Full holomorphic maps from the Riemann sphere to complex projective spaces (Q688611)

We study the topology of \(FRat_ k({\mathbf {CP}}^ n)\), the space of full, based, holomorphic maps of degree \(k\) from \(S^ 2\) to \({\mathbf {CP}}^ n\); that is,those based holomorphic maps whose image does not lie in any proper projective subspace of \({\mathbf {CP}}^ n\). We prove that the natural inclusion of \(FRat_ k({\mathbf {CP}}^ n)\) into \(Rat_ k({\mathbf {CP}}^ n)\), the space of all based holomorphic maps, is a homotopy equivalence through dimension \(2(k-n)\). We compute \(H_* \bigl( FRat_ k({\mathbf {CP}}^ 2);{\mathbf Z}/p \bigr)\) and \(H_* \bigl( FRat_ k({\mathbf {CP}}^ n);{\mathbf Q} \bigr)\) completely. We also obtain partial results for \(H_* \bigl( FRat_ k({\mathbf {CP}}^ n);{\mathbf Z}/p \bigr)\) for \(n>2\).