The modulo 2 homology groups of the space of rational functions (Q1191032)

The space of rational functions of degree \(k\) of the 2-sphere \(S^ 2\) into complex projective \(m\)-space \(\mathbb{C} P^ m\) is naturally a subspace of the space of all based continuous maps of degree \(k\) of \(S^ 2\) into \(\mathbb{C} P^ m\) by an inclusion map that \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] proved to be a homotopy equivalence up to dimension \(k(2m-1)\). By results of \textit{C. P. Boyer} and \textit{B. M. Mann} [Commun. Math. Phys. 115, 571-594 (1988; Zbl 0656.58049)], the space of rational functions as above admits a loop sum and a \(C_ 2\) structure compatible with the similar structures on the space of continuous maps. Exploiting these structures, the author constructs (with certain restrictions) a basis for the modulo 2 homology groups of the space of rational functions of degree \(k\) as above.