Complexification of real cycles and Lawson suspension theorem

We embed the space of totally real $r$-cycles of a totally real projective variety into the space of complex $r$-cycles by complexification. We provide a proof of the holomorphic taffy argument in the proof of Lawson suspension theorem by using Chow forms and this proof gives us an analogous result for totally real cycle spaces. We use Sturm theorem to derive a criterion for a real polynomial of degree $d$ to have $d$ distinct real roots and use it to prove the openness of some subsets of real divisors. This enables us to prove that the suspension map induces a weak homotopy equivalence between two enlarged spaces of totally real cycle spaces.