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Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces. (English) Zbl 1311.55009
The inclusion map of the space of holomorphic maps $$\text{Hol}(\mathbb{C}\text{P}^m,\mathbb{C}\text{P}^n)$$, between compex projective spaces of dimensions $$1\leq m\leq n$$, into the corresponding space of continuous maps $$\text{Map}(\mathbb{C}\text{P}^m,\mathbb{C}\text{P}^n)$$, is a homotopy equivalence up to a certain dimension. The history of such results dates back to a fundamental paper by G. Segal [Acta Math. 143, 39–72 (1979; Zbl 0427.55006)].
A similar result for the homotopy type of the space of algebraic maps $$\text{Alg}(\mathbb{R}\text{P}^m,\mathbb{R}\text{P}^n)$$, between real projective spaces of dimensions $$2\leq m\leq n$$, compared to the homotopy type of the space of continuous maps $$\text{Map}(\mathbb{R}\text{P}^m,\mathbb{R}\text{P}^n)$$, was obtained by the authors in collaboration with M. Adamaszek [Q. J. Math. 62, No. 4, 771–790 (2011; Zbl 1245.14060)].
In this paper, the authors improve and extend similar results in their earlier paper [Contemp. Math. 519, 145–164 (2010; Zbl 1209.55005)] approximating the homotopy type of the space of real algebraic maps $$\text{Alg}(\mathbb{R}\text{P}^m,\mathbb{C}\text{P}^n)$$, $$2\leq m\leq 2n$$, with the homotopy type of the space of continuous maps $$\text{Map}(\mathbb{R}\text{P}^m,\mathbb{C}\text{P}^n)$$.

##### MSC:
 55P10 Homotopy equivalences in algebraic topology 55R80 Discriminantal varieties and configuration spaces in algebraic topology 55P35 Loop spaces
##### Keywords:
algebraic map; homotopy equivalence
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