve this exercise?

Answer 1

We know that $X=\mathfrak{M}(3,\mathbb{R})$ with rank $1$ is of the form $\{uv^{T}, u,v \in \mathbb{R}^3, u,v \ne 0\}.$ We also note that choosing $\lambda \in \mathbb{R}^{*}$, we have $$ uv^T=\lambda u \frac{1}{\lambda}v^T, $$ so we can take $u \in S^2$ and $v \in \mathbb{R}^3 \setminus \{0\}$. In this way, we can define the covering map $p: E \mapsto X$ as the hint of the book.