Spaces of real polynomials with common roots

Let Ratk(CP) denote the space of based holomorphic maps of degree k from the Riemannian sphere S2 = C ∪∞ to the complex projective space CPn . The basepoint condition we assume is that f (∞) = [1, . . . , 1]. Such holomorphic maps are given by rational functions: Ratk(CP) = {(p0(z), . . . , pn(z)) : each pi(z) is a monic polynomial over C of degree k and such that there are no roots common to all pi(z)}. There is an inclusion Ratk(CP) ↪→ ΩkCP ' Ω2S2n+1 . Segal [6] proved that the inclusion is a homotopy equivalence up to dimension k(2n − 1). Later, the stable homotopy type of Ratk(CP) was described by Cohen et al [2, 3] as follows. Let Ω2S2n+1 ' s ∨ 1≤q Dq(S 2n−1) be Snaith’s stable splitting of Ω2S2n+1 . Then