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 $S^{2n+1}$ and from the real projective space ${\mathbb {R}\mathrm {P}}^{2n+1}$ into the complex projective space ${\mathbb {C}\mathrm {P}}^n$, for $n\geq 1$. 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 $S^{4n+3}$ and ${\mathbb {R}\mathrm {P}}^{4n+3}$ into the orbit space ${\mathbb {C}\mathrm {P}}^{2n+1}\!/\tau$, for $n\geq 0$, where $\tau$ is a free involution on ${\mathbb {C}\mathrm {P}}^{2n+1}$. In this setting, we prove that the disjoint union of two circles is a minimal root set for any non-trivial homotopy class.