Polynomial and Regular Maps into Grassmannians

Abstract

We find a regular deformation retractionn,r (K) :I dem n,r (K) → Gn,r (K) from the manifold Idemn,r (K) of idempotent n × n matrices with rank r to the Grassmannian manifold Gn,r (K) over K the reals, complex numbers or quaternions. Then we derive an injection P C(VC, Idemn,r (K)) → RR(V, Gn,r (K)) from the sets of homotopy classes of complex-valued polynomial to such a set of real-valued regular maps, where VC denotes the Zariski closure in the affine space C n of a subset V ⊆ R n . Furthermore, we list complex-valued polynomial maps S 2 → S 2 of any Brouwer degree and deduce that the map � 2,1(C) :I dem2,1(C) → G2,1(C) yields an isomorphism PC(S 2 , S 2 ) ∼ −→ RR(S 2 , S 2 ) of cyclic infinite homotopy groups. Finally, we show that every nonzero even Brouwer degree of the spheres S n and S n cannot be realized by a real-valued (resp. complex-valued) homogeneous polynomial map provided that n is even.