Minimizing roots of maps between spheres and projective spaces in codimension one

Abstract

We determine a lower bound for the dimension of the Čech cohomology of the root sets of maps from the sphere S 2 n + 1 S^{2n+1} and from the real projective space R P 2 n + 1 {\mathbb {R}\mathrm {P}}^{2n+1} into the complex projective space C P n {\mathbb {C}\mathrm {P}}^n , for n ≥ 1 n\geq 1 . For each such a map, we construct a representative of its homotopy class which realize the lower bound and whose root set is minimal in the class. We prove that the circle is a minimal root set for any non-trivial homotopy class. We present analogous results for maps from both S 4 n + 3 S^{4n+3} and R P 4 n + 3 {\mathbb {R}\mathrm {P}}^{4n+3} into the orbit space C P 2 n + 1 / τ {\mathbb {C}\mathrm {P}}^{2n+1}\!/\tau , for n ≥ 0 n\geq 0 , where τ \tau is a free involution on C P 2 n + 1 {\mathbb {C}\mathrm {P}}^{2n+1} . In this setting, we prove that the disjoint union of two circles is a minimal root set for any non-trivial homotopy class.