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Spaces of rational loops on a real projective space. (English) Zbl 0979.55008
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 $$n$$. 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 rat$$_n(m)$$ the closure of $$\mathbb{R} \text{Rat}_n(m)$$ in $$\Omega \mathbb{R} P^m$$. The author proves that the space rat$$_n(1)$$ consists of $$n+1$$ contractible components and if $$m>1$$ the natural inclusion 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,…) seems to be of independent interest.

##### MSC:
 55P35 Loop spaces 26C15 Real rational functions 55P10 Homotopy equivalences in algebraic topology
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##### References:
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