Abstract
The space of totally real r-cycles of a totally real projective variety is embedded into the space of complex r-cycles by complexiflcation. The holomorphic taffy argument in the proof of Lawson's suspension theorem is proved by using Chow forms, and this proof gives an analogous result for totally real cycle spaces. The Sturm theorem is used to derive a criterion for real polynomials of degree d to have d distinct real roots, and this criterion is used 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. © 2007 London Mathematical Society.