Spaces of rational loops on a real projective space (Q2701689)

Let \(\text{Rat}_n(m)\) denote the space of rational maps from \(\mathbb{C} P^1\) to \(\mathbb{C} P^m\) given by a polynomial of degree \(n\) that sends the point \(\infty\in \mathbb{C} P^1\) to a fixed point in \(\mathbb{C} P^m\). Let \((\Omega^2 \mathbb{C} P^m)_n\) be the component of the double loop space \(\Omega^2 \mathbb{C} P^m\) which parametrizes maps of degree \(n\). \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)], proved that the natural inclusion of \(\text{Rat}_n(m)\) in \((\Omega^2 \mathbb{C} P^m)_n\) is a homotopy equivalence up to dimension \(n(2m-1)\). Denote by \(\mathbb{R} \text{Rat}_n(m)\) the subspace of \(\text{Rat}_n(m)\) of maps which commute with complex conjugation and by \textbf{rat}\(_n(m)\) the closure of \(\mathbb{R} \text{Rat}_n(m)\) in \(\Omega \mathbb{R} P^m\). The author proves that the space \textbf{rat}\(_n(1)\) consists of \(n+1\) contractible components and if \(m>1\) the natural inclusion \textbf{rat}\(_n(m) \subset(\Omega \mathbb{R} P^m)_{n\bmod 2}\) is a homotopy equivalence up to dimension \(n(m-1)\). The proof goes along the line of Segal's proof, however the technique involved (configuration spaces, action of the \(\pi_1(\mathbf{rat}_n(2)\)) on higher homotopy groups,\dots) seems to be of independent interest.